A024537 a(n) = floor( a(n-1)/(sqrt(2) - 1) ), with a(0) = 1.

1, 2, 4, 9, 21, 50, 120, 289, 697, 1682, 4060, 9801, 23661, 57122, 137904, 332929, 803761, 1940450, 4684660, 11309769, 27304197, 65918162, 159140520, 384199201, 927538921, 2239277042, 5406093004, 13051463049, 31509019101, 76069501250, 183648021600

Offset

0,2

Comments

a(n) = A048739(n-1)+1 = 1/2 * (P(n)+P(n-1)+1), with P(n) = Pell numbers (A000129).

Number of (3412,#)-avoiding involutions in S_{n+1}, where # can be one of 22 patterns, see Egge reference.

Number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+1, s(0) = 1, s(n+1) = 1. - Herbert Kociemba, Jun 02 2004

Define the sequence S(a_0,a_1) by a_{n+2} is the least integer such that a_{n+2}/a_{n+1} > a_{n+1}/a_n for n >= 0 . This is S(2,4). (For proof, see the Alekseyev link.) - R. K. Guy

This sequence occurs in the lower bound of the order of the set of equivalent resistances of n equal resistors combined in series and in parallel (A048211). - Sameen Ahmed Khan, Jun 28 2010

Partial sums of the Pell numbers prefaced with a 1: (1, 1, 2, 5, 12, 29, 70, ...). - Gary W. Adamson, Feb 15 2012

The number of ways to write an n-bit binary sequence and then give runs of ones weakly incrementing labels starting with 1, e.g., 0011010011022203003330044040055555. - Andrew Woods, Jan 03 2015

Sums of the positive coefficients in Chebyshev polynomials of the first kind, beginning with T_1. a(n+1)/a(n) approaches 1/(sqrt(2)-1). - Gregory Gerard Wojnar, Mar 19 2018

Links

Clark Kimberling, Table of n, a(n) for n = 0..250

Max Alekseyev, Notes on A024537

Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2) (2000) 175-179. [From Sameen Ahmed Khan, Jun 28 2010]

Michael D. Barrus, Weakly threshold graphs, arXiv preprint arXiv:1608.01358 [math.CO], 2016.

D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993

E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, arXiv:math/0307050 [math.CO], 2003; section 8.

S. Felsner, D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.

Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.

Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346 [physics.gen-ph], 2010. [Sameen Ahmed Khan, Jun 28 2010]

J. V. Leyendekkers and A. G. Shannon, Pellian sequence relationships among pi, e, sqrt(2), Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 2, 58-62. See {u_n}. - N. J. A. Sloane, Dec 23 2012

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Formula

a(n) = 2*a(n-1) + a(n-2) - 1. - Christian G. Bower

a(n) = 3*a(n-1) - a(n-2) - a(n-3).

From Paul Barry, Dec 25 2003: (Start)

G.f.: (1 - x - x^2)/((1-x)*(1 - 2*x - x^2)) = (1 - x - x^2)/(1 - 3*x + x^2 + x^3).

E.g.f.: exp((1+sqrt(2))*x)*(1+sqrt(2))/4+exp((1-sqrt(2))*x)*(1-sqrt(2))/4+exp(x)/2. (End)

a(n) = (1/4)*(2 + (1-sqrt(2))^(n+1) + (1+sqrt(2))^(n+1)). - Herbert Kociemba, Jun 02 2004

Let M = a tridiagonal matrix with all 1's in the super and main diagonals and [1,1,0,0,0,...] in the subdiagonal, and let V = vector [1,0,0,0,...], and the rest zeros. The sequence is generated as the leftmost column from iterates of M*V. - Gary W. Adamson, Jun 07 2011

G.f.: (1 + Q(0)*x/2)/(1-x), where Q(k) = 1 + 1/(1 - x*(4*k+2 + x)/( x*(4*k+4 + x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 06 2013

a(n) = A171842(n+1), n>=0. That sequence starts with an extra 1. - Andrew Woods, Jan 03 2015

a(n) = 1 + sum_{k=1..floor((n+1)/2)} C(n+1,2*k)*2^(k-1). - Andrew Woods, Jan 03 2015

Mathematica

NestList[Floor[#/(Sqrt[2]-1)]&, 1, 40] (* Harvey P. Dale, Apr 01 2012 *)
LinearRecurrence[{3, -1, -1}, {1, 2, 4}, 31] (* Jean-François Alcover, Jan 07 2019 *)

Prog

(PARI) a=vector(99); a[1]=1; for(n=2, #a, a[n]=a[n-1]\(sqrt(2) - 1)); a \\ Charles R Greathouse IV, Jun 14 2011
(PARI) x='x+O('x^99); Vec((1-x-x^2)/((1-x)*(1-2*x-x^2))) \\ Altug Alkan, Mar 19 2018

Crossrefs

Cf. A048211, A153588, A174283-A174286, A176499-A176502. - Sameen Ahmed Khan, Jun 28 2010

Cf. A171842. - Andrew Woods, Jan 03 2015

Sequence in context: A275864 A343264 A018905 * A171842 A296201 A027826

Adjacent sequences: A024534 A024535 A024536 * A024538 A024539 A024540

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

Edited by N. J. A. Sloane at the suggestion of Max Alekseyev, Aug 24 2007

Status

approved