A049678 a(n) = F(8*n+4)/3, where F=A000045 (the Fibonacci sequence).

1, 48, 2255, 105937, 4976784, 233802911, 10983760033, 516002918640, 24241153416047, 1138818207635569, 53500214605455696, 2513371268248782143, 118074949393087305025, 5547009250206854554032, 260591359810329076734479, 12242246901835259751966481

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Examples

			a(2) = F(8 * 2 + 4) / 3 = F(20) / 3 = 6765 / 3 = 2255. - _Indranil Ghosh_, Feb 04 2017
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..596
• Tanya Khovanova, Recursive Sequences
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
• Index entries for linear recurrences with constant coefficients, signature (47,-1).

Crossrefs

Cf. A004187, A099483, A100047.

Programs

Magma

[Fibonacci(8*n+4)/3: n in [0..30]]; // G. C. Greubel, Dec 02 2017

Mathematica

CoefficientList[Series[(1+x)/(1-47x+x^2),{x,0,20}],x]  (* Harvey P. Dale, Feb 18 2011 *)
Table[Fibonacci[8*n+4]/3, {n,0,30}] (* G. C. Greubel, Dec 02 2017 *)

PARI

for(n=0,30, print1(fibonacci(8*n+4)/3, ", ")) \\ G. C. Greubel, Dec 02 2017

Formula

a(n) = 47*a(n-1) - a(n-2), n>1. a(0)=1, a(1)=48.

G.f.: (1+x)/(1-47*x+x^2).

From Peter Bala, Mar 23 2015: (Start)

a(n) = A004187(2*n + 1); a(n) = A099483(4*n + 1).

a(n) = ( Fibonacci(8*n + 8 - 2*k) + Fibonacci(8*n + 2*k) )/( Fibonacci(8 - 2*k) + Fibonacci(2*k) ), for k an arbitrary integer.

a(n) = ( Fibonacci(8*n + 8 - 2*k - 1) - Fibonacci(8*n + 2*k + 1) )/( Fibonacci(8 - 2*k - 1) - Fibonacci(2*k + 1) ), for k an arbitrary integer.

The aerated sequence (b(n))n>=1 = [1, 0, 48, 0, 2255, 0, 105937, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -45, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)

Extensions

Better description and more terms from Michael Somos

2 more terms from Indranil Ghosh, Feb 04 2017