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

Author

Benedek Nagy, Apr 04 2018

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A273322.

Programs

Mathematica

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

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

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

Extensions

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