OFFSET
0,3
COMMENTS
From Gary W. Adamson, Mar 14 2009: (Start)
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)
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
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