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A296636 Sequences n*(n+1)*(6*n+1)/2 and n*(n+1)*(7*n+1)/2 interleaved.

%I A296636 #32 Feb 16 2025 08:33:52
%S A296636 0,7,8,39,45,114,132,250,290,465,540,777,903,1204,1400,1764,2052,2475,
%T A296636 2880,3355,3905,4422,5148,5694,6630,7189,8372,8925,10395,10920,12720,
%U A296636 13192,15368,15759,18360,18639,21717,21850,25460,25410,29610,29337,34188,33649,39215,38364,44712
%N A296636 Sequences n*(n+1)*(6*n+1)/2 and n*(n+1)*(7*n+1)/2 interleaved.
%C A296636 Difference between these subsequences is A002411.
%C A296636 This sequence gives numbers of triangles all sizes in every n-th stage [of what? - _N. J. A. Sloane_, Feb 09 2018].
%H A296636 Colin Barker, <a href="/A296636/b296636.txt">Table of n, a(n) for n = 0..1000</a>
%H A296636 Luce ETIENNE, <a href="/A296636/a296636.pdf">Illustration</a>
%H A296636 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polyiamond.html">Polyiamond</a>
%H A296636 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-6,0,4,0,-1).
%F A296636 a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9).
%F A296636 a(n) = (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128.
%F A296636 From _Colin Barker_, Dec 18 2017: (Start)
%F A296636 G.f.: x*(7 + 8*x + 11*x^2 + 13*x^3) / ((1 - x)^4*(1 + x)^4).
%F A296636 a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8) for n>7.
%F A296636 (End)
%t A296636 CoefficientList[Series[x (7 + 8 x + 11 x^2 + 13 x^3)/((1 - x)^4*(1 + x)^4), {x, 0, 46}], x] (* _Michael De Vlieger_, Dec 18 2017 *)
%t A296636 LinearRecurrence[{0,4,0,-6,0,4,0,-1},{0,7,8,39,45,114,132,250},50] (* _Harvey P. Dale_, May 01 2018 *)
%t A296636 Rest[Flatten[Table[With[{c=(n(n+1))/2},{c*(6n+1),c*(7n+1)}],{n,0,30}]]] (* _Harvey P. Dale_, Oct 11 2020 *)
%o A296636 (PARI) concat(0, Vec(x*(7 + 8*x + 11*x^2 + 13*x^3) / ((1 - x)^4*(1 + x)^4) + O(x^80))) \\ _Colin Barker_, Dec 18 2017
%o A296636 (GAP) List([0..50], n -> (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128); # _Bruno Berselli_, Feb 12 2018
%o A296636 (Magma) [(2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128: n in [0..50]]; // _Bruno Berselli_, Feb 12 2018
%Y A296636 Cf. A002411, A002717, A006331, A033581, A033582, A255211.
%K A296636 nonn,easy
%O A296636 0,2
%A A296636 _Luce ETIENNE_, Dec 17 2017