A005319 a(n) = 6*a(n-1) - a(n-2). (Formerly M3599)

0, 4, 24, 140, 816, 4756, 27720, 161564, 941664, 5488420, 31988856, 186444716, 1086679440, 6333631924, 36915112104, 215157040700, 1254027132096, 7309005751876, 42600007379160, 248291038523084, 1447146223759344, 8434586304032980

Offset

0,2

Comments

Solutions y of the equation 2x^2-y^2=2; the corresponding x values are given by A001541. - N-E. Fahssi, Feb 25 2008

The lower intermediate convergents to 2^(1/2) beginning with 4/3, 24/17, 140/99, 816/577, form a strictly increasing sequence; essentially, numerators=A005319 and denominators=A001541. - Clark Kimberling, Aug 26 2008

Numbers n such that (ceiling(sqrt(n*n/2)))^2 = 1 + n*n/2. - Ctibor O. Zizka, Nov 09 2009

All nonnegative solutions of the indefinite binary quadratic form X^2 + 4*X*Y -4*Y^2 of discriminant 32, representing -4 are (X(n), Y(n)) = (a(n), A001653(n+1)), for n >= 0. - Wolfdieter Lang, Jun 13 2018

Also the number of edge covers in the n-triangular snake graph. - Eric W. Weisstein, Jun 08 2019

All of the positive integer solutions of a*b+1=x^2, a*c+1=y^2, b*c+1=z^2, x+z=2*y, 0<a<b<c are given by a=A001542(n), b=A005319(n), c=A001542(n+1), x=A001541(n), y=A001653(n+1), z=A002315(n) with 0<n. - Michael Somos, Jun 26 2022

a(n) is the sum of 4*n consecutive powers of the silver ratio 1+sqrt(2), starting at (1+sqrt(2))^(-2*n) and ending at (1+sqrt(2))^(2*n-1). - Greg Dresden and Ruxin Sheng, Jul 25 2024

References

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 160, middle display.

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

Links

Marius A. Burtea, Table of n, a(n) for n = 0..100

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.

Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.

Tanya Khovanova, Recursive Sequences

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

Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, Integer sequences and ellipse chains inside a hyperbola, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.

Eric Weisstein's World of Mathematics, Edge Cover

Eric Weisstein's World of Mathematics, Triangular Snake Graph

Index entries for linear recurrences with constant coefficients, signature (6,-1).

Formula

G.f.: 4*x / ( 1-6*x+x^2 ). - Simon Plouffe in his 1992 dissertation.

G.f. for signed version beginning with 1: (1+2*x+x^2)/(1+6*x+x^2).

For any term n of the sequence, 2*n^2 + 4 is a perfect square. Limit_{n->infinity} a(n)/a(n-1) = 3 + 2*sqrt(2). - Gregory V. Richardson, Oct 06 2002

a(n) = ((3+2*sqrt(2))^n - (3-2*sqrt(2))^n) / sqrt(2). - Gregory V. Richardson, Oct 06 2002

(-1)^(n+1) = A090390(n+1) + A001542(n+1) + A046729(n) - a(n) (conjectured). - Creighton Dement, Nov 17 2004

For n > 0, a(n) = A000129(n+1)^2 - A000129(n-1)^2; a(n) = A046090(n-1) + A001652(n); e.g., 816 = 120 + 696; a(n) = A001653(n) - A001653(n-1); e.g., 816 = 985 - 169. - Charlie Marion Jul 22 2005

a(n) = 4*A001109(n). - M. F. Hasler, Mar 2009

For n > 1, a(n) is the denominator of continued fraction [1,4,1,4,...,1,4] with (n-1) repetitions of 1,4. For the numerators, see A001653. - Greg Dresden, Sep 10 2019

1/a(n) - 1/a(n+1) = 1/(Pell(2*n+1) - 1/Pell(2*n+1)) for n >= 1, where Pell(n) = A000129(n). - Peter Bala, Aug 21 2022

E.g.f.: sqrt(2)*exp(3*x)*sinh(2*sqrt(2)*x). - Stefano Spezia, Nov 25 2022

a(n) = 2*A000129(2*n). - Tanya Khovanova and MIT PRIMES STEP senior group, Apr 17 2024

Example

G.f. = 4*x + 24*x^2 + 140*x^3 + 816*x^4 + 4756*x^5 + ... - Michael Somos, Jun 26 2022

Mathematica

LinearRecurrence[{6, -1}, {0, 4}, 22] (* Jean-François Alcover, Sep 26 2017 *)
Table[((3 + 2 Sqrt[2])^n - (3 - 2 Sqrt[2])^n)/Sqrt[2], {n, 20}] // Expand (* Eric W. Weisstein, Jun 08 2019 *)
CoefficientList[Series[(4 x)/(1 - 6 x + x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 08 2019 *)
a[ n_] := 4*ChebyShevU[n-1, 3]; (* Michael Somos, Jun 26 2022 *)

Prog

(Magma) a:=[0, 4]; [n le 2 select a[n] else 6*Self(n-1) - Self(n-2):n in [1..22]]; // Marius A. Burtea, Sep 19 2019
(PARI) {a(n) = 4*polchebyshev(n-1, 2, 3)}; /* Michael Somos, Jun 26 2022 */

Crossrefs

Cf. A000129, A001109, A001541, A001542, A001652, A001653, A046090, A046729, A090390.

Sequence in context: A364277 A262376 A204199 * A155119 A370234 A114169

Adjacent sequences: A005316 A005317 A005318 * A005320 A005321 A005322

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved