%I #21 Sep 08 2022 08:46:14
%S 0,-11,45,168,358,615,939,1330,1788,2313,2905,3564,4290,5083,5943,
%T 6870,7864,8925,10053,11248,12510,13839,15235,16698,18228,19825,21489,
%U 23220,25018,26883,28815,30814,32880,35013,37213,39480,41814,44215,46683,49218,51820
%N a(n) = n*(67*n - 89)/2.
%C For n>=3, a(n) = the hyper-Wiener index of the Jahangir graph J_{3,n}. The Jahangir graph J_{3,n} is a connected graph consisting of a cycle graph C(3n) and one additional center vertex that is adjacent to n vertices of C(3n) at distances 3 to each other on C(3n).
%C The Hosoya polynomial of J_(3,n) is 4nx + (1/2)n(n+9)x^2 + 2n(n-1)x^3 + n(2n-5)x^4.
%H M. R. Farahani, <a href="http://frdint.com/the_wiener_index_and.pdf">The Wiener index and Hosoya polynomial of a class of Jahangir graphs J_{3,m}</a>, Fundamental J. Math. and Math. Sci., 3 (1), 91-96, 2015.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(-11+78*x)/(1-x)^3. - _Vincenzo Librandi_, Oct 13 2015
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Oct 13 2015
%p seq((1/2)*n*(67*n-89), n = 0 .. 40);
%t Table[n (67 n - 89)/2, {n, 0, 40}] (* _Vincenzo Librandi_, Oct 13 2015
%o (PARI) vector(50, n, n--; n*(67*n-89)/2) \\ _Altug Alkan_, Oct 12 2015
%o (Magma) [n*(67*n-89)/2: n in [0..40]]; // _Bruno Berselli_, Oct 15 2015
%Y Cf. A049598, A263226, A263228, A263229, A263231.
%K sign,easy
%O 0,2
%A _Emeric Deutsch_, Oct 12 2015