A169711 The function W_n(6) (see Borwein et al. reference for definition).

1, 20, 93, 256, 545, 996, 1645, 2528, 3681, 5140, 6941, 9120, 11713, 14756, 18285, 22336, 26945, 32148, 37981, 44480, 51681, 59620, 68333, 77856, 88225, 99476, 111645, 124768, 138881, 154020, 170221, 187520, 205953, 225556, 246365, 268416, 291745, 316388

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Jonathan M. Borwein, Dirk Nuyens, Armin Straub, and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, May 2011.

Pakawut Jiradilok and Elchanan Mossel, Gaussian Broadcast on Grids, arXiv:2402.11990 [cs.IT], 2024. See p. 27.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = 6*n^3 - 9*n^2 + 4*n. - Peter Luschny, May 27 2017

G.f.: x*(1+16*x+19*x^2)/(1-x)^4. - Vincenzo Librandi, May 28 2017

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, May 28 2017

Maple

A169711 := proc(n)
        W(n, 6) ;
end proc:
seq(A169711(n), n=1..20) ; # uses W from A169715; R. J. Mathar, Mar 28 2012
a := n -> 6*n^3 - 9*n^2 + 4*n: seq(a(n), n=1..33); # Peter Luschny, May 27 2017

Mathematica

CoefficientList[Series[(1 + 16 x + 19 x^2) / (1 - x)^4, {x, 0, 50}], x] (* or *) Table[6 n^3 - 9 n^2 + 4 n, {n, 1, 40}] (* Vincenzo Librandi, May 28 2017 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 20, 93, 256}, 40] (* Harvey P. Dale, Feb 27 2023 *)

Prog

(Magma) [6*n^3-9*n^2+4*n: n in [1..40]]; // Vincenzo Librandi, May 28 2017
(PARI) a(n)=6*n^3-9*n^2+4*n \\ Charles R Greathouse IV, Oct 18 2022

Crossrefs

The sequence in Table 1 of the Borwein et al. reference are A000384, A109711-A109713; A000984, A002893, A002895, A169714, A169715.

Column 3 of A287316.

Cf. A287314.

Sequence in context: A200431 A172200 A177291 * A281391 A071092 A144359

Adjacent sequences: A169708 A169709 A169710 * A169712 A169713 A169714

Keyword

nonn,easy

Author

N. J. A. Sloane, Apr 17 2010

Status

approved