A133476 a(n) = Sum_{k>=0} binomial(n,5*k+1).

0, 1, 2, 3, 4, 5, 7, 14, 36, 93, 220, 474, 948, 1807, 3381, 6385, 12393, 24786, 50559, 103702, 211585, 427351, 854702, 1698458, 3368259, 6690150, 13333932, 26667864, 53457121, 107232053, 214978335, 430470899, 860941798, 1720537327, 3437550076, 6869397265

Offset: 0

Author

Paul Curtz, Nov 29 2007

Keywords

• nonn
• easy

Comments

From Gary W. Adamson, Mar 14 2009: (Start)

M^n * [1,0,0,0,0] = [A139398(n), A139761(n), A139748(n), A139714(n), a(n)]

where M = a 5 X 5 matrix [1,1,0,0,0; 0,1,1,0,0; 0,0,1,1,0; 0,0,0,1,1; 1,0,0,0,1]

Sum of terms = 2^n. Example: M^6 * [1,0,0,0,0] = [7, 15, 20, 15, 7] = 2^6 = 64. (End)

{A139398, A133476, A139714, A139748, A139761} is the difference analog of the hyperbolic functions of order 5, {h_1(x), h_2(x), h_3(x), h_4(x), h_5 (x)}. For a definition see the reference "Higher Transcendental Functions" and the Shevelev link. - Vladimir Shevelev, Jun 18 2017

References

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

Links

• Robert Israel, Table of n, a(n) for n = 0..3260
• 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 (5,-10,10,-5,2).

Crossrefs

Cf. A049016.

Programs

Maple

f:= gfun:-rectoproc({a(n)=5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+2*a(n-5),
seq(a(i)=i,i=0..4)},a(n),remember):
map(f, [$0..30]); # Robert Israel, Dec 20 2015

Mathematica

LinearRecurrence[{5, -10, 10, -5, 2}, Range[0, 4], 40] (* Jean-François Alcover, Jul 10 2018 *)

PARI

a(n) = sum(k=0, n\5, binomial(n,5*k+1)); \\ Michel Marcus, Dec 21 2015

Formula

a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + 2a(n-5).

Sequence is identical to its fifth differences.

O.g.f.: x*(x-1)^3/((2*x-1)*(x^4-2*x^3+4*x^2-3*x+1)) = (1/5)*(3*x^3-7*x^2+6*x-1)/(x^4-2*x^3+4*x^2-3*x+1)-(1/5)/(2*x-1). - R. J. Mathar, Nov 30 2007

Starting (1, 2, 3, 4, 5, 7, ...) = binomial transform of (1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, ...). - Gary W. Adamson, Jul 03 2008

a(n) = round((2/5)*(2^(n-1)+phi^n*cos(Pi*(n-2)/5))), where phi is the golden ratio, round(x) is the nearest to x integer. - Vladimir Shevelev, Jun 18 2017

a(n+m) = a(n)*H_1(m) + H_1(n)*H_2(m) + H_5(n)*H_3(m) + H_4(n)*H_4(m) + H_3(n)*H_5(m), where H_1=A139398, H_3=A139714, H_4=A139748, H_5=A139761. - Vladimir Shevelev, Jun 18 2017

Extensions

Better definition from N. J. A. Sloane, Jun 13 2008

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar