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

0,3

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).

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

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 *)

Prog

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

Crossrefs

Cf. A049016.

Sequence in context: A048317 A037398 A048331 * A131023 A069514 A249155

Adjacent sequences: A133473 A133474 A133475 * A133477 A133478 A133479

Keyword

nonn,easy

Author

Paul Curtz, Nov 29 2007

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

Status

approved