A064043 Number of length 3 walks on an n-dimensional hypercubic lattice starting at the origin and staying in the nonnegative part.

0, 3, 18, 51, 108, 195, 318, 483, 696, 963, 1290, 1683, 2148, 2691, 3318, 4035, 4848, 5763, 6786, 7923, 9180, 10563, 12078, 13731, 15528, 17475, 19578, 21843, 24276, 26883, 29670, 32643, 35808, 39171, 42738, 46515, 50508, 54723, 59166, 63843, 68760, 73923, 79338

Offset

0,2

Links

Harry J. Smith, Table of n, a(n) for n = 0..1000

D. R. L. Brown, Bounds on surmising remixed keys, IACR, Report 2015/375, 2015-2016. See Table 1.

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

Formula

a(n) = n*(n^2 + 3*n - 1) = n*A014209(n) = A064044(n, 3).

a(n) = a(n-1) + 3*A002378(n-1) + 6*A001477(n-1) + 3*A000012(n-1).

G.f.: 3*x*(1+2*x-x^2)/(1-x)^4. - Colin Barker, Apr 19 2012

E.g.f.: (x^3 + 6*x^2 + 3*x)*exp(x). - G. C. Greubel, Jul 20 2017

a(n) = A084990(n)/3. - Alois P. Heinz, Jul 21 2017

Maple

seq(sum(3*n+n^2-1, k=1..n), n=0..39); # Zerinvary Lajos, Jan 28 2008

Mathematica

Table[n*(n^2 + 3n -1), {n, 0, 50}] (* G. C. Greubel, Jul 20 2017 *)

Prog

(PARI) a(n) = { n*(n^2 + 3*n - 1) } \\ Harry J. Smith, Sep 06 2009

Crossrefs

Number of walks length 0, 1 and 2 are A000012, A001477 and A002378.

Cf. A084990.

Sequence in context: A275038 A304976 A364599 * A267639 A238649 A375764

Adjacent sequences: A064040 A064041 A064042 * A064044 A064045 A064046

Keyword

nonn,easy

Author

Henry Bottomley, Aug 23 2001

Status

approved