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 A275709 a(n) = 2*n^3 + 3*n^2. 2
 0, 5, 28, 81, 176, 325, 540, 833, 1216, 1701, 2300, 3025, 3888, 4901, 6076, 7425, 8960, 10693, 12636, 14801, 17200, 19845, 22748, 25921, 29376, 33125, 37180, 41553, 46256, 51301, 56700, 62465, 68608, 75141, 82076, 89425, 97200, 105413, 114076, 123201, 132800, 142885 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Apart from the initial zero this sequence gives the 2nd pentagonal number, the 4th hexagonal number, the 6th heptagonal number, the 8th octagonal number, the 10th nonagonal number, etc. as well as the 5th nonnegative number, the 7th triangular number, the 9th square, the 11th pentagonal number, the 13th hexagonal number, etc. This is a reliable pattern that does not seem to appear on any other pairs of polygonal numbers (see link). LINKS Carauleanu Marc and Colin Barker, Table of n, a(n) for n = 0..3030 (first 1000 terms from Colin Barker) Joshua Giambalvo, Illustration of initial terms in a square array, Imgur, (2016) Petro Kolosov, Another power identity involving binomial theorem and Faulhaber's formula, arXiv:1603.02468 [math.NT], 2018. (In particular p. 3) Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA From Colin Barker, Aug 06 2016: (Start) a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. G.f.: x*(5+8*x-x^2) / (1-x)^4. (End) a(n) = A033431(n) + A033428(n). - Omar E. Pol, Aug 09 2016 a(n) = A000290(n) * A005408(n+1). - Robert Israel, Aug 09 2016 a(n) = A320047(1, n, 0). - Kolosov Petro, Oct 04 2018 E.g.f.: x*(5 + 9*x + 2*x^2)*exp(x). - G. C. Greubel, Oct 19 2018 MAPLE seq(2*n^3+3*n^2, n=0..30); # Robert Israel, Aug 09 2016 MATHEMATICA Table[2 n^3 + 3 n^2, {n, 0, 41}] (* or *) CoefficientList[Series[x (5 + 8 x - x^2)/(1 - x)^4, {x, 0, 41}], x] (* Michael De Vlieger, Aug 11 2016 *) PROG (PARI) concat(0, Vec(x*(5+8*x-x^2)/(1-x)^4 + O(x^50))) \\ Colin Barker, Aug 28 2016 (PARI) a(n)=n^2*(2*n+3) \\ Charles R Greathouse IV, Aug 28 2016 (MAGMA) [n^2*(2*n + 3): n in [0..30]]; // G. C. Greubel, Oct 19 2018 (Python) for n in range(0, 50): print(n**2*(2*n+3), end=' ') # Stefano Spezia, Oct 19 2018 CROSSREFS Cf. A000290, A005408, A033428, A033431, A057145, A139600. Sequence in context: A161165 A063140 A316560 * A257093 A225261 A088727 Adjacent sequences:  A275706 A275707 A275708 * A275710 A275711 A275712 KEYWORD nonn,easy AUTHOR Joshua Giambalvo, Aug 06 2016 STATUS approved

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Last modified October 15 12:31 EDT 2019. Contains 328026 sequences. (Running on oeis4.)