A131708 A024494 prefixed by a 0.

0, 1, 2, 3, 5, 10, 21, 43, 86, 171, 341, 682, 1365, 2731, 5462, 10923, 21845, 43690, 87381, 174763, 349526, 699051, 1398101, 2796202, 5592405, 11184811, 22369622, 44739243, 89478485, 178956970, 357913941, 715827883, 1431655766, 2863311531, 5726623061, 11453246122

Offset: 0

Author

Paul Curtz, Sep 14 2007, Mar 01 2008

Keywords

• nonn
• easy

Comments

Binomial transform of 0, 1, 0. Also A024495 = first differences.

Recurrence: a(n+1) - 2*a(n) = 1, 0, -1, -1, 0, 1, 1.

{A024493, A131708, A024495} is the difference analog of the hyperbolic functions {h_1(x), h_2(x), h_3(x)} of order 3. For the definitions of {h_i(x)} and the difference analog {H_i(n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Aug 01 2017

References

• A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Antoine-Augustin Cournot, Solution d'un problème d'analyse combinatoire, Bulletin des Sciences Mathématiques, Physiques et Chimiques, item 34, volume 11, 1829, pages 93-97. Also at Google Books. Page 97 case p=3 formula y^(1) = a(n).
• Christian Ramus, Solution générale d'un problème d'analyse combinatoire, Journal für die Reine und Angewandte Mathematik (Crelle's journal), volume 11, 1834, pages 353-355. Page 353 case p=3 formula y^(1) = a(n).
• Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Crossrefs

Cf. A024493, A024494, A024495, A111927, A113405, A135350, A137221.

Programs

Magma

[n le 3 select n-1 else 3*Self(n-1) -3*Self(n-2) +2*Self(n-3): n in [1..40]]; // G. C. Greubel, Jan 23 2023

Mathematica

LinearRecurrence[{3,-3,2}, {0,1,2}, 40] (* Harvey P. Dale, Nov 27 2013 *)

PARI

v=vector(99,i,i);for(i=4,#v,v[i]=3*v[i-1]-3*v[i-2]+2*v[i-3]);v \\ Charles R Greathouse IV, Jun 01 2011

SageMath

def A131708(n): return (1/3)*(2^n -chebyshev_U(n,1/2) +2*chebyshev_U(n-1,1/2))
[A131708(n) for n in range(41)] # G. C. Greubel, Jan 23 2023

Formula

G.f.: x*(1-x)/((1-2*x)*(1-x+x^2)). - R. J. Mathar, Nov 14 2007

Recurrences:

a(n) = k*a(n-1) + (6-3*k)*a(n-2) + (3*k-7)*a(n-3) + (6-2*k)*a(n-4).

k = 0: a(n) = 6*a(n-2) - 7*a(n-3) + 6*a(n-4).

k = 1: a(n) = a(n-1) + 3*a(n-2) - 4*a(n-3) + 4*a(n-4).

k = 2: a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4), cf. A113405, A135350.

k = 3: a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3), this sequence.

k = 4: a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 2*a(n-4), cf. A111927.

k = 5: a(n) = 5*a(n-1) - 9*a(n-2) + 8*a(n-3) - 4*a(n-4), cf. A137221.

The sum of coefficients = 5 - k. Of the family k=3 gives the best recurrence.

a(n+m) = a(n)*A024493(m) + A024493(n)*a(m) + A024495(n)*A024495(m). - Vladimir Shevelev, Aug 01 2017

From Kevin Ryde, Sep 24 2020: (Start)

a(n) = (1/3)*2^n - (1/3)*cos((1/3)*Pi*n) + (1/sqrt(3))*sin((1/3)*Pi*n). [Cournot]

a(n) + A024495(n) + A111927(n) = 2^n - 1. [Cournot, page 96 last formula, but misprint should be 2^x - 1 rather than 2^p - 1]. (End)

a(n) = C(n,1) + C(n,4) + ... + C(n, 3*floor(n/3)+1). - Jianing Song, Oct 04 2021

E.g.f.: exp(x/2)*(exp(3*x/2) - cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Feb 06 2025