eis replacement start seq Author alt rule GF rule record separator A000045 "k/. {1->{2,3},2->{1,3},3->{1}}" 1 "{1, 2, 3, 5, 8, 13, 21, 34}" Fibonacci numbers. "{1 -> {2, 3}, 2 -> {1, 3}, 3 -> {1}}" record separator A000108 "Range[0,k+1]" 0 "{1, 2, 5, 14, 42, 132, 429, 1430}" Catalans; G.f.: (1-sqrt(1-4*x)) / (2*x). "{0 -> {0, 1}, 1 -> {0, 1, 2}, 2 -> {0, 1, 2, 3}, 3 -> {0, 1, 2, 3, 4}, 4 -> {0, 1, 2, 3, 4, 5}}" record separator A000125 "Cases[Range[ k, k+2],_?PrimeQ]" 1 "{1, 2, 4, 8, 15, 26, 42, 64}" njas "{1 -> {2, 3}, 2 -> {2, 3}, 3 -> {3, 5}, 5 -> {5, 7}, 7 -> {7}}" "Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3)+n+1." "{1 -> {2, 3}, 2 -> {2, 3}, 3 -> {3, 5}, 5 -> {5, 7}, 7 -> {7}}" record separator A000245 "Range[0,k+1]" 1 "{1, 3, 9, 28, 90, 297, 1001, 3432}" a(n)=3*n*Catalan(n)/(n+2) "{0 -> {0, 1}, 1 -> {0, 1, 2}, 2 -> {0, 1, 2, 3}, 3 -> {0, 1, 2, 3, 4}, 4 -> {0, 1, 2, 3, 4, 5}, 5 -> {0, 1, 2, 3, 4, 5, 6}}" record separator A000290 "k/. {1->{1,2,3,4},2->{2,3,4},3->{3,4},4->{}}" 1 "{1, 4, 9, 16, 25, 36, 49, 64}" n^2 "{1 -> {1, 2, 3, 4}, 2 -> {2, 3, 4}, 3 -> {3, 4}, 4 -> {}}" record separator A000984 "Range[-1/2,Floor[Abs[k]+3/2]]" 0 "{1, 2, 6, 20, 70, 252, 924, 3432, 12870}" " Central binomial coefficients : C(2n, n) = (2n)!/(n!)^2." "{-1/2 -> {-1/2, 1/2, 3/2}, 0 -> {-1/2, 1/2}, 1/2 -> {-1/2, 1/2, 3/2}, 3/2 -> {-1/2, 1/2, 3/2, 5/2}, 5/2 -> {-1/2, 1/2, 3/2, 5/2, 7/2}, 7/2 -> {-1/2, 1/2, 3/2, 5/2, 7/2, 9/2}}" record separator A001003 "Join[Range[0,k+1],Range[k,0,-1]]" 0 "{1, 3, 11, 45, 197, 903, 4279, 20793}" G.f.: (1/4)*(1+x-sqrt(1-6*x+x^2)). "{0 -> {0, 1, 0}, 1 -> {0, 1, 2, 1, 0}, 2 -> {0, 1, 2, 3, 2, 1, 0}, 3 -> {0, 1, 2, 3, 4, 3, 2, 1, 0}, 4 -> {0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0}}" record separator A001003 "Range[1,k+1+Mod[k,2]]" 1 "{1, 3, 11, 45, 197, 903, 4279, 20793}" njas G.f.: (1/4)*(1+x-sqrt(1-6*x+x^2)). "{1 -> {1, 2, 3}, 2 -> {1, 2, 3}, 3 -> {1, 2, 3, 4, 5}, 4 -> {1, 2, 3, 4, 5}, 5 -> {1, 2, 3, 4, 5, 6, 7}, 6 -> {1, 2, 3, 4, 5, 6, 7}, 7 -> {1, 2, 3, 4, 5, 6, 7, 8, 9}, 8 -> {1, 2, 3, 4, 5, 6, 7, 8, 9}, 9 -> {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}}" record separator A001333 "k/. {1->{1,2,3},2->{1,3},3->{1,2}}" 1 "{1, 3, 7, 17, 41, 99, 239, 577}" G.f.: (1-x)/(1-2*x-x^2). "{1 -> {1, 2, 3}, 2 -> {1, 3}, 3 -> {1, 2}}" record