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A208588
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Row square-sums of triangle A185384.
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3
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1, 5, 65, 979, 15345, 247535, 4069155, 67773805, 1139789185, 19311870095, 329149434263, 5637030686105, 96925730626035, 1672193347218577, 28932082285914005, 501821453320612915, 8722842168045249345, 151912536408383664095, 2650102280875677625415
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = sum(binomial(n,k)^2*F(2n-k+1)^2, k=0..n), where F(n) are the Fibonacci numbers (A000045).
Recurrence: (n-2)*(n-1)*n*(1305*n^4 - 14094*n^3 + 57321*n^2 - 104304*n + 71944)*a(n) = 4*(n-2)*(n-1)*(6525*n^5 - 74385*n^4 + 327843*n^3 - 685683*n^2 + 653864*n - 201144)*a(n-1) - 6*(n-2)*(9135*n^6 - 125193*n^5 + 695046*n^4 - 1997365*n^3 + 3132821*n^2 - 2544304*n