A264891 a(n) = n*(5*n - 3)*(25*n^2 - 15*n - 6)/8.

0, 1, 112, 783, 2839, 7480, 16281, 31192, 54538, 89019, 137710, 204061, 291897, 405418, 549199, 728190, 947716, 1213477, 1531548, 1908379, 2350795, 2865996, 3461557, 4145428, 4925934, 5811775, 6812026, 7936137, 9193933, 10595614, 12151755, 13873306

Offset: 0

Author

Ilya Gutkovskiy, Nov 27 2015

Keywords

• nonn
• easy

Comments

Doubly heptagonal numbers.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1200
• OEIS Wiki, Figurate numbers
• Eric Weisstein's World of Mathematics, Heptagonal Number
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Crossrefs

Cf. A000566, A002817, A000583, A232713, A063249.

Programs

Magma

[n*(5*n-3)*(25*n^2-15*n-6)/8: n in [0..30]]; // Vincenzo Librandi, Nov 28 2015

Maple

seq(n*(5*n - 3)*(25*n^2 - 15*n - 6)/8, n=0..100); # Robert Israel, Dec 02 2015

Mathematica

Table[n (5 n - 3) (25 n^2 - 15 n - 6)/8, {n, 0, 35}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,112,783,2839},40] (* Harvey P. Dale, Nov 19 2019 *)

PARI

vector(100, n, n--; n*(5*n-3)*(25*n^2-15*n-6)/8) \\ Altug Alkan, Nov 27 2015

Formula

G.f.: x*(1 + 107*x + 233*x^2 + 34*x^3)/(1 - x)^5.

a(n) = A000566(A000566(n)).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 28 2015

Sum_{n>0} 1/a(n) = (4*(sqrt(33)*gamma + sqrt(33)*polygamma(0, 2/5) - 3*polygamma(0, (1/10)*(7 - sqrt(33))) + 3 polygamma(0, (1/10)* (7 + sqrt(33)))))/(9*sqrt(33)) = 1.0108420043...., where gamma is the Euler-Mascheroni constant (A001620), and polygamma is the derivative of the logarithm of the gamma function.

E.g.f.: x*(8 + 440*x + 600*x^2 + 125*x^3)*exp(x)/8, - Robert Israel, Dec 02 2015