separator A001519 "Range[1,k+1+Mod[k+1,3],2]" 0 "{1, 1, 2, 5, 13, 34, 89, 233}" njas G.f.: (1-x)/(1-3*x+x^2) "{0 -> {1}, 1 -> {1, 3}, 3 -> {1, 3, 5}, 5 -> {1, 3, 5}}" record separator A001700 "Range[-1/2,Floor[Abs[k]+3/2]]" 1 "{1, 3, 10, 35, 126, 462, 1716, 6435, 24310}" njas G.f. : (1/sqrt(1-4*x)-1)/(2*x) "{-1/2 -> {-1/2, 1/2, 3/2}, 1/2 -> {-1/2, 1/2, 3/2}, 1 -> {-1/2, 1/2, 3/2}, 3/2 -> {-1/2, 1/2, 3/2, 5/2}, 5/2 -> {-1/2, 1/2, 3/2, 5/2, 7/2}, 7/2 -> {-1/2, 1/2, 3/2, 5/2, 7/2, 9/2}, 9/2 -> {-1/2, 1/2, 3/2, 5/2, 7/2, 9/2, 11/2}}" record separator A001906 Range[k+1+MoebiusMu[k]] 1 "{1, 3, 8, 21, 55, 144, 377, 987}" "{0->{1},1->{1,2,3},2->{1,2},3->{1,2,3}}" G.f. : x/(1-3x+x^2) "{1 -> {1, 2, 3}, 2 -> {1, 2}, 3 -> {1, 2, 3}}" record separator A001911 "k/. {1->{1,2,3},2->{2,3},3->{2}}" 1 "{1, 3, 6, 11, 19, 32, 53, 87}" njas Extension : More terms and better description from Michael Somos "{0->{0},1->{0,1,2},2->{0,1}}" "Fibonacci numbers - 2; G.f. : (x+x^2)/(1-2*x+x^3). a(n) = a(n-1)+a(n-2)+2, a(0) = 0, a(1) = 1. " "{1 -> {1, 2, 3}, 2 -> {2, 3}, 3 -> {2}}" record separator A002478 "Range[1,1+Mod[k+1,3]]" 1 "{1, 3, 6, 13, 28, 60, 129, 277}" "{1->{1,2,3},2->{1},3->{1,2}}" a(n) = a(n-1)+2a(n-2)+a(n-3);a(1)=1;a(2)=1;a(3)=3; "{1 -> {1, 2, 3}, 2 -> {1}, 3 -> {1, 2}}" record separator A002522 "k/. {0->{0,1},1->{1,2,3},2->{2,3},3->{}}" 0 "{1, 2, 5, 10, 17, 26, 37, 50}" n^2 + 1 "{0 -> {0, 1}, 1 -> {1, 2, 3}, 2 -> {2, 3}, 3 -> {}}" record separator A003945 "k/. {1->{1,2,3},2->{2,3},3->{1}}" 1 "{1, 3, 6, 12, 24, 48, 96, 192}" "G.f.: (1+x)/(1-2*x) ; a(n)=2a(n-1), n>1; a(0)=1, a(1)=3." "{1 -> {1, 2, 3}, 2 -> {2, 3}, 3 -> {1}}" record separator A003946 "k/. {1->{1,2,3,4},2->{1,2,3,4},3->{1,2,3,4},4->{}}" 1 "{1, 4, 12, 36, 108, 324, 972, 2916}" G.f.: (1+x)/(1-3x). "{1 -> {1, 2, 3, 4}, 2 -> {1, 2, 3, 4}, 3 -> {1, 2, 3, 4}, 4 -> {}}" record separator A006318 "Range[0,k+1+Mod[k,2]]" 0 "{1, 2, 6, 22, 90, 394, 1806, 8558}" G.f. : (1-x-(1-6*x+x^2)^(1/2))/(2*x). "{0 -> {0, 1}, 1 -> {0, 1, 2, 3}, 2 -> {0, 1, 2, 3}, 3 -> {0, 1, 2, 3, 4, 5}, 4 -> {0, 1, 2, 3, 4, 5}, 5 -> {0, 1, 2, 3, 4, 5, 6, 7}, 6 -> {0, 1, 2, 3, 4, 5, 6, 7}, 7 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}}" record separator A006319 "Range[-Abs[k],Abs[k]+1]" 1 "{1, 4, 16, 68, 304, 1412, 6752, 33028}" "a(1)=1 and, for