A117717 -id:A117717 - cp's OEIS frontend

A117727 Partial sums of A051109.

1, 3, 8, 18, 38, 88, 188, 388, 888, 1888, 3888, 8888, 18888, 38888, 88888, 188888, 388888, 888888, 1888888, 3888888, 8888888, 18888888, 38888888, 88888888, 188888888, 388888888, 888888888, 1888888888, 3888888888, 8888888888

Offset: 0

N. J. A. Sloane, Apr 14 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,10,-10).

Cf. A051109, A117717.

I:=[1,3,8,18]; [n le 4 select I[n] else Self(n-1) +10*Self(n-3) -10*Self(n-4): n in [1..40]]; // G. C. Greubel, Jul 23 2023

LinearRecurrence[{1,0,10,-10}, {1,3,8,18}, 41] (* G. C. Greubel, Jul 23 2023 *)

[sum((1 + (j%3)^2)*10^(j//3) for j in range(n+1)) for n in range(41)] # G. C. Greubel, Jul 23 2023

a(n) = Sum_{j=0..n} A051109(j).
From G. C. Greubel, Jul 23 2023: (Start)
a(n) = (1/9)*( -8 + 17*b(n) + 35*b(n-1) + 80*b(n-2) ), where b(n) = 10^floor(n/3)*floor((n-1 mod 3)/2).
a(n) = a(n-1) + 10*a(n-3) - 10*a(n-4).
G.f.: (1 + 2*x + 5*x^2)/((1 - x)*(1 - 10*x^3)). (End)

A123350 a(n) = (n^4 + 2n^3 + 5n^2 + 4)/4.

Original entry on oeis.org

1, 3, 14, 46, 117, 251, 478, 834, 1361, 2107, 3126, 4478, 6229, 8451, 11222, 14626, 18753, 23699, 29566, 36462, 44501, 53803, 64494, 76706, 90577, 106251, 123878, 143614, 165621, 190067, 217126, 246978, 279809, 315811, 355182, 398126, 444853

Offset: 0

N. J. A. Sloane, Oct 10 2006

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Oct 16 2006

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

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

[(n^4 + 2*n^3 + 5*n^2 + 4)/4: n in [0..30]]; // G. C. Greubel, Oct 12 2017

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

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

for(n=0,50, print1((n^4 + 2*n^3 + 5*n^2 + 4)/4, ", ")) \\ G. C. Greubel, Oct 12 2017

G.f.: (-1 + 2*x - 9*x^2 + 4*x^3 - 2*x^4) / (x-1)^5 . - R. J. Mathar, Oct 19 2012
a(n) = 1 + A117717(n+1). - R. J. Mathar, Sep 15 2013
E.g.f.: (x^4 + 8*x^3 + 18*x^2 + 8*x + 4)*exp(x)/4. - G. C. Greubel, Oct 12 2017

More terms from Emeric Deutsch, Oct 16 2006

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A117727 Partial sums of A051109.

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A123350 a(n) = (n^4 + 2n^3 + 5n^2 + 4)/4.

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