A288913 a(n) = Lucas(4*n + 3).

4, 29, 199, 1364, 9349, 64079, 439204, 3010349, 20633239, 141422324, 969323029, 6643838879, 45537549124, 312119004989, 2139295485799, 14662949395604, 100501350283429, 688846502588399, 4721424167835364, 32361122672259149, 221806434537978679, 1520283919093591604

Offset: 0

Author

Bruno Berselli, Jun 19 2017

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000032, A000045, A004187, A101000.

Cf. A033891: fourth quadrisection of A000045.

Partial sums are in A081007 (after 0).

Positive terms of A098149, and subsequence of A001350, A002878, A016897, A093960, A068397.

Quadrisection of A000032: A056854 (first), A056914 (second), A246453 (third, without 11), this sequence (fourth).

Programs

Magma

[Lucas(4*n + 3): n in [0..30]]; // G. C. Greubel, Dec 22 2017

Mathematica

LucasL[4 Range[0, 21] + 3]
LinearRecurrence[{7,-1}, {4,29}, 30] (* G. C. Greubel, Dec 22 2017 *)

PARI

Vec((4 + x)/(1 - 7*x + x^2) + O(x^30)) \\ Colin Barker, Jun 20 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

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

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) = 4*A004187(n+1) + A004187(n).

a(n) = 5*A003482(n) + 4 = 5*A081016(n) - 1.

a(n) = A002878(2*n+1) = A093960(2*n+3) = A001350(4*n+3) = A068397(4*n+3).

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.