A004254 a(n) = 5*a(n-1) - a(n-2) for n > 1, a(0) = 0, a(1) = 1. (Formerly M3930)

0, 1, 5, 24, 115, 551, 2640, 12649, 60605, 290376, 1391275, 6665999, 31938720, 153027601, 733199285, 3512968824, 16831644835, 80645255351, 386394631920, 1851327904249, 8870244889325, 42499896542376, 203629237822555, 975646292570399, 4674602225029440

Offset

0,3

Comments

Nonnegative values of y satisfying x^2 - 21*y^2 = 4; values of x are in A003501. - Wolfdieter Lang, Nov 29 2002

a(n) is equal to the permanent of the (n-1) X (n-1) Hessenberg matrix with 5's along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else. - John M. Campbell, Jun 09 2011

For n >= 1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,2,3,4}. - Milan Janjic, Jan 25 2015

Lim_{n->oo} a(n+1)/a(n) = (5 + sqrt(21))/2 = A107905. - Wolfdieter Lang, Nov 15 2023

References

F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Indranil Ghosh, Table of n, a(n) for n = 0..1467 (terms 0..200 from T. D. Noe)

Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.

Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.

D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 , Example 12

Chair, Noureddine Exact two-point resistance, and the simple random walk on the complete graph minus N edges, Ann. Phys. 327, No. 12, 3116-3129 (2012), B(7).

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.

Dale Gerdemann, Fractal images from (5,-1) recursion, YouTube Video, Nov 05 2014.

Dale Gerdemann, Fractal images from (5,-1) recursion: Selections in detail, YouTube Video, Nov 05 2014.

Frank A. Haight, Letter to N. J. A. Sloane, Sep 06 1976

Frank A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971. [Annotated scanned copy]

A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=5, q=-1.

A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Tanya Khovanova, Recursive Sequences

Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=7.

Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

F. M. van Lamoen, Square wreaths around hexagons, Forum Geometricorum, 6 (2006) 311-325.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (5,-1).

Formula

G.f.: x/(1-5*x+x^2).

a(n) = ((5+sqrt(21))/2)^n-((5-sqrt(21))/2)^n)/sqrt(21). - Barry E. Williams, Aug 29 2000

a(n) = S(2*n-1, sqrt(7))/sqrt(7) = S(n-1, 5); S(n, x)=U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.

A003501(n) = sqrt(21*a(n)^2 + 4).

a(n) = Sum_{k=0..n-1} binomial(n+k, 2*k+1)*2^k. - Paul Barry, Nov 30 2004

[A004253(n), a(n)] = [1,3; 1,4]^n * [1,0]. - Gary W. Adamson, Mar 19 2008

a(n+1) = Sum_{k=0..n} Gegenbauer_C(n-k,k+1,2). - Paul Barry, Apr 21 2009

a(n+1) = Sum_{k=0..n} A101950(n,k)*4^k. - Philippe Deléham, Feb 10 2012

Product {n >= 1} (1 + 1/a(n)) = (1/3)*(3 + sqrt(21)). - Peter Bala, Dec 23 2012

Product {n >= 2} (1 - 1/a(n)) = (1/10)*(3 + sqrt(21)). - Peter Bala, Dec 23 2012

A054493(2*n - 1) = 7 * a(n)^2 for all n in Z. - Michael Somos, Jan 22 2017

a(n) = -a(-n) for all n in Z. - Michael Somos, Jan 22 2017

0 = -1 + a(n)*(+a(n) - 5*a(n+1)) + a(n+1)*(+a(n+1)) for all n in Z. - Michael Somos, Jan 22 2017

Example

G.f. = x + 5*x^2 + 24*x^3 + 115*x^4 + 551*x^5 + 2640*x^6 + 12649*x^7 + ...

Maple

A004254:=1/(1-5*z+z**2); # Simon Plouffe in his 1992 dissertation

Mathematica

a[n_]:=(MatrixPower[{{1, 3}, {1, 4}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
a[ n_] := ChebyshevU[2 n - 1, Sqrt[7]/2] / Sqrt[7]; (* Michael Somos, Jan 22 2017 *)

Prog

(PARI) {a(n) = subst(4*poltchebi(n+1) - 10*poltchebi(n), x, 5/2) / 21}; /* Michael Somos, Dec 04 2002 */
(PARI) {a(n) = imag((5 + quadgen(84))^n) / 2^(n-1)}; /* Michael Somos, Dec 04 2002 */
(PARI) {a(n) = polchebyshev(n - 1, 2, 5/2)}; /* Michael Somos, Jan 22 2017 */
(PARI) {a(n) = simplify( polchebyshev( 2*n - 1, 2, quadgen(28)/2) / quadgen(28))}; /* Michael Somos, Jan 22 2017 */
(Sage) [lucas_number1(n, 5, 1) for n in range(27)] # Zerinvary Lajos, Jun 25 2008
(Magma) [ n eq 1 select 0 else n eq 2 select 1 else 5*Self(n-1)-Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 19 2011

Crossrefs

Partial sums of A004253.

Cf. A000027, A001906, A001353, A003501, A030221. a(n) = sqrt((A003501(n)^2 - 4)/21).

First differences of a(n) are in A004253, partial sums in A089817.

Cf. A004253.

INVERT transformation yields A001109. - R. J. Mathar, Sep 11 2008

Cf. A054493, A107905.

Sequence in context: A026388 A242509 A057969 * A086347 A200739 A026707

Adjacent sequences: A004251 A004252 A004253 * A004255 A004256 A004257

Keyword

easy,nonn,easy

Author

N. J. A. Sloane

Status

approved