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

Author

Creighton Dement, Sep 04 2004; corrected Sep 16 2004

Keywords

• nonn
• easy
• changed

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.

Examples

			G.f. = 2 + 7*x + 30*x^2 + 127*x^3 + 538*x^4 + 2279*x^5 + 9654*x^6 + 40895*x^7 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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
• Index entries for linear recurrences with constant coefficients, signature (4,1).

Crossrefs

Cf. A001076, A001077, A097924.

Programs

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

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 *)

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

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

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