A003501 a(n) = 5*a(n-1) - a(n-2), with a(0) = 2, a(1) = 5. (Formerly M1540)

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

0,1

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

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).

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A004252, A004253, A217787.

Sequence in context: A038833 A279819 A249606 * A006990 A358608 A242227

Adjacent sequences: A003498 A003499 A003500 * A003502 A003503 A003504

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Chebyshev comments from Wolfdieter Lang, Oct 31 2002

Status

approved