n>1, a(n)=4a(n-1)+Sum[a(k)a(n-k-1),k= 2,...,n-2]" "{-4 -> {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5}, -3 -> {-3, -2, -1, 0, 1, 2, 3, 4}, -2 -> {-2, -1, 0, 1, 2, 3}, -1 -> {-1, 0, 1, 2}, 0 -> {0, 1}, 1 -> {-1, 0, 1, 2}, 2 -> {-2, -1, 0, 1, 2, 3}, 3 -> {-3, -2, -1, 0, 1, 2, 3, 4}, 4 -> {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5}, 5 -> {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}}" record separator A006319 "Range[0,k+1+Mod[k,2]]" 1 "{1, 4, 16, 68, 304, 1412, 6752, 33028}" njas "All listed terms satisfy the recurrence a(1) = 1 and, for n > 1, a(n) = 4a(n-1)+Sum[a(k)a(n-k-1), k = 2, ..., n-2] " "{0 -> {0, 1}, 1 -> {0, 1, 2, 3}, 2 -> {0, 1, 2, 3}, 3 -> {0, 1, 2, 3, 4, 5}, 4 -> {0, 1, 2, 3, 4, 5}, 5 -> {0, 1, 2, 3, 4, 5, 6, 7}, 6 -> {0, 1, 2, 3, 4, 5, 6, 7}, 7 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 8 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 9 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}}" record separator A006356 Range[4-k] 1 "{1, 3, 6, 14, 31, 70, 157, 353}" a(n)=2*a(n-1)+a(n-2)-a(n-3). "{1 -> {1, 2, 3}, 2 -> {1, 2}, 3 -> {1}}" record separator A006357 "k/. {1->{1,2,3,4},2->{1,2,3},3->{1,2},4->{1}}" 1 "{1, 4, 10, 30, 85, 246, 707, 2037}" njas a(n)=2*a(n-1)+3*a(n-2)-a(n-3)-a(n-4). "{1 -> {1, 2, 3, 4}, 2 -> {1, 2, 3}, 3 -> {1, 2}, 4 -> {1}}" record separator A006358 Range[6-k] 1 "{1, 5, 15, 55, 190, 671, 2353, 8272}" a(n)=3*a(n-1)+3*a(n-2)-4*a(n-3)-a(n-4)+a(n-5). "{1 -> {1, 2, 3, 4, 5}, 2 -> {1, 2, 3, 4}, 3 -> {1, 2, 3}, 4 -> {1, 2}, 5 -> {1}}" record separator A006629 "Range[0,Ceiling[Abs[k]+3/2]]" 1 "{1, 4, 18, 88, 455, 2448, 13566, 76912}" "a(n)=2*binomial(3n-3,n-2)/n (from Emeric Deutsch). G.f.: HypergeometricPFQ[{2, 5/3, 4/3}, {3, 5/2}, (27*x)/4]" "{0 -> {0, 1, 2}, 1 -> {0, 1, 2, 3}, 2 -> {0, 1, 2, 3, 4}, 3 -> {0, 1, 2, 3, 4, 5}, 4 -> {0, 1, 2, 3, 4, 5, 6}, 5 -> {0, 1, 2, 3, 4, 5, 6, 7}, 6 -> {0, 1, 2, 3, 4, 5, 6, 7, 8}, 7 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 8 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 9 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}}" record separator A007051 "Floor[Select[Range[2k+1],PrimeQ]/2+1/2]" 1 "{1, 2, 5, 14, 41, 122, 365, 1094}" "Colin Mallows, njas, Simon Plouffe, Robert G. Wilson v (rgwv(AT)rgwv.com)" (3^n + 1)/2. "{1 -> {1, 2}, 2 -> {1, 2, 3}, 3 -> {1, 2, 3, 