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a(n) = (n^4 + 2n^3 + 5n^2 + 4)/4.
2

%I #35 Sep 08 2022 08:45:28

%S 1,3,14,46,117,251,478,834,1361,2107,3126,4478,6229,8451,11222,14626,

%T 18753,23699,29566,36462,44501,53803,64494,76706,90577,106251,123878,

%U 143614,165621,190067,217126,246978,279809,315811,355182,398126,444853

%N a(n) = (n^4 + 2n^3 + 5n^2 + 4)/4.

%C Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Oct 16 2006

%C Form the 2 X 3 matrix with first row C(n,0), C(n,1), and C(n,2) and second row C(n+1,0), C(n+1,1), and C(n+1,2), multiply it by its transpose to get a 2 X 2 matrix: its determinant = a(n). - _J. M. Bergot_, Sep 05 2013

%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).

%H G. C. Greubel, <a href="/A123350/b123350.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F G.f.: (-1 + 2*x - 9*x^2 + 4*x^3 - 2*x^4) / (x-1)^5 . - _R. J. Mathar_, Oct 19 2012

%F a(n) = 1 + A117717(n+1). - _R. J. Mathar_, Sep 15 2013

%F E.g.f.: (x^4 + 8*x^3 + 18*x^2 + 8*x + 4)*exp(x)/4. - _G. C. Greubel_, Oct 12 2017

%p a:=n->(n^4+2*n^3+5*n^2+4)/4: seq(a(n),n=0..40); # _Emeric Deutsch_, Oct 16 2006

%t Table[(n^4 + 2*n^3 + 5*n^2 + 4)/4, {n,0,50}] (* _G. C. Greubel_, Oct 12 2017 *)

%o (PARI) for(n=0,50, print1((n^4 + 2*n^3 + 5*n^2 + 4)/4, ", ")) \\ _G. C. Greubel_, Oct 12 2017

%o (Magma) [(n^4 + 2*n^3 + 5*n^2 + 4)/4: n in [0..30]]; // _G. C. Greubel_, Oct 12 2017

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Oct 10 2006

%E More terms from _Emeric Deutsch_, Oct 16 2006