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 A097924 a(n) = 4*a(n-1) + a(n-2), n>=2, a(0) = 2, a(1) = 7. 7
 2, 7, 30, 127, 538, 2279, 9654, 40895, 173234, 733831, 3108558, 13168063, 55780810, 236291303, 1000946022, 4240075391, 17961247586, 76085065735, 322301510526, 1365291107839, 5783465941882, 24499154875367, 103780085443350, 439619496648767, 1862258072038418 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Previous name was:  Sequence relates numerators and denominators in the continued fraction convergents to sqrt(5). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials, arXiv:1502.03085 [math.NT], 2015 (see p. 31). Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (4,1). FORMULA a(n) = A001077(n+1) - 2*A001076(n). A048875(n) + A001077(n+1)/2 = a(n)/2 + A048876(n). a(n) = ((2*sqrt(5)+3)*(2+sqrt(5))^n + (2*sqrt(5)-3)*(2-sqrt(5))^n)/(2*sqrt(5)). a(n+1) = A001077(n+1) + A015448(n+2) - Creighton Dement, Mar 08 2005 a(n) = 4*a(n-1) + a(n-2) for n>=2, a(0)=2, a(1)=7. G.f.: (2-x)/(1-4*x-x^2). - Philippe Deléham, Nov 20 2008 G.f.: G(0)*(2-x)/2, where G(k) = 1 + 1/(1 - x*(8*k + 4 +x)/(x*(8*k + 8 +x) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 15 2014 a(-1 - n) = -(-1)^n * A048875(n). - Michael Somos, Feb 23 2014 EXAMPLE G.f. = 2 + 7*x + 30*x^2 + 127*x^3 + 538*x^4 + 2279*x^5 + 9654*x^6 + 40895*x^7 + ... MATHEMATICA a[n_] := Expand[((2Sqrt[5] + 3)*(2 + Sqrt[5])^n + (2Sqrt[5] - 3)*(2 - Sqrt[5])^n)/(2Sqrt[5])]; Table[ a[n], {n, 0, 20}] (* Robert G. Wilson v, Sep 17 2004 *) a[ n_] := (3 I ChebyshevT[ n + 1, -2 I] + 4 ChebyshevT[ n, -2 I]) I^n / 5; (* Michael Somos, Feb 23 2014 *) a[ n_] := If[ n < 0, SeriesCoefficient[ (2 + 7 x) / (1 + 4 x - x^2), {x, 0, -n}], SeriesCoefficient[ (2 - x) / (1 - 4 x - x^2), {x, 0, n}]]; (* Michael Somos, Feb 23 2014 *) LinearRecurrence[{4, 1}, {2, 7}, 50] (* G. C. Greubel, Dec 20 2017 *) PROG Floretion Algebra Multiplication Program, FAMP Code: 2lesforcycseq[ ( - 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj' )*( .5'i + .5i' ) ], 2vesforcycseq = A000004. (Dement) (PARI) {a(n) = ( 3*I*polchebyshev( n+1, 1, -2*I) + 4*polchebyshev( n, 1, -2*I)) * I^n / 5}; \\ Michael Somos, Feb 23 2014 (PARI) {a(n) = if( n<0, polcoeff( (2 + 7*x) / (1 + 4*x - x^2) + x * O(x^-n), -n), polcoeff( (2 - x) / (1 - 4*x - x^2) + x * O(x^n), n))}; \\ Michael Somos, Feb 23 2014 (MAGMA) I:=[2, 7]; [n le 2 select I[n] else 4*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 20 2017 CROSSREFS Cf. A001076, A001077, A097924. Sequence in context: A173233 A074416 A325577 * A027136 A299296 A116363 Adjacent sequences:  A097921 A097922 A097923 * A097925 A097926 A097927 KEYWORD nonn,easy AUTHOR Creighton Dement, Sep 04 2004; corrected Sep 16 2004 EXTENSIONS Edited, corrected and extended by Robert G. Wilson v, Sep 17 2004 Better name (using formula from Philippe Deléham) from Joerg Arndt, Feb 16 2014 STATUS approved

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Last modified December 7 12:17 EST 2021. Contains 349581 sequences. (Running on oeis4.)