A108649 - OEIS

%I #23 Oct 19 2023 07:37:00

%S 1,17,111,457,1428,3710,8442,17382,33099,59191,100529,163527,256438,

%T 389676,576164,831708,1175397,1630029,2222563,2984597,3952872,5169802,

%U 6684030,8551010,10833615,13602771,16938117,20928691,25673642,31282968

%N a(n) = (n+1)*(n+2)*(n+3)*(13*n^3 + 69*n^2 + 113*n + 60)/360.

%C Kekulé numbers for certain benzenoids.

%H Colin Barker, <a href="/A108649/b108649.txt">Table of n, a(n) for n = 0..1000</a>

%H S. J. Cyvin and I. Gutman, <a href="https://doi.org/10.1007/978-3-662-00892-8">Kekulé structures in benzenoid hydrocarbons</a>, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 25).

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(0)=1, a(1)=17, a(2)=111, a(3)=457, a(4)=1428, a(5)=3710, a(6)=8442, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - _Harvey P. Dale_, Jul 01 2012

%F G.f.: (1 + 10*x + 13*x^2 + 2*x^3) / (1 - x)^7. - _Colin Barker_, Apr 22 2020

%F E.g.f.: (1/360)*(360 + 5760*x + 14040*x^2 + 10440*x^3 + 2985*x^4 + 342*x^5 + 13*x^6)*exp(x). - _G. C. Greubel_, Oct 19 2023

%p a:=(n+1)*(n+2)*(n+3)*(13*n^3+69*n^2+113*n+60)/360: seq(a(n),n=0..36);

%t Table[(n+1)(n+2)(n+3)(13n^3+69n^2+113n+60)/360,{n,0,30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1}, {1,17,111,457,1428,3710, 8442},30] (* _Harvey P. Dale_, Jul 01 2012 *)

%o (PARI) Vec((1+10*x+13*x^2+2*x^3)/(1-x)^7 + O(x^40)) \\ _Colin Barker_, Apr 22 2020

%o (Magma) [(13*n^3+69*n^2+113*n+60)*Binomial(n+3,3)/60: n in [0..40]]; // _G. C. Greubel_, Oct 19 2023

%o (SageMath) [(13*n^3+69*n^2+113*n+60)*binomial(n+3,3)/60 for n in range(41)] # _G. C. Greubel_, Oct 19 2023

%Y Cf. A108645, A108646, A108647, A108648, A108650.

%K nonn,easy

%O 0,2

%A _Emeric Deutsch_, Jun 13 2005