A097924 a(n) = 4*a(n-1) + a(n-2), n>=2, a(0) = 2, a(1) = 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).

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 A353288 A299296

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