A110035 Row sums of an unsigned characteristic triangle for the Fibonacci numbers.

1, 2, 5, 12, 31, 80, 209, 546, 1429, 3740, 9791, 25632, 67105, 175682, 459941, 1204140, 3152479, 8253296, 21607409, 56568930, 148099381, 387729212, 1015088255, 2657535552, 6957518401, 18215019650, 47687540549, 124847601996

Offset

0,2

Comments

Rows sums of abs(A110033).

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Formula

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

a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4);

a(n) = F(2n) + 1 + Sum_{k=0..n-1} F(k)*F(k+1).

From R. J. Mathar, Jul 22 2010: (Start)

a(n) = Sum_{i=0..n} A061646(i).

a(n) = (5 + (-1)^n + 4*A002878(n))/10. (End)

a(n) = A110034(-n) = 1 - A110034(1+n) = A236438(n) + (n mod 2) = (1 + F(n+1)*F(n+2) + F(2*n))/2 for all n in Z. - Michael Somos, Mar 03 2023

Example

G.f. = 1 + 2*x + 5*x^2 + 12*x^3 + 31*x^4 + 80*x^5 + 209*x^6 + ... - Michael Somos, Mar 03 2023

Mathematica

LinearRecurrence[{3, 0, -3, 1}, {1, 2, 5, 12}, 50] (* Harvey P. Dale, May 01 2022 *)
a[ n_] := With[{F = Fibonacci}, (1 + F[n+1]*F[n+2] + F[n+n])/2]; (* Michael Somos, Mar 03 2023 *)

Prog

(PARI) {a(n) = my(F = fibonacci); (1 + F(n+1)*F(n+2) + F(n+n))/2}; /* Michael Somos, Mar 03 2023 */

Crossrefs

Cf. A110034, A236438.

Sequence in context: A317882 A335457 A290616 * A000635 A317098 A238427

Adjacent sequences: A110032 A110033 A110034 * A110036 A110037 A110038

Keyword

easy,nonn

Author

Paul Barry, Jul 08 2005

Status

approved