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a(n) = 2*n*(16*n - 13).
4

%I #22 Sep 08 2022 08:46:14

%S 0,6,76,210,408,670,996,1386,1840,2358,2940,3586,4296,5070,5908,6810,

%T 7776,8806,9900,11058,12280,13566,14916,16330,17808,19350,20956,22626,

%U 24360,26158,28020,29946,31936,33990,36108,38290,40536,42846,45220,47658,50160

%N a(n) = 2*n*(16*n - 13).

%C For n>=3, a(n) = the 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). In the Farahani reference the expression in Theorem 2 is accidentally incorrect; it should be 2m(16m - 13).

%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. = 2*x*(3+29*x)/(1-x)^3.

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

%p seq(32*n^2 - 26*n, n=0..40);

%t Table[2 n (16 n - 13), {n, 0, 40}] (* _Bruno Berselli_, Oct 15 2015 *)

%o (Magma) [2*n*(16*n-13): n in [0..60]]; // _Vincenzo Librandi_, Oct 15 2015

%o (PARI) vector(50, n, n--; 2*n*(16*n-13)) \\ _Altug Alkan_, Oct 15 2015

%Y Cf. A049598, A263226, A263227, A263229, A263231.

%K nonn,easy

%O 0,2

%A _Emeric Deutsch_, Oct 13 2015