4}, 4 -> {1, 2, 3, 4}}" record separator A008934 "Range[ 2k,(k+1),-1]" 1 "{1, 1, 2, 7, 41, 397, 6377, 171886}" "torelli(AT)hermes.mc.dsi.unimi.it (Mauro Torelli), Jeffrey Shallit (shallit(AT)graceland.uwaterloo.ca)" " Number of tournament sequences: : sequences (a_1, a_2, ..., a_n) with a_1 = 1 such that a_i < a_{i+1} <= 2*a_i for all i." "{1 -> {2}, 2 -> {4, 3}, 3 -> {6, 5, 4}, 4 -> {8, 7, 6, 5}, 5 -> {10, 9, 8, 7, 6}, 6 -> {12, 11, 10, 9, 8, 7}, 7 -> {14, 13, 12, 11, 10, 9, 8}, 8 -> {16, 15, 14, 13, 12, 11, 10, 9}, 9 -> {18, 17, 16, 15, 14, 13, 12, 11, 10}, 10 -> {20, 19, 18, 17, 16, 15, 14, 13, 12, 11}, 11 -> {22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12}, 12 -> {24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13}, 13 -> {26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14}, 14 -> {28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15}, 15 -> {30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16}, 16 -> {32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17}}" record separator A016121 "Range[ 2k-1,(k),-1]" 2 "{1, 2, 5, 17, 86, 698, 9551, 226592}" Jeffrey Shallit (shallit(AT)graceland.uwaterloo.ca) "Number of sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i." "{2 -> {3, 2}, 3 -> {5, 4, 3}, 4 -> {7, 6, 5, 4}, 5 -> {9, 8, 7, 6, 5}, 6 -> {11, 10, 9, 8, 7, 6}, 7 -> {13, 12, 11, 10, 9, 8, 7}, 8 -> {15, 14, 13, 12, 11, 10, 9, 8}, 9 -> {17, 16, 15, 14, 13, 12, 11, 10, 9}, 10 -> {19, 18, 17, 16, 15, 14, 13, 12, 11, 10}, 11 -> {21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11}, 12 -> {23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12}, 13 -> {25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13}, 14 -> {27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14}, 15 -> {29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15}, 16 -> {31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16}, 17 -> {33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17}}" record separator A024493 "k/. {0->{0,1},1->{1,2,3},2->{2,3},3->{0}}" 0 "{1, 2, 5, 11, 22, 43, 85, 170}" "C(n,0) + C(n,3) + ... + C(n,3[n/3]).; a(n) = (1/3)*(2^n+2*cos( n*Pi/3 ))" "{0 -> {0, 1}, 1 -> {1, 2, 3}, 2 -> {2, 3}, 3 -> {0}}" record separator A026016 Sum in A000245 "{1, 3, 10, 34, 117, 407, 1430, 5070}" "G.f.: (1+x^1*C^3)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers" nihil