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Row sums of an unsigned characteristic triangle for the Fibonacci numbers.
6

%I #25 Jan 01 2024 11:37:49

%S 1,2,5,12,31,80,209,546,1429,3740,9791,25632,67105,175682,459941,

%T 1204140,3152479,8253296,21607409,56568930,148099381,387729212,

%U 1015088255,2657535552,6957518401,18215019650,47687540549,124847601996

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

%C Rows sums of abs(A110033).

%H Harvey P. Dale, <a href="/A110035/b110035.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).

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

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

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

%F From _R. J. Mathar_, Jul 22 2010: (Start)

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

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

%F 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

%e 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

%t LinearRecurrence[{3,0,-3,1},{1,2,5,12},50] (* _Harvey P. Dale_, May 01 2022 *)

%t a[ n_] := With[{F = Fibonacci}, (1 + F[n+1]*F[n+2] + F[n+n])/2]; (* _Michael Somos_, Mar 03 2023 *)

%o (PARI) {a(n) = my(F = fibonacci); (1 + F(n+1)*F(n+2) + F(n+n))/2}; /* _Michael Somos_, Mar 03 2023 */

%Y Cf. A110034, A236438.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 08 2005