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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 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 *) 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 Cf. A004187, A099483, A100047. Sequence in context: A165043 A301859 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 23 16:47 EDT 2024. Contains 374552 sequences. (Running on oeis4.)