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A024537 a(n) = floor( a(n-1)/(sqrt(2) - 1) ), with a(0) = 1. 23
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 (list; graph; refs; listen; history; text; internal format)
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
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
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.
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
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. A171842. - Andrew Woods, Jan 03 2015
Sequence in context: A275864 A343264 A018905 * A171842 A296201 A027826
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane at the suggestion of Max Alekseyev, Aug 24 2007
STATUS
approved

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Last modified May 22 06:48 EDT 2024. Contains 372743 sequences. (Running on oeis4.)