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%I #31 Jan 20 2024 18:20:30
%S 213,942,2956,7326,15447,29038,50142,81126,124681,183822,261888,
%T 362542,489771,647886,841522,1075638,1355517,1686766,2075316,2527422,
%U 3049663,3648942,4332486,5107846,5982897,6965838,8065192,9289806,10648851,12151822
%N Hyper-Wiener index of rows of unit cells on the face-centered cubic lattice.
%C 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).
%H Colin Barker, <a href="/A302256/b302256.txt">Table of n, a(n) for n = 1..1000</a>
%H Hamzeh Mujahed, Benedek Nagy: <a href="https://doi.org/10.2478/auom-2018-0011">Exact Formula for Computing the Hyper-Wiener Index on Rows of Unit Cells of the Face-Centred Cubic Lattice</a>, Analele Universitatii "Ovidius" Constanta - Seria Matematica 26/1 (2018), 169-187.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hyper-Wiener_index">Hyper-Wiener index</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = (81*n^4+261*n^3+264*n^2+540*n+132)/6. Proved in the Hamzeh Mujahed - Benedek Nagy paper.
%F 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.
%F G.f.: x*(213 - 123*x + 376*x^2 - 164*x^3 + 22*x^4) / (1 - x)^5. - _Colin Barker_, Jun 11 2018
%t Table[(81*n^4 + 261*n^3 + 264*n^2 + 540*n + 132)/6, {n, 30}] (* _Wesley Ivan Hurt_, Jan 20 2024 *)
%o (PARI) a(n) = (81*n^4+261*n^3+264*n^2+540*n+132)/6; \\ _Altug Alkan_, Apr 04 2018
%o (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
%Y Cf. A273322.
%K nonn,easy
%O 1,1
%A _Benedek Nagy_, Apr 04 2018
%E a(5) corrected by _Altug Alkan_, Apr 04 2018
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