A264891 - OEIS

%I #23 Feb 16 2025 08:33:27

%S 0,1,112,783,2839,7480,16281,31192,54538,89019,137710,204061,291897,

%T 405418,549199,728190,947716,1213477,1531548,1908379,2350795,2865996,

%U 3461557,4145428,4925934,5811775,6812026,7936137,9193933,10595614,12151755,13873306

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

%C Doubly heptagonal numbers.

%H G. C. Greubel, <a href="/A264891/b264891.txt">Table of n, a(n) for n = 0..1200</a>

%H OEIS Wiki, <a href="https://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeptagonalNumber.html">Heptagonal Number</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1)

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

%F a(n) = A000566(A000566(n)).

%F 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

%F 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.

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

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

%t Table[n (5 n - 3) (25 n^2 - 15 n - 6)/8, {n, 0, 35}]

%t LinearRecurrence[{5,-10,10,-5,1},{0,1,112,783,2839},40] (* _Harvey P. Dale_, Nov 19 2019 *)

%o (PARI) vector(100, n, n--; n*(5*n-3)*(25*n^2-15*n-6)/8) \\ _Altug Alkan_, Nov 27 2015

%o (Magma) [n*(5*n-3)*(25*n^2-15*n-6)/8: n in [0..30]]; // _Vincenzo Librandi_, Nov 28 2015

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

%K nonn,easy,changed

%O 0,3

%A _Ilya Gutkovskiy_, Nov 27 2015