A263229 - OEIS

%I #21 Sep 08 2022 08:46:14

%S 0,-20,128,444,928,1580,2400,3388,4544,5868,7360,9020,10848,12844,

%T 15008,17340,19840,22508,25344,28348,31520,34860,38368,42044,45888,

%U 49900,54080,58428,62944,67628,72480,77500,82688,88044,93568,99260,105120,111148,117344,123708,130240

%N a(n) = 4*n*(21*n - 26).

%C For n>=3, a(n) = the hyper-Wiener index of the Jahangir graph J_{4,n}. The Jahangir graph J_{4,n} is a connected graph consisting of a cycle graph C(4n) and one additional center vertex that is adjacent to n vertices of C(4n) at distances 4 to each other on C(4n).

%C The Hosoya polynomial of J_{4,n} is 5nx + n(n+1))x^2/2 + n(2n+1)x^3 +n(3n-4)x^4 + 2n(n-2)x^5 + n(n-3)x^6/2 (see the Farahani reference, p. 234, last line; however, the expression in Theorem 1, p. 233, is accidentally incorrect).

%H M. R. Farahani, <a href="http://gpcpublishing.com/index.php?journal=gjm&amp;page=article&amp;op=view&amp;path%5B%5D=89">Hosoya polynomial and of Jahangir graphs J_{4,m}</a>, Global J. Math, 3 (1), 232-236, 2015.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f. = 4*x*(47*x-5)/(1-x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%p seq(84*n^2 - 104*n, n=0..40);

%t Table[4 n (21 n - 26), {n, 0, 40}] (* _Bruno Berselli_, Oct 15 2015 *)

%o (Magma) [4*n*(21*n-26): n in [0..20]]; // _Vincenzo Librandi_, Oct 15 2015

%o (PARI) vector(50, n, n--; 4*n*(21*n-26)) \\ _Altug Alkan_, Oct 15 2015

%Y Cf. A049598, A263226, A263227, A263228, A263231.

%K sign,easy

%O 0,2

%A _Emeric Deutsch_, Oct 13 2015