A008749 Expansion of (1+x^6)/((1-x)*(1-x^2)*(1-x^3)).

1, 1, 2, 3, 4, 5, 8, 9, 12, 15, 18, 21, 26, 29, 34, 39, 44, 49, 56, 61, 68, 75, 82, 89, 98, 105, 114, 123, 132, 141, 152, 161, 172, 183, 194, 205, 218, 229, 242, 255, 268, 281, 296, 309, 324, 339, 354, 369

Offset

0,3

Comments

Conjecture: For n >= 1, A067628(a(n+2)) appears for the first time in A067628. Equivalently, A067628(a(n+2)) is the first T such that the minimal perimeter of polyiamonds of T triangles is a(n+2). - Winston C. Yang (winston(AT)cs.wisc.edu), Feb 05 2002

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Formula

Conjecture: Let b(n>=0) = (0, 1, 1, 1, 1, 3, 1, 3, 3, 3, 3, 5, 3, 5, 5, 5, 5, 7, 3, ...). Equivalently, let b(0) = 0, b(n>=1) = 2*floor((n-1)/6) + 1 + (2 if n+1=0 mod 6; 0 else). Then a(0) = 1, a(n>=1) = a(n-1) + b(n-1). - Winston C. Yang (winston(AT)cs.wisc.edu), Feb 05 2002

a(n) = (47 + 6*n^2 + 9*(-1)^n + 8*A099837(n+3))/36, n>0. - R. J. Mathar, Jun 24 2009

Example

Let n = 8. Then a(n+2) = a(10) = 18. Note A067628(18) = 12 and is the first appearance of 12 in A067628. Equivalently, 12 is the first T such that the min perimeter of polyiamonds of T triangles is 18.

Mathematica

CoefficientList[Series[(1+x^6)/((1-x)*(1-x^2)*(1-x^3)), {x, 0, 60}], x] (* G. C. Greubel, Aug 03 2019 *)

Prog

(PARI) my(x='x+O('x^60)); Vec((1+x^6)/((1-x)*(1-x^2)*(1-x^3))) \\ G. C. Greubel, Aug 03 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^6)/((1-x)*(1-x^2)*(1-x^3)) )); // G. C. Greubel, Aug 03 2019
(Sage) ((1+x^6)/((1-x)*(1-x^2)*(1-x^3))).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019
(GAP) a:=[1, 1, 2, 3, 4, 5];; for n in [7..60] do a[n]:=a[n-1]+a[n-2]-a[n-4] -a[n-5]+a[n-6]; od; a; # G. C. Greubel, Aug 03 2019

Crossrefs

Cf. A067628.

Sequence in context: A118956 A372586 A109850 * A029000 A042962 A027584

Adjacent sequences: A008746 A008747 A008748 * A008750 A008751 A008752

Keyword

nonn

Author

N. J. A. Sloane

Status

approved