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URL: https://oeis.org/A097924

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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
OFFSET
0,1
COMMENTS
Previous name was: Sequence relates numerators and denominators in the continued fraction convergents to sqrt(5).
Floretion Algebra Multiplication Program, FAMP Code: 2lesforcycseq[ ( - 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj' )*( .5'i + .5i' ) ], 2vesforcycseq = A000004.
LINKS
Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, and 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
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
From Philippe Deléham, Nov 20 2008: (Start)
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). (End)
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
E.g.f.: exp(2*x)*(10*cosh(sqrt(5)*x) + 3*sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, Aug 21 2025
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
(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
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