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%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_
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