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

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

Author

Ilya Gutkovskiy, Feb 04 2016

Keywords

• nonn
• easy

Comments

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.

Links

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

Crossrefs

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

Programs

Magma

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

Mathematica

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

PARI

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

Formula

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)

Extensions

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