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

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, 22397364832576801

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

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

From Klaus Purath, Jul 26 2024: (Start)

For any three consecutive terms (x, y, z), y^2 - xz = 1 always applies.

a(n) = (t(i+2n) - t(i))/(t(i+n+1) - t(i+n-1)) where (t) is any recurrence t(k) = 4t(k-1) + 4t(k-2) - t(k-3) or t(k) = 5t(k-1) - t(k-2) without regard to initial values.

In particular, if the recurrence (t) of the form (4,4,-1) has the same three initial values as the current sequence, a(n) = t(n) applies.

a(n) = (t(k+1)*t(k+n) - t(k)*t(k+n+1))/(y^2 - xz) where (t) is any recurrence of the current family with signature (5,-1) and (x, y, z) are any three consecutive terms of (t), for integer k >= 0. (End)

Examples

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

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.
• K. Andersen, L. Carbone, and D. Penta, 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
• Noureddine Chair, 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.
• Milan 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).

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.

Programs

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

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 *)

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

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

From Peter Bala, Dec 23 2012: (Start)

Product {n >= 1} (1 + 1/a(n)) = (1/3)*(3 + sqrt(21)).

Product {n >= 2} (1 - 1/a(n)) = (1/10)*(3 + sqrt(21)). (End)

From Michael Somos, Jan 22 2017: (Start)

A054493(2*n - 1) = 7 * a(n)^2 for all n in Z.

a(n) = -a(-n) for all n in Z.

0 = -1 + a(n)*(+a(n) - 5*a(n+1)) + a(n+1)*(+a(n+1)) for all n in Z. (End)

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

From Klaus Purath, Jul 26 2024: (Start)

a(n) = 4(a(n-1) + a(n-2)) - a(n-3).

a(n) = 6(a(n-1) - a(n-2)) + a(n-3).

In general, for all sequences of the form U(n) = P*U(n-1) - U(n-2) the following applies:

U(n) = (P-1)*U(n-1) + (P-1)*U(n-2) - U(n-3).

U(n) = (P+1)*U(n-1) - (P+1)*U(n-2) + U(n-3). (End)

a(n) = (5*a(n-1)+sqrt(21*a(n-1)^2+4))/2 for n>0. - Alexandru Petrescu, Apr 15 2025

From Peter Bala, May 22 2025: (Start)

Product_{n >= 1} ((a(2*n) + 1)/(a(2*n) - 1))^2 = 7/3.

Product_{n >= 1} ((a(2*n+1) + 1)/(a(2*n+1) - 1))^2 = 25/21.

The o.g.f. A(x) satisfies A(x) + A(-x) + 10*A(x)*A(-x) = 0. The o.g.f. for A097778 equals -1/x * A(sqrt(x))*A(-sqrt(x)). (End)

E.g.f.: 2*exp(5*x/2)*sinh(sqrt(21)*x/2)/sqrt(21). - Stefano Spezia, Jul 02 2025