record separator A026671 "Range[-1,Floor[Abs[k]+3/2]]" 0 "{1, 3, 11, 43, 173, 707, 2917, 12111, 50503}" G.f. : 1/(sqrt(1-4*x)-x) "{-1 -> {-1, 0, 1, 2}, 0 -> {-1, 0, 1}, 1 -> {-1, 0, 1, 2}, 2 -> {-1, 0, 1, 2, 3}, 3 -> {-1, 0, 1, 2, 3, 4}, 4 -> {-1, 0, 1, 2, 3, 4, 5}}" record separator A026672 "Range[-1,Floor[Abs[k]+3/2]]" 3/2 "{1, 5, 22, 94, 398, 1680, 7085, 29877, 126021}" "Clark Kimberling, ck6(AT)evansville.edu " "[-n+(1+5n-n^2)a(n)+(-2n+4n^2)a(n)^2+n^2a(n)^3,revogf];" "{-1 -> {-1, 0, 1, 2}, 0 -> {-1, 0, 1}, 1 -> {-1, 0, 1, 2}, 3/2 -> {-1, 0, 1, 2, 3}, 2 -> {-1, 0, 1, 2, 3}, 3 -> {-1, 0, 1, 2, 3, 4}, 4 -> {-1, 0, 1, 2, 3, 4, 5}, 5 -> {-1, 0, 1, 2, 3, 4, 5, 6}, 6 -> {-1, 0, 1, 2, 3, 4, 5, 6, 7}}" record separator A026674 "Range[-1,Ceiling[Abs[k]+1/2]]" 1 "{1, 4, 16, 65, 267, 1105, 4597, 19196}" "Clark Kimberling, ck6(AT)evansville.edu " "{-1 -> {-1, 0, 1, 2}, 0 -> {-1, 0, 1}, 1 -> {-1, 0, 1, 2}, 2 -> {-1, 0, 1, 2, 3}, 3 -> {-1, 0, 1, 2, 3, 4}, 4 -> {-1, 0, 1, 2, 3, 4, 5}, 5 -> {-1, 0, 1, 2, 3, 4, 5, 6}}" record separator A047749 "Range[k+1,0,-2]" 0 "{1, 1, 2, 3, 7, 12, 30, 55}" njas "If n=2m then C(3m,m)/(2m+1); if n=2m+1 then C(3m+1,m+1)/(2m+1)" "{0 -> {1}, 1 -> {2, 0}, 2 -> {3, 1}, 3 -> {4, 2, 0}, 4 -> {5, 3, 1}}" record separator A052529 "Range[0,k+1+MoebiusMu[k]]" 1 "{1, 4, 13, 41, 129, 406, 1278, 4023}" "encyclopedia(AT)pommard.inria.fr, Jan 25 2000 " "{{0,1},{0,1,2,3},{0,1,2},{0,1,2,3}}" G.f. : (-1+x)^3/(-1+4*x-3*x^2+x^3) "{0 -> {0, 1}, 1 -> {0, 1, 2, 3}, 2 -> {0, 1, 2}, 3 -> {0, 1, 2, 3}}" record separator A052544 "Range[0,k+1+MoebiusMu[k]]" 0 "{1, 2, 6, 19, 60, 189, 595, 1873}" "encyclopedia(AT)pommard.inria.fr, Jan 25 2000 " "{{0,1},{0,1,2,3},{0,1,2},{0,1,2,3}}" G.f. : -(-1+x)^2/(-1+4*x-3*x^2+x^3) "{0 -> {0, 1}, 1 -> {0, 1, 2, 3}, 2 -> {0, 1, 2}, 3 -> {0, 1, 2, 3}}" record separator A052709 "Range[-Abs[k+1],Abs[-k+1]]" 0 "{1, 3, 9, 31, 113, 431, 1697, 6847}" "encyclopedia(AT)pommard.inria.fr, Jan 25 2000 Extension: Better g.f. and recurrence from Michael Somos, Aug 03 2000 More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000 " G.f.: (1-sqrt(1-4*x*(1+x)))/(2*(1+x)) "{-4 -> {-3, -2, -1, 0, 1, 2, 3, 4, 5}, -3 -> {-2, -1, 0, 1, 2, 3, 4}, -2 -> {-1, 0, 1, 2, 3}, -1 -> {0, 1, 2}, 0 -> {-1, 0, 1}, 1 -> {-2, -1, 0}, 2 -> {-3, -2, -1, 0, 1}, 