Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #17 Sep 08 2022 08:44:36
%S 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,
%T 105,114,123,132,141,152,161,172,183,194,205,218,229,242,255,268,281,
%U 296,309,324,339,354,369
%N Expansion of (1+x^6)/((1-x)*(1-x^2)*(1-x^3)).
%C 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
%H G. C. Greubel, <a href="/A008749/b008749.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F 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
%F a(n) = (47 + 6*n^2 + 9*(-1)^n + 8*A099837(n+3))/36, n>0. - _R. J. Mathar_, Jun 24 2009
%e 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.
%t CoefficientList[Series[(1+x^6)/((1-x)*(1-x^2)*(1-x^3)), {x,0,60}], x] (* _G. C. Greubel_, Aug 03 2019 *)
%o (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
%o (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
%o (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
%o (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
%Y Cf. A067628.
%K nonn
%O 0,3
%A _N. J. A. Sloane_