A024490 a(n) = C(n-1,1) + C(n-3,3) + ... + C(n-2*m-1,2*m+1), where m = floor((n-2)/4).

1, 2, 3, 4, 6, 10, 17, 28, 45, 72, 116, 188, 305, 494, 799, 1292, 2090, 3382, 5473, 8856, 14329, 23184, 37512, 60696, 98209, 158906, 257115, 416020, 673134, 1089154, 1762289, 2851444, 4613733, 7465176, 12078908, 19544084, 31622993, 51167078, 82790071

Offset: 2

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Essentially both the first difference sequence and partial sum of A005252, so its own shifted second difference and indeed virtually the same as A005252, so close to being its own shifted first difference.

From Paul Curtz, Jun 22 2011: (Start)

b(n) = 0,0,0,1,2,3,4,6, and differences are

0, 0, 0, 1, 2, 3, 4, 6,

0, 0, 1, 1, 1, 1, 2, 4,

0, 1, 0, 0, 0, 1, 2, 3,

1, -1, 0, 0, 1, 1, 1, 1,

-2, 1, 0, 1, 0, 0, 0, 1,

3, -1, 1 -1, 0, 0, 1, 1,

-4, 2, -2, 1, 0, 1, 0, 0,

6, -4, 3, -1, 1, -1, 0, 0;

b(n) is an autosequence (sequence identical to its inverse binomial transform signed) of first kind, i.e., its main diagonal is A000004.

Examples: A000045, A001045, A113405, A191754 (array). (End)

a(n) is the number of vertices of the Fibonacci cube Gamma(n-1) having an odd number of ones. The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's and in which two vertices are adjacent when their Hamming distance is exactly 1. Example: a(4) = 3; indeed, the Fibonacci cube Gamma(3) has the five vertices 000, 010, 001, 100, 101, three of which have an odd number of ones. See the E. Munarini et al. reference, p. 323. - Emeric Deutsch, Jun 28 2015

a(n) is the number of odd permutations p of 1..n such that |p(i)-i| <= 1 for i=1..n. - Dmitry Efimov, Jan 08 2016

Links

• T. D. Noe, Table of n, a(n) for n = 2..502
• Shishuo Fu and Dazhao Tang, Partitions with fixed largest hook length, arXiv:1604.04028 [math.CO], 2016.
• Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 2.1.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 886
• S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
• E. Munarini and N. Z. Salvi, Structural and enumerative properties of the Fibonacci cubes, Discrete Math., 255, 2002, 317-324.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1).

Crossrefs

Cf. A000004, A000045, A001045, A005252, A010892, A113405, A191754.

Programs

Magma

[n le 4 select n else 2*Self(n-1)-Self(n-2)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jan 09 2016

Mathematica

LinearRecurrence[{2,-1,0,1},{1,2,3,4},39] (* Ray Chandler, Sep 23 2015 *)
CoefficientList[Series[1/((1-x-x^2)(1-x+x^2)), {x,0,40}], x] (* Vincenzo Librandi, Jan 09 2016 *)

PARI

Vec(x^2/((1-x-x^2)*(1-x+x^2)) + O(x^50)) \\ Michel Marcus, Feb 03 2016

SageMath

def A024490(n): return (fibonacci(n+1) -chebyshev_U(n,1/2))/2
[A024490(n) for n in range(2,60)] # G. C. Greubel, Apr 10 2023

Formula

a(n) = A000045(n+1) - A005252(n).

a(n) = (A000045(n+1) - A010892(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003

a(n) = Sum_{k=0..n} Fibonacci(k+1)*2*sin(Pi*(n-k)/3 + Pi/3)/sqrt(3). - Paul Barry, May 18 2004

G.f.: x^2/((1-x-x^2)(1-x+x^2)). - Jon Perry, Jun 22 2004

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+1,k+1)*(1+(-1)^k)/2. - Paul Barry, Jul 05 2007

G.f.: (1 + Q(0)*x^4/2)/(1-x)^2, where Q(k) = 1 + 1/(1 - x*( 4*k + 2 - x + x^3)/( x*(4*k + 4 - x + x^3) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jan 07 2014

E.g.f.: exp(x/2)*(15*cosh(sqrt(5)*x/2) - 5*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/30. - Stefano Spezia, Aug 03 2022

Extensions

Additional comments from Henry Bottomley, Apr 07 2000

Corrected by Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003

Further corrections from Hugo van der Sanden, Oct 05 2006