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A120537 Sum of all matrix elements of n X n matrix M[i,j] = Lucas[i+j-1], (i,j = 1..n), where Lucas[n] = A000032[n] = Fibonacci[n-1] + Fibonacci[n+1].

%I A120537 #7 Aug 14 2012 06:16:44
%S A120537 1,11,44,145,431,1216,3329,8955,23836,63041,166079,436480,1145441,
%T A120537 3003211,7869644,20614545,53988271,141373376,370169249,969194875,
%U A120537 2537513276,6643503361,17393253119,45536670720,119217430081
%N A120537 Sum of all matrix elements of n X n matrix M[i,j] = Lucas[i+j-1], (i,j = 1..n), where Lucas[n] = A000032[n] = Fibonacci[n-1] + Fibonacci[n+1].
%C A120537 5 divides a(4k). a(n) is prime for n = {2,5,7,17,19,31,439,545,...}. p^2 divides a(p-1) for p = {11,19,29,31,41,59,61,71,...} = A045468[n] Primes congruent to {1, 4} mod 5, also odd primes where 5 is a square mod p except 5. Square prime divisors of a(n) up to n=70 are p = {2,3,7,11,13,19,23,29,31,41,47,59,61,71,89,101,139,151,199,233,281,461,521,911,1597,2207,3571,5779,9349,9901,19489,3010349,...} that appear to be the prime factors of Fibonacci numbers.
%H A120537 Vincenzo Librandi, <a href="/A120537/b120537.txt">Table of n, a(n) for n = 1..200</a>
%F A120537 a(n) = Sum[ Sum[ Fibonacci[i+j-2] + Fibonacci[i+j],{i,1,n}],{j,1,n}]. a(n) = Lucas[2n+3] - 2*Lucas[n+3] + 4, where Lucas[k] = Fibonacci[k-1] + Fibonacci[k+1].
%F A120537 G.f.:(1+x^3-4*x^2+6*x)/((x-1)*(x^2+x-1)*(x^2-3*x+1)) [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009]
%e A120537 Matrix begins:
%e A120537 1 3 4 7 11...
%e A120537 3 4 7 11 18...
%e A120537 4 7 11 18 29...
%e A120537 7 11 18 29 47...
%e A120537 11 18 29 47 76...
%e A120537 ...
%t A120537 Table[Sum[Sum[Fibonacci[i+j-2]+Fibonacci[i+j],{i,1,n}],{j,1,n}],{n,1,70}] Table[(Fibonacci[2n+2]+Fibonacci[2n+4])-2(Fibonacci[n+2]+Fibonacci[n+4])+4,{n,1,70}]
%Y A120537 Cf. A120297, A000032, A000045, A045468.
%K A120537 nonn
%O A120537 1,2
%A A120537 _Alexander Adamchuk_, Aug 07 2006