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A120537
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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].
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1
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1, 11, 44, 145, 431, 1216, 3329, 8955, 23836, 63041, 166079, 436480, 1145441, 3003211, 7869644, 20614545, 53988271, 141373376, 370169249, 969194875, 2537513276, 6643503361, 17393253119, 45536670720, 119217430081
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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].
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]
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EXAMPLE
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Matrix begins:
1 3 4 7 11...
3 4 7 11 18...
4 7 11 18 29...
7 11 18 29 47...
11 18 29 47 76...
...
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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