A067988 Row sums of triangle A067330; also of triangle A067418.

1, 3, 10, 25, 60, 133, 284, 585, 1175, 2310, 4464, 8502, 15995, 29775, 54920, 100487, 182556, 329555, 591550, 1056405, 1877821, 3323868, 5860800, 10297500, 18033925, 31487643, 54824854, 95211205, 164948700, 285121105, 491804144, 846631137, 1454746355, 2495275650

Offset: 0

Author

Wolfdieter Lang, Feb 15 2002

Keywords

• nonn
• easy
• changed

Comments

a(n) is the sum of the positions of the 0's in all Fibonacci binary words of length n+1. A Fibonacci binary word is a binary word having no 00 subword. Example: a(3)=25 because the Fibonacci binary words of length 4 are 1110, 1111, 1101, 1010, 1011, 0110, 0111 and 0101, the positions of the 0's being 4, 3, 2, 4, 2, 1, 4, 1, 1 and 3 (their sum is 25). - Emeric Deutsch, Jan 04 2009

Crossrefs

Cf. A000045, A001628, A067330, A067418.

Programs

Maple

a:=n->sum(binomial(n-j,j)*n*j/2,j=0..n): seq(a(n), n=2..30); # Zerinvary Lajos, Oct 19 2006

Mathematica

Table[((n+2)((3n+5)Fibonacci[n+1]+(n+1)Fibonacci[n]))/10,{n,0,30}] (* Harvey P. Dale, Feb 02 2020 *)

Formula

a(n) = (n+2)*((3*n+5)*F(n+1)+(n+1)*F(n))/10, with F(n) := A000045(n) (Fibonacci).

G.f.: (1+x^2)/(1-x-x^2)^3.

Sum_{j=0..n} binomial(n-j,j)*n*j/2. - Zerinvary Lajos, Oct 19 2006

E.g.f.: exp(x/2)*(5*(10 + 18*x + 7*x^2)*cosh(sqrt(5)*x/2) + sqrt(5)*(14 + 46*x + 15*x^2)*sinh(sqrt(5)*x/2))/50. - Stefano Spezia, Aug 30 2025

Extensions

a(29)-a(33) from Stefano Spezia, Aug 30 2025