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

Author

N. J. A. Sloane, Apr 17 2010

Keywords

• nonn
• easy

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).

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.

Programs

Magma

[6*n^3-9*n^2+4*n: n in [1..40]]; // 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 *)

PARI

a(n)=6*n^3-9*n^2+4*n \\ Charles R Greathouse IV, Oct 18 2022

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