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A049678 a(n) = F(8n+4)/3, where F=A000045 (the Fibonacci sequence). 2
1, 48, 2255, 105937, 4976784, 233802911, 10983760033, 516002918640, 24241153416047, 1138818207635569, 53500214605455696, 2513371268248782143, 118074949393087305025, 5547009250206854554032, 260591359810329076734479, 12242246901835259751966481 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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)

EXAMPLE

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

MATHEMATICA

CoefficientList[Series[(1+x)/(1-47x+x^2), {x, 0, 20}], x]  (* Harvey P. Dale, Feb 18 2011 *)

CROSSREFS

Cf. A004187, A099483, A100047.

Sequence in context: A274958 A165043 A233259 * A162913 A156093 A163266

Adjacent sequences:  A049675 A049676 A049677 * A049679 A049680 A049681

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

EXTENSIONS

Better description and more terms from Michael Somos

2 more terms from Indranil Ghosh, Feb 04 2017

STATUS

approved

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Last modified July 20 16:53 EDT 2017. Contains 289628 sequences.