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 A276917 Numbers obtained by alternatively adding centered pentagonal layers of 5*(2^n-1) and 5*(3^n-1) elements. 1
 1, 6, 16, 31, 71, 106, 236, 311, 711, 866, 2076, 2391, 6031, 6666, 17596, 18871, 51671, 54226, 152636, 157751, 452991, 463226, 1348956, 1369431, 4026631, 4067586, 12039196, 12121111, 36035951, 36199786, 107944316, 108271991, 323505591, 324160946, 969861756 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. a(13) is the first palindromic number in the sequence. a(19) is the second one. First prime terms are a(3), a(4), a(7), a(31), a(100) and a(115). LINKS Daniel Poveda Parrilla, Table of n, a(n) for n = 0..1000 Daniel Poveda Parrilla, Illustration of initial terms Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-1,12,-6). FORMULA 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. a(n) = a(n-1) + 5*((2+((n+1) mod 2))^((n+(n mod 2))/2) - 1) for n>0. G.f.: (1+4*x-15*x^3+6*x^4-6*x^5)/((-1+x)^2*(1-5*x^2+6*x^4)). From Colin Barker, Dec 30 2016: (Start) a(n) = (-10*n + 5*3^(n/2+1) + 5*2^(n/2+2) - 33)/2 for n even. a(n) = (-10*n + 5*3^(n/2+1/2) + 5*2^(n/2+5/2) - 33)/2 for n odd. (End) MATHEMATICA 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 *) 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 *) LinearRecurrence[{2, 4, -10, -1, 12, -6}, {1, 6, 16, 31, 71, 106}, 29] PROG (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 CROSSREFS Cf. A005891. Sequence in context: A092286 A301723 A288113 * A097118 A369548 A296957 Adjacent sequences: A276914 A276915 A276916 * A276918 A276919 A276920 KEYWORD nonn,easy AUTHOR Daniel Poveda Parrilla, Dec 29 2016 STATUS approved

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Last modified June 16 11:31 EDT 2024. Contains 373429 sequences. (Running on oeis4.)