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A110035
Row sums of an unsigned characteristic triangle for the Fibonacci numbers.
5
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).
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
Sequence in context: A317882 A335457 A290616 * A000635 A317098 A238427
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jul 08 2005
STATUS
approved