A115730 a(n) = a(n-3) + A001654(n-1) with a(0)=0, a(1)=0 and a(2)=1.

0, 0, 1, 2, 6, 16, 42, 110, 289, 756, 1980, 5184, 13572, 35532, 93025, 243542, 637602, 1669264, 4370190, 11441306, 29953729, 78419880, 205305912, 537497856, 1407187656, 3684065112, 9645007681, 25250957930, 66107866110

Offset: 0

Author

Roger L. Bagula, Mar 13 2006

Keywords

• nonn
• easy

Comments

The a(n+1) represent the Ca2 and Ze4 sums of the Golden Triangle A180662. Furthermore the a(3*n) represent the Ze1 (terms doubled) and Ca3 sums of the Golden triangle. See A180662 for more information about these and other triangle sums.

Examples

			G.f. = x^2 + 2*x^3 + 6*x^4 + 16*x^5 + 42*x^6 + 110*x^7 + 289*x^8 + ... - _Michael Somos_, Sep 05 2023
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,0,-2,-2,1).

Crossrefs

Cf. A000045, A001654, A064831, A079962, A180662, A180664, A180665, A180666, A182890.

Programs

Magma

function A115730(n)
  if n lt 3 then return Floor(n/2);
  else return A115730(n-3) + Fibonacci(n-1)*Fibonacci(n);
  end if; return A115730;
end function;
[A115730(n): n in [0..40]]; // G. C. Greubel, Jan 20 2022

Maple

nmax:=31: with(combinat): for n from 0 to nmax do A001654(n):=fibonacci(n) * fibonacci(n+1) od: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):=a(n-3) + A001654(n-1) od: seq(a(n),n=0..nmax);

Mathematica

LinearRecurrence[{2,2,0,-2,-2,1}, {0,0,1,2,6,16}, 40] (* modified by G. C. Greubel, Jan 20 2022 *)
a[ n_] := Floor[(2*Fibonacci[2*n+1] + Fibonacci[2*n+2] + 2)/20]; (* Michael Somos, Sep 05 2023 *)

PARI

{a(n) = (2*fibonacci(2*n+1) + fibonacci(2*n+2) + 2)\20}; /* Michael Somos, Sep 05 2023 */

Sage

U=chebyshev_U
def A115730(n): return (1/60)*((-1)^n*(6 - 5*U(n, 1/2) + 10*U(n-1, 1/2)) - (10 - 9*U(n, 3/2) + 6*U(n-1, 3/2)))
[A115730(n) for n in (0..40)] # G. C. Greubel, Jan 20 2022

Formula

a(n) = -floor(g(Fibonacci(n+1))) where g(x) = (1-x^2)^2/(-4*x^2).

G.f.: x^2/( (1-x)*(1+x)*(1+x+x^2)*(1-3*x+x^2) ). - R. J. Mathar, Jun 20 2015

a(n) - a(n-2) = A182890(n-1). - R. J. Mathar, Jun 20 2015

a(n) = (1/60)*((-1)^n*(6 - 5*ChebyshevU(n, 1/2) + 10*ChebyshevU(n-1, 1/2)) - (10 - 9*ChebyshevU(n, 3/2) + 6*ChebyshevU(n-1, 3/2))). - G. C. Greubel, Jan 20 2022

a(n) = floor((2*Fibonacci(2*n+1) + Fibonacci(2*n+2) + 2)/20). - Michael Somos, Sep 05 2023

Extensions

Corrected and information added by Johannes W. Meijer, Sep 22 2010

Edited by Editors-in-Chief. - N. J. A. Sloane, Jun 20 2015