OFFSET
0,3
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).
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 = 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
MATHEMATICA
LinearRecurrence[{3, -3, 2}, {0, 1, 2}, 40] (* Harvey P. Dale, Nov 27 2013 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Sep 14 2007, Mar 01 2008
STATUS
approved