A003501 a(n) = 5*a(n-1) - a(n-2), with a(0) = 2, a(1) = 5.

2, 5, 23, 110, 527, 2525, 12098, 57965, 277727, 1330670, 6375623, 30547445, 146361602, 701260565, 3359941223, 16098445550, 77132286527, 369562987085, 1770682648898, 8483850257405, 40648568638127, 194758992933230, 933146396028023, 4470972987206885

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Positive values of x satisfying x^2 - 21*y^2 = 4; values of y are in A004254. - Wolfdieter Lang, Nov 29 2002

Except for the first term, positive values of x (or y) satisfying x^2 - 5xy + y^2 + 21 = 0. - Colin Barker, Feb 08 2014

Examples

			G.f. = 2 + 5*x + 23*x^2 + 110*x^3 + 527*x^4 + 2525*x^5 + ... - _Michael Somos_, Oct 25 2022
		

References

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

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Peter Bala, Some simple continued fraction expansions for an infinite product, Part 1
• 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), P(7).
• Tanya Khovanova, Recursive Sequences
• Lisa Lokteva, Constructing Rational Homology 3-Spheres That Bound Rational Homology 4-Balls, arXiv:2208.14850 [math.GT], 2022.
• 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
• Jeffrey Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 207-211.
• Jeffrey Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 207-211. [Annotated scanned copy]
• Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (5,-1).

Crossrefs

Cf. A004252, A004253, A217787.

Programs

GAP

a:=[2,5];; for n in [4..30] do a[n]:=5*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 16 2020

Magma

I:=[2,5]; [n le 2 select I[n] else 5*Self(n-1) -Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2020

Magma

R:=PowerSeriesRing(Integers(), 25); Coefficients(R!((2-5*x)/(1-5*x+x^2))); // Marius A. Burtea, Jan 16 2020

Maple

seq( simplify(2*ChebyshevT(n, 5/2)), n=0..30); # G. C. Greubel, Jan 16 2020

Mathematica

a[0]=2; a[1]=5; a[n_]:= 5a[n-1] -a[n-2]; Table[a[n], {n,0,30}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{5,-1},{2,5},30] (* Harvey P. Dale, May 12 2019 *)
2*ChebyshevT[Range[0, 30], 5/2] (* G. C. Greubel, Jan 16 2020 *)
a[ n_] := LucasL[n, 5*I]/I^n; (* Michael Somos, Oct 25 2022 *)

PARI

{a(n) = subst(poltchebi(n),x,5/2)*2};

PARI

{a(n) = polchebyshev(n,1,5/2)*2 }; /* Michael Somos, Oct 25 2022 */

Sage

[lucas_number2(n,5,1) for n in range(37)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = 5*S(n-1, 5) - 2*S(n-2, 5) = S(n, 5) - S(n-2, 5) = 2*T(n, 5/2), with S(n, x)=U(n, x/2), S(-1, x)=0, S(-2, x)=-1. U(n, x), resp. T(n, x), are Chebyshev's polynomials of the second, resp. first, kind. S(n-1, 5) = A004254(n), n>=0.

G.f.: (2-5*x)/(1-5*x+x^2). - Simon Plouffe in his 1992 dissertation.

a(n) ~ (1/2*(5 + sqrt(21)))^n. - Joe Keane (jgk(AT)jgk.org), May 16 2002

a(n) = ap^n + am^n, with ap=(5+sqrt(21))/2 and am=(5-sqrt(21))/2.

a(n) = sqrt(4 + 21*A004254(n)^2).

From Peter Bala, Jan 06 2013: (Start)

Let F(x) = Product_{n=0..inf} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let alpha = 1/2*(5 - sqrt(21)). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.19827 65373 95327 17782 ... = 2 + 1/(5 + 1/(23 + 1/(110 + ...))).

Also F(-alpha) = 0.79824 49142 28050 93561 ... has the continued fraction representation 1 - 1/(5 - 1/(23 - 1/(110 - ...))) and the simple continued fraction expansion 1/(1 + 1/((5-2) + 1/(1 + 1/((23-2) + 1/(1 + 1/((110-2) + 1/(1 + ...))))))).

F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((5^2-4) + 1/(1 + 1/((23^2-4) + 1/(1 + 1/((110^2-4) + 1/(1 + ...))))))).

(End)

a(n) = (A217787(k+3n) + A217787(k-3n))/A217787(k) for k>=3n. - Bruno Berselli, Mar 25 2013

Extensions

Chebyshev comments from Wolfdieter Lang, Oct 31 2002