A120297 - OEIS

%I #11 Sep 24 2018 02:40:11

%S 1,5,20,65,193,544,1489,4005,10660,28193,74273,195200,512257,1343077,

%T 3519412,9219105,24144289,63224096,165544721,433437125,1134810436,

%U 2971065025,7778499265,20364618240,53315655553,139582833989

%N Sum of all matrix elements of n X n matrix M(i,j) = Fibonacci(i+j-1).

%C p^2 divides a(p-1) for p = 5, 11, 19, 29, 31, 41, 59, 61, 71, ... = A038872 (Primes congruent to {0, 1, 4} mod 5), also odd primes p such that where 5 is a square mod p. All squared prime divisors of a(n) also belong to A038872.

%H Vincenzo Librandi, <a href="/A120297/b120297.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) = Sum_{j=1..n} Sum_{i=1..n} Fibonacci(i+j-1).

%F a(n) = Fibonacci(2*n+3) - 2*Fibonacci(n+3) + 2. - _Vladeta Jovovic_, Jul 21 2006

%F G.f.: (1 - x^3 + 2*x^2)/((x-1)*(x^2 + x - 1)*(x^2 - 3*x + 1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009

%e Matrix begins:

%e   1  1  2  3  5

%e   1  2  3  5  8

%e   2  3  5  8 13

%e   3  5  8 13 21

%e   5  8 13 21 34

%t Table[Sum[Sum[Fibonacci[i+j-1],{i,1,n}],{j,1,n}],{n,1,50}]

%Y Cf. A000045, A038872, A001924, A062381.

%K nonn

%O 1,2

%A _Alexander Adamchuk_, Jul 11 2006