OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434.
Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).
FORMULA
a(n) = 2*a(n-1) + a(n-2) + (7*n+1), with a(0)=1, a(1)=10.
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).
a(n) = ( (25 + 17*sqrt(2))*(1+sqrt(2))^n - (25 - 17*sqrt(2))*(1-sqrt(2))^n )/(4*sqrt(2)) - (7*n + 15)/2.
a(n) = (1/2)*(4*Pell(n+2) - 3*Pell(n) - 7*n - 15), with Pell(n) = A000129(n). - Ralf Stephan, May 15 2007
E.g.f.: (1/4)*exp(x)*(-30 - 14*x + 25*sqrt(2)*sinh(sqrt(2)*x) + 34*cosh(sqrt(2)*x)). - G. C. Greubel, Nov 11 2024
MATHEMATICA
LinearRecurrence[{4, -4, 0, 1}, {1, 10, 36, 104}, 40] (* Vincenzo Librandi, Jun 22 2012 *)
PROG
(Magma) I:=[1, 10, 36, 104]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2) +Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012
(SageMath)
def A051959(n):
@CachedFunction
def a(n):
if n<4: return (1, 10, 36, 104)[n]
else: return 4*a(n-1) -4*a(n-2) +a(n-4)
return a(n)
[A051959(n) for n in range(41)] # G. C. Greubel, Nov 11 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Jan 04 2000
STATUS
approved