3 -> {-4, -3, -2, -1, 0, 1, 2}, 4 -> {-5, -4, -3, -2, -1, 0, 1, 2, 3}}" record separator A052925 A055588 "Select[Range[2k+1],PrimeQ]/2+1/2" 1 "{1, 2, 4, 9, 22, 56, 145, 378}" " encyclopedia(AT)pommard.inria.fr, Jan 25 2000 ;; Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) May 30 2000, Barry E. Williams, Jun 04 2000" "G.f.: (-2+6*x-4*x^2+x^3)/(-1+x)/(1-3*x+x^2) Recurrence: {a(2)=4,a(1)=2,a(0)=2,a(3)=9,-a(n)+3*a(n+1)-a(n+2)-1} 1+Sum(-1/5*(3*_alpha-2)*_alpha^(-1-n),_alpha=RootOf(_Z^2-3*_Z+1)) ; a(n)=3a(n-1)-a(n-2)-1; a(0)=1, a(1)=2." "{1 -> {3/2, 2}, 3/2 -> {3/2, 2}, 2 -> {3/2, 2, 3}, 3 -> {3/2, 2, 3, 4}, 4 -> {3/2, 2, 3, 4}}" record separator A052980 "Range[0,Mod[k+1,3]]" 0 "{1, 2, 5, 11, 24, 53, 117, 258}" "encyclopedia(AT)pommard.inria.fr, Jan 25 2000 " "{0->{0,1},1->{0,1,2},2->{0},3->{0,1}}" G.f.: (-1+x)/(-1+2*x+x^3) "{0 -> {0, 1}, 1 -> {0, 1, 2}, 2 -> {0}}" record separator A062992 A064062 "Range[-1-Abs[k],Abs[k]+1,2]" 0 "{1, 2, 6, 20, 76, 296, 1240, 5200}" "Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 12 2001 " G.f.: (2*c(2*x)-1)/(1+x) with c(x) g.f. of A000108. "{-4 -> {-5, -3, -1, 1, 3, 5}, -3 -> {-4, -2, 0, 2, 4}, -2 -> {-3, -1, 1, 3}, -1 -> {-2, 0, 2}, 0 -> {-1, 1}, 1 -> {-2, 0, 2}, 2 -> {-3, -1, 1, 3}, 3 -> {-4, -2, 0, 2, 4}, 4 -> {-5, -3, -1, 1, 3, 5}}" record separator A068921 "k/. {1->{1,2},2->{3},3->{1}}" 1 "{1, 2, 3, 4, 6, 9, 13, 19}" "G.f.: 1/(1-x-x^3) ; a(n) = a(n-1) + a(n-3).," "{1 -> {1, 2}, 2 -> {3}, 3 -> {1}}" record separator A078039 "Range[1,1+Mod[k+1,3]]" 3 "{1, 2, 4, 9, 19, 41, 88, 189}" "njas, Nov 17 2002 " "{1->{1,2,3},2->{1},3->{1,2}}" G.f.: (1-x)/(1+x-2*x^2+x^3) "{1 -> {1, 2, 3}, 2 -> {1}, 3 -> {1, 2}}" record separator A078482 "Range[1,k+Mod[k,3]]" 1 "{1, 2, 6, 20, 70, 254, 948, 3618}" "njas, Jan 04 2003 " G.f.: (1-3*x+x^2-(1-6*x+7*x^2-2*x^3+x^4)^(1/2))/(2*x) "{1 -> {1, 2}, 2 -> {1, 2, 3, 4}, 3 -> {1, 2, 3}, 4 -> {1, 2, 3, 4, 5}, 5 -> {1, 2, 3, 4, 5, 6, 7}, 6 -> {1, 2, 3, 4, 5, 6}, 7 -> {1, 2, 3, 4, 5, 6, 7, 8}}" record separator A083329 "Range[-1-Abs@k-MoebiusMu[Abs@k],k+1-MoebiusMu[Abs@k],2]" 0 "{1, 2, 5, 11, 23, 47, 95, 191}" "Paul Barry (pbarry(AT)wit.ie), Apr 27 2003 " "{-3->{-3,-1},-2->{-2,0},-1->{-3,-1},0->{-1,1},1->{-3,-1,1}}" G.f. (1+(1-x)/(2(1-x)(1-2x)) correction to 1+x(2-x)/( (1-x)(1-2x)) "{-3 -> {-3, -1}, -1 -> {-3, -1}, 