OFFSET
0,1
COMMENTS
a(n) mod 4 gives A101000.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (7,-1).
FORMULA
G.f.: (4 + x)/(1 - 7*x + x^2).
a(n) = 7*a(n-1) - a(n-2) for n>1, with a(0)=4, a(1)=29.
a(n) = ((sqrt(5) + 1)^(4*n + 3) - (sqrt(5) - 1)^(4*n + 3))/(8*16^n).
a(n) = Fibonacci(4*n+4) + Fibonacci(4*n+2).
a(n+1)*a(n+k) - a(n)*a(n+k+1) = 15*Fibonacci(4*k). Example: for k=6, a(n+1)*a(n+6) - a(n)*a(n+7) = 15*Fibonacci(24) = 695520.
MATHEMATICA
LucasL[4 Range[0, 21] + 3]
LinearRecurrence[{7, -1}, {4, 29}, 30] (* G. C. Greubel, Dec 22 2017 *)
PROG
(PARI) Vec((4 + x)/(1 - 7*x + x^2) + O(x^30)) \\ Colin Barker, Jun 20 2017
(Sage)
def L():
x, y = -1, 4
while True:
yield y
x, y = y, 7*y - x
r = L(); [next(r) for _ in (0..21)] # Peter Luschny, Jun 20 2017
(Magma) [Lucas(4*n + 3): n in [0..30]]; // G. C. Greubel, Dec 22 2017
(Python)
from sympy import lucas
def a(n): return lucas(4*n + 3)
print([a(n) for n in range(22)]) # Michael S. Branicky, Apr 29 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Jun 19 2017
STATUS
approved