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

0,4

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.

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).

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A025169 A111282 A358464 * A191694 A224232 A217661

Adjacent sequences: A115727 A115728 A115729 * A115731 A115732 A115733

Keyword

nonn,easy

Author

Roger L. Bagula, Mar 13 2006

Extensions

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

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

Status

approved