A014827 -id:A014827 - cp's OEIS frontend

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

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

Offset: 0

Yahia Kahloune, Sep 24 2013

This sequence was chosen to illustrate a method of solution.

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

Index entries for linear recurrences with constant coefficients, signature (28,-53,26).

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

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

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)

A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.

Original entry on oeis.org

1, 3, 11, 49, 237, 1175, 5863, 29301, 146489, 732427, 3662115, 18310553, 91552741, 457763679, 2288818367, 11444091805, 57220458993, 286102294931, 1430511474619, 7152557373057, 35762786865245, 178813934326183, 894069671630871, 4470348358154309, 22351741790771497, 111758708953857435

Offset: 0

Ilya Gutkovskiy, Feb 04 2016

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*n, with n > 0 and b(0)=1, is (1 - (m + 2)*x + x^2)/((1 - x)^2*(1 - k*x)). This recurrence gives the closed form b(n) = ((k^2 - k*(m + 2) + 1)*k^n + m*((k - 1)*n + k))/(k - 1)^2.

Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A014827, A024050, A094195, A104745, A107585, A164045, A176916, A221907.

[(4*n + 3*5^n + 5)/8: n in [0..30]]; // Vincenzo Librandi, Feb 06 2016

Table[(4 n + 3 5^n + 5)/8, {n, 0, 23}]
LinearRecurrence[{7, -11, 5}, {1, 3, 11}, 24]

Vec((1-4*x+x^2)/((1-x)^2*(1-5*x)) + O(x^100)) \\ Altug Alkan, Feb 04 2016

G.f.: (1 - 4*x + x^2)/((1 - x)^2*(1 - 5*x)).
a(n) = (4*n + 3*5^n + 5)/8.
Sum_{n>=0} 1/a(n) = 1.449934283402232875...
Lim_{n -> oo} a(n + 1)/a(n) = 5.
From Elmo R. Oliveira, Sep 10 2024: (Start)
E.g.f.: exp(x)*(3*exp(4*x) + 4*x + 5)/8.
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3) for n > 2. (End)

a(24)-a(25) from Elmo R. Oliveira, Sep 10 2024

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

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A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.

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cp's OEIS Frontend

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

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A268414 a(n) = 5*a(n-1) - 2*n for n > 0, a(0) = 1.

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

A268414 a(n) = 5a(n-1) - 2n for n > 0, a(0) = 1.