A133476 - OEIS

%I #43 Oct 21 2022 21:59:15

%S 0,1,2,3,4,5,7,14,36,93,220,474,948,1807,3381,6385,12393,24786,50559,

%T 103702,211585,427351,854702,1698458,3368259,6690150,13333932,

%U 26667864,53457121,107232053,214978335,430470899,860941798,1720537327,3437550076,6869397265

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

%C From _Gary W. Adamson_, Mar 14 2009: (Start)

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

%C 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]

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

%C {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

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

%H Robert Israel, <a href="/A133476/b133476.txt">Table of n, a(n) for n = 0..3260</a>

%H Vladimir Shevelev, <a href="https://arxiv.org/abs/1706.01454">Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n</a>, arXiv:1706.01454 [math.CO], 2017.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,2).

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

%F Sequence is identical to its fifth differences.

%F 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

%F 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

%F 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

%F 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

%p 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),

%p seq(a(i)=i,i=0..4)},a(n),remember):

%p map(f, [$0..30]); # _Robert Israel_, Dec 20 2015

%t LinearRecurrence[{5, -10, 10, -5, 2}, Range[0, 4], 40] (* _Jean-François Alcover_, Jul 10 2018 *)

%o (PARI) a(n) = sum(k=0, n\5, binomial(n,5*k+1)); \\ _Michel Marcus_, Dec 21 2015

%Y Cf. A049016.

%K nonn,easy

%O 0,3

%A _Paul Curtz_, Nov 29 2007

%E Better definition from _N. J. A. Sloane_, Jun 13 2008

%E Edited by _N. J. A. Sloane_, Jul 02 2008 at the suggestion of _R. J. Mathar_