A116521 Binomial transform of tetranacci sequence A000078.

0, 0, 0, 1, 5, 17, 51, 148, 429, 1250, 3655, 10701, 31336, 91752, 268623, 786414, 2302262, 6739984, 19731685, 57765711, 169112717, 495088023, 1449400960, 4243211207, 12422263776, 36366946961, 106466490879, 311687250156

Offset: 0

Author

Jonathan Vos Post, Mar 10 2006

Keywords

• easy
• nonn

Comments

See also A115390, the binomial transform of tribonacci sequence A000073. Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4), with a(0) = a(1) = a(2) = 0, a(3) = 1.

Examples

			Table shows the tetranacci numbers multiplied into rows of Pascal's triangle.
1*0 = 0.
1*0 + 1*0 = 0.
1*0 + 2*0 + 1*0 = 0.
1*0 + 3*0 + 3*0 + 1* 1 = 1.
1*0 + 4*0 + 6*0 + 4*1 + 1*1 = 5.
1*0 + 5*0 + 10*0 + 10*1 + 5*1 + 1*2 = 17.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-8,6,-1).

Crossrefs

Cf. A000073, A000078, A115390.

Programs

Maple

t[0]:=0: t[1]:=0: t[2]:=0: t[3]:=1: for n from 4 to 35 do t[n]:=t[n-1]+t[n-2]+t[n-3]+t[n-4] od: seq(add(binomial(n,k)*t[k],k=0..n),n=0..30); # end of first Maple program
G:=x^3/(1-5*x+8*x^2-6*x^3+x^4): Gser:=series(G,x=0,33): seq(coeff(Gser,x,n),n=0..30); # Emeric Deutsch, Apr 09 2006

Mathematica

LinearRecurrence[{5,-8,6,-1}, {0,0,0,1}, 25] (* G. C. Greubel, Nov 03 2016 *)

PARI

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,6,-8,5]^n*[0;0;0;1])[1,1] \\ Charles R Greathouse IV, Jun 28 2017

Formula

a(n) = Sum_{k=0..n} C(n,k) * A000078(k).

G.f.: x^3/(1-5*x+8*x^2-6*x^3+x^4). - Emeric Deutsch, Apr 09 2006

a(n) = 5*a(n-1) - 8*a(n-2) + 6*a(n-3) - a(n-4). - G. C. Greubel, Nov 03 2016

Extensions

Definition corrected by Franklin T. Adams-Watters, Mar 13 2006

More terms from Emeric Deutsch, Apr 09 2006