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A276917 Numbers obtained by alternatively adding centered pentagonal layers of 5*(2^n-1) and 5*(3^n-1) elements.

%I A276917 #14 Dec 31 2016 01:52:52
%S A276917 1,6,16,31,71,106,236,311,711,866,2076,2391,6031,6666,17596,18871,
%T A276917 51671,54226,152636,157751,452991,463226,1348956,1369431,4026631,
%U A276917 4067586,12039196,12121111,36035951,36199786,107944316,108271991,323505591,324160946,969861756
%N A276917 Numbers obtained by alternatively adding centered pentagonal layers of 5*(2^n-1) and 5*(3^n-1) elements.
%C A276917 a(0), a(1), a(2) and a(3) are the first four centered pentagonal numbers, as they match the same pattern. From a(4) onwards all terms are a different kind of centered pentagonal numbers, as the number of elements in subsequent layers doesn't increase uniformly.
%C A276917 a(13) is the first palindromic number in the sequence. a(19) is the second one.
%C A276917 First prime terms are a(3), a(4), a(7), a(31), a(100) and a(115).
%H A276917 Daniel Poveda Parrilla, <a href="/A276917/b276917.txt">Table of n, a(n) for n = 0..1000</a>
%H A276917 Daniel Poveda Parrilla, <a href="/A276917/a276917.gif">Illustration of initial terms</a>
%H A276917 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,-10,-1,12,-6).
%F A276917 a(n) = 5*(Sum_{i=0..((n+(n mod 2))/2)} 2^i + Sum_{j=0..((n-(n mod 2))/2)} 3^j) - 5*n - 9.
%F A276917 a(n) = a(n-1) + 5*((2+((n+1) mod 2))^((n+(n mod 2))/2) - 1) for n>0.
%F A276917 G.f.: (1+4*x-15*x^3+6*x^4-6*x^5)/((-1+x)^2*(1-5*x^2+6*x^4)).
%F A276917 From _Colin Barker_, Dec 30 2016: (Start)
%F A276917 a(n) = (-10*n + 5*3^(n/2+1) + 5*2^(n/2+2) - 33)/2 for n even.
%F A276917 a(n) = (-10*n + 5*3^(n/2+1/2) + 5*2^(n/2+5/2) - 33)/2 for n odd.
%F A276917 (End)
%t A276917 Table[5 (Sum[2^i, {i, 0, ((n + Mod[n, 2])/2)}] + Sum[3^j, {j, 0, ((n - Mod[n, 2])/2)}]) - 5 n - 9, {n, 0, 28}] (* or *)
%t A276917 CoefficientList[Series[(1 + 4 x - 15 x^3 + 6 x^4 - 6 x^5)/((-1 + x)^2 (1 - 5 x^2 + 6 x^4)), {x, 0, 28}], x] (* or *)
%t A276917 LinearRecurrence[{2, 4, -10, -1, 12, -6}, {1, 6, 16, 31, 71, 106}, 29]
%o A276917 (PARI) Vec((1+4*x-15*x^3+6*x^4-6*x^5) / ((-1+x)^2*(1-5*x^2+6*x^4)) + O(x^40)) \\ _Colin Barker_, Dec 30 2016
%Y A276917 Cf. A005891.
%K A276917 nonn,easy
%O A276917 0,2
%A A276917 _Daniel Poveda Parrilla_, Dec 29 2016