0 -> {-1, 1}, 1 -> {-3, -1, 1}}" record separator A083691 "Range[-1-Abs[k],Abs[k]+1,2]" 0 "{1, 2, 6, 20, 76, 296, 1240, 5200}" "InverseSeries[Series[(-4n-5n^2+n^2Sqrt[1+8n+8n^2])/(2(-2-6n-6n^2-2n^3)),{n,0,28}]]" "{-4 -> {-5, -3, -1, 1, 3, 5}, -3 -> {-4, -2, 0, 2, 4}, -2 -> {-3, -1, 1, 3}, -1 -> {-2, 0, 2}, 0 -> {-1, 1}, 1 -> {-2, 0, 2}, 2 -> {-3, -1, 1, 3}, 3 -> {-4, -2, 0, 2, 4}, 4 -> {-5, -3, -1, 1, 3, 5}}" record separator A083692 "Range[-1-Abs[k],Abs[k]+1,2]" 1 "{1, 3, 10, 38, 148, 620, 2600, 11480}" A083691/2 "{-5 -> {-6, -4, -2, 0, 2, 4, 6}, -4 -> {-5, -3, -1, 1, 3, 5}, -3 -> {-4, -2, 0, 2, 4}, -2 -> {-3, -1, 1, 3}, -1 -> {-2, 0, 2}, 0 -> {-1, 1}, 1 -> {-2, 0, 2}, 2 -> {-3, -1, 1, 3}, 3 -> {-4, -2, 0, 2, 4}, 4 -> {-5, -3, -1, 1, 3, 5}, 5 -> {-6, -4, -2, 0, 2, 4, 6}}" record separator A083693 Sum in A083692 "{1, 4, 18, 72, 324, 1360, 6280, 27296}" A083691(n+2)/2-A083691(n+1) nihil record separator A084075 "Range[-Abs[k+1],Abs[k-1],2]" 1 "{1, 2, 5, 12, 33, 86, 249, 680}" "a(2n-1)=A027307/2 ; Rest @ CoefficientList[InverseSeries[Series[(-1-6n-8n^2+(1+2n)^2Sqrt[1+4n])/(2(n+4n^2+4n^3)),{n,0,28}]],n]" "{-4 -> {-3, -1, 1, 3, 5}, -3 -> {-2, 0, 2, 4}, -2 -> {-1, 1, 3}, -1 -> {0, 2}, 0 -> {-1, 1}, 1 -> {-2, 0}, 2 -> {-3, -1, 1}, 3 -> {-4, -2, 0, 2}, 5 -> {-6, -4, -2, 0, 2, 4}}" record separator A084076 "Range[-1-Abs[k],Abs[k]+1]" 1 "{1, 5, 27, 157, 963, 6141, 40323, 270845}" "Rest@CoefficientList[ InverseSeries[ Series[-((1+5*n+2*n^2-(1+2*n)*Sqrt[1+6*n+n^2])/(4*n^2)), {n, 0, 28}]], n]" "{-5 -> {-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}, -4 -> {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}, -3 -> {-4, -3, -2, -1, 0, 1, 2, 3, 4}, -2 -> {-3, -2, -1, 0, 1, 2, 3}, -1 -> {-2, -1, 0, 1, 2}, 0 -> {-1, 0, 1}, 1 -> {-2, -1, 0, 1, 2}, 2 -> {-3, -2, -1, 0, 1, 2, 3}, 3 -> {-4, -3, -2, -1, 0, 1, 2, 3, 4}, 4 -> {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}, 5 -> {-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}}" record separator A084077 "Range[-Abs[k+1],Abs[k-1]]" 1 "{1, 3, 11, 41, 159, 633, 2575, 10657}" "invOGF Solve[n+(-1-3 n) a[n]+(-2 n-2 n^2) a[n]^2-2 n^2 a[n]^3 == 0, a[n]]" "{-4 -> {-3, -2, -1, 0, 1, 2, 3, 4, 5}, -3 -> {-2, -1, 0, 1, 2, 3, 4}, -2 -> {-1, 0, 1, 2, 3}, -1 -> {0, 1, 2}, 0 -> {-1, 0, 1}, 1 -> {-2, -1, 0}, 2 -> {-3, -2, -1, 0, 1}, 3 -> {-4, -3, -2, -1, 0, 1, 2}, 4 -> {-5, -4, -3, -2, -1, 0, 1, 2, 3}, 5 -> {-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4}}" record separator A084078 "Range[ -Abs[k+1],Abs[k-1],2]" 0 "{1, 2, 4, 10, 24, 66, 172, 498}" a(2n-1)=A027307 "{-4 -> {-3, -1, 1, 3, 5}, -3 -> {-2, 0, 2, 4}, -2 -> {-1, 1, 3}, -1 -> {0, 2}, 0 -> {-1, 1}, 1 -> {-2, 0}, 2 -> {-3, -1, 1}, 3 -> {-4, -2, 0, 2}, 4 -> {-5, -3, -1, 1, 3}}" record separator A084079 Sum Abs in A084075 "{1, 2, 7, 16, 53, 130, 431, 1104}" nihil record separator A084080 "Range[k+1,1,-3]" 1 "{1, 1, 1, 2, 3, 4, 9, 15}" "I am sorry, but the terms ; Many tests are carried out, but " "{1 -> {2}, 2 -> {3}, 3 -> {4, 1}, 4 -> {5, 2}, 5 -> {6, 3}}" record separator A084081 Sum in A047749 "{0, 1, 2, 5, 10, 24, 50, 121}" "Many tests are carried out, but" nihil record separator A084082 "Range[-Floor[Abs[k]/2],Floor[Abs[k]+3/2]]" 0 "{1, 2, 5, 15, 49, 169, 604, 2212}" "Many tests are carried out, but " "{-1 -> {0, 1, 2}, 0 -> {0, 1}, 1 -> {0, 1, 2}, 2 -> {-1, 0, 1, 2, 3}, 3 -> {-1, 0, 1, 2, 3, 4}, 4 -> {-2, -1, 0, 1, 2, 3, 4, 5}}" record separator A084083 "Range[k+1,-Abs[k],-2]" 1 "{1, 2, 4, 9, 19, 44, 96, 227}" "Many tests are carried out, but " "{-3 -> {-2}, -2 -> {-1}, -1 -> {0}, 0 -> {1}, 1 -> {2, 0}, 2 -> {3, 1, -1}, 3 -> {4, 2, 0, -2}, 4 -> {5, 3, 1, -1, -3}, 5 -> {6, 4, 2, 0, -2, -4}}" record separator A084084 "Range[0,1+Mod[k+1,3]]" 0 "{1, 3, 9, 28, 86, 265, 816, 2513}" "{0->{0,1,2},1->{0,1,2,3},2->{0,1},3->{0,1,2}}" G.f.: -(-1-x)/(1-2x-3x^2-x^3) ; Transformation invert T109 gave a match with A078039 ; T100 binomial gave a match with A012781 "{0 -> {0, 1, 2}, 1 -> {0, 1, 2, 3}, 2 -> {0, 1}, 3 -> {0, 1, 2}}" record separator A084085 "Range[0,Mod[k+1,4]]" 0 "{1, 2, 5, 14, 38, 102, 273, 731}" "{0->{0,1},1->{0,1,2},2->{0,1,2,3},3->{0}}" G.f.: -(1-x+x^3)/(-x^2-1+3 x+x^4)) "{0 -> {0, 1}, 1 -> {0, 1, 2}, 2 -> {0, 1, 2, 3}, 3 -> {0}}" record separator A084086 "Range[-1-Abs[k]+MoebiusMu[Abs[k]],k+1+MoebiusMu[Abs[k]],2]" -1 "{1, 2, 5, 15, 44, 125, 347, 948}" "{-10->{-10,-8}; -9->{-10,-8}; -8->{-9,-7}; -7->{-9,-7}; -5->{-7,-5}; -3->{-5,-3}; -1->{-1,1}; 1->{-1,1,3}; 3->{-5,-3,-1,1,3}}" Gf: -(1-3 x+2 x^2+2 x^3)/(-7 x^2-1+5 x+2 x^3) "{-7 -> {-9, -7}, -5 -> {-7, -5}, -3 -> {-5, -3}, -1 -> {-1, 1}, 1 -> {-1, 1, 3}, 3 -> {-5, -3, -1, 1, 3}}" record separator