A091903 -id:A091903 - cp's OEIS frontend

A091883 Expansion of (1-5x)/((1+5x)(1-10x)).

1, 0, 50, 250, 3750, 31250, 343750, 3281250, 33593750, 332031250, 3339843750, 33300781250, 333496093750, 3332519531250, 33337402343750, 333312988281250, 3333435058593750, 33332824707031250, 333335876464843750

Offset: 0

Paul Barry, Feb 10 2004

A091903(n)=a(n+1)/50. a(n)/a(n+1)->1/10.

Index entries for linear recurrences with constant coefficients, signature (5,50).

a(n)=10^n/3+2(-5)^n/3.

a(n) = 5^n*A078008(n). - R. J. Mathar, Mar 08 2021

A099138 a(n) = 6^(n-1)*J(n), where J(n) = A001045(n).

Original entry on oeis.org

0, 1, 6, 108, 1080, 14256, 163296, 2006208, 23794560, 287214336, 3436494336, 41298398208, 495217981440, 5944792559616, 71324450021376, 855971764420608, 10271190988062720, 123257112966660096, 1479068428940476416

Offset: 0

Paul Barry, Sep 29 2004

In general k^(n-1)*J(n), where J(n) = A001045(n), is given by ((2*k)^n - (-k)^n)/(3*k) with g.f. x/((1+k*x)*(1-2*k*x)).

G. C. Greubel, Table of n, a(n) for n = 0..925
Index entries for linear recurrences with constant coefficients, signature (6,72).

Cf. A001045, A003683, A080424, A091903, A091904.

[(12^n - (-6)^n)/18: n in [0..40]]; // G. C. Greubel, Feb 18 2023

LinearRecurrence[{6,72}, {0,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

[(12^n - (-6)^n)/18 for n in range(41)] # G. C. Greubel, Feb 18 2023

G.f.: x/((1+6*x)*(1-12*x)).
a(n) = 6^(n-1)*Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k) * 2^k.
a(n) = (12^n - (-6)^n)/18.
a(n) = 6^(n-1)*A001045(n).
E.g.f.: (1/18)*(exp(12*x) - exp(-6*x)). - G. C. Greubel, Feb 18 2023

cp's OEIS Frontend

A091883 Expansion of (1-5x)/((1+5x)(1-10x)).

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A099138 a(n) = 6^(n-1)*J(n), where J(n) = A001045(n).

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