OFFSET
0,2
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).
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)
MATHEMATICA
Table[(4 n + 3 5^n + 5)/8, {n, 0, 23}]
LinearRecurrence[{7, -11, 5}, {1, 3, 11}, 24]
PROG
(PARI) Vec((1-4*x+x^2)/((1-x)^2*(1-5*x)) + O(x^100)) \\ Altug Alkan, Feb 04 2016
(Magma) [(4*n + 3*5^n + 5)/8: n in [0..30]]; // Vincenzo Librandi, Feb 06 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Feb 04 2016
EXTENSIONS
a(24)-a(25) from Elmo R. Oliveira, Sep 10 2024
STATUS
approved