A302256 Hyper-Wiener index of rows of unit cells on the face-centered cubic lattice.

213, 942, 2956, 7326, 15447, 29038, 50142, 81126, 124681, 183822, 261888, 362542, 489771, 647886, 841522, 1075638, 1355517, 1686766, 2075316, 2527422, 3049663, 3648942, 4332486, 5107846, 5982897, 6965838, 8065192, 9289806, 10648851, 12151822

Offset

1,1

Comments

This sequence is related to the Wiener-index of the FCC grid (cf. A273322). Now the second order distances are also counted (see definition of Hyper-Wiener index).

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Hamzeh Mujahed, Benedek Nagy: Exact Formula for Computing the Hyper-Wiener Index on Rows of Unit Cells of the Face-Centred Cubic Lattice, Analele Universitatii "Ovidius" Constanta - Seria Matematica 26/1 (2018), 169-187.

Wikipedia, Hyper-Wiener index

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = (81*n^4+261*n^3+264*n^2+540*n+132)/6. Proved in the Hamzeh Mujahed - Benedek Nagy paper.

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5); with a(1)=213, a(2)=942, a(3)=2956, a(4)=7326 and a(5)=15447.

G.f.: x*(213 - 123*x + 376*x^2 - 164*x^3 + 22*x^4) / (1 - x)^5. - Colin Barker, Jun 11 2018

Mathematica

Table[(81*n^4 + 261*n^3 + 264*n^2 + 540*n + 132)/6, {n, 30}] (* Wesley Ivan Hurt, Jan 20 2024 *)

Prog

(PARI) a(n) = (81*n^4+261*n^3+264*n^2+540*n+132)/6; \\ Altug Alkan, Apr 04 2018
(PARI) Vec(x*(213 - 123*x + 376*x^2 - 164*x^3 + 22*x^4) / (1 - x)^5 + O(x^40)) \\ Colin Barker, Jun 11 2018

Crossrefs

Cf. A273322.

Sequence in context: A251146 A085309 A252024 * A092127 A082967 A212312

Adjacent sequences: A302253 A302254 A302255 * A302257 A302258 A302259

Keyword

nonn,easy

Author

Benedek Nagy, Apr 04 2018

Extensions

a(5) corrected by Altug Alkan, Apr 04 2018

Status

approved