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A049678 a(n) = F(8*n+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
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
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 *)
Table[Fibonacci[8*n+4]/3, {n, 0, 30}] (* G. C. Greubel, Dec 02 2017 *)
PROG
(PARI) for(n=0, 30, print1(fibonacci(8*n+4)/3, ", ")) \\ G. C. Greubel, Dec 02 2017
(Magma) [Fibonacci(8*n+4)/3: n in [0..30]]; // G. C. Greubel, Dec 02 2017
CROSSREFS
Sequence in context: A165043 A301859 A233259 * A162913 A156093 A163266
KEYWORD
nonn,easy
AUTHOR
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 23 16:47 EDT 2024. Contains 374552 sequences. (Running on oeis4.)