cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A218722 a(n) = (19^n-1)/18.

Original entry on oeis.org

0, 1, 20, 381, 7240, 137561, 2613660, 49659541, 943531280, 17927094321, 340614792100, 6471681049901, 122961939948120, 2336276859014281, 44389260321271340, 843395946104155461, 16024522975978953760, 304465936543600121441
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 19 (A001029); q-integers for q=19: diagonal k=1 in triangle A022183.
Partial sums are in A014903. Also, the sequence is related to A014936 by A014936(n) = n*a(n)-sum(a(i), i=0..n-1) for n>0. - Bruno Berselli, Nov 06 2012

Links

Crossrefs

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.
Cf. A001029, A014903, A014936, A022183.

Programs

Formula

a(n) = floor(19^n/18).
G.f.: x/((1-x)*(1-19*x)). - Bruno Berselli, Nov 06 2012
a(n) = 20*a(n-1) - 19*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(10*x)*sinh(9*x)/9. - Stefano Spezia, Mar 11 2023

A229463 Expansion of g.f. 1/((1-x)^2*(1-26*x)).

Original entry on oeis.org

1, 28, 731, 19010, 494265, 12850896, 334123303, 8687205886, 225867353045, 5872551179180, 152686330658691, 3969844597125978, 103215959525275441, 2683614947657161480, 69773988639086198495, 1814123704616241160886, 47167216320022270183053, 1226347624320579024759396
Offset: 0

Views

Author

Yahia Kahloune, Sep 24 2013

Keywords

Comments

This sequence was chosen to illustrate a method of solution.

Examples

			a(3) = (26^5 - 25*3 - 51)/625 = 19010.

Links

Crossrefs

Cf. A000295, A000340, A014825, A014827, A014936.

Programs

  • PARI
    my(x='x+O('x^18)); Vec(1/((1-26*x)*(1-x)^2)) \\ Elmo R. Oliveira, May 24 2025

Formula

a(n) = (26^(n+2) - 25*n - 51)/625.
In general, for the expansion of 1/((1-s*x)^2*(1-r*x)) with r>s>=1 we have the formula: a(n) = (r^(n+2)- s^(n+1)*((r-s)*n +(2*r-s)))/(r-s)^2.
From Elmo R. Oliveira, May 24 2025: (Start)
E.g.f.: exp(x)*(-51 - 25*x + 676*exp(25*x))/625.
a(n) = 28*a(n-1) - 53*a(n-2) + 26*a(n-3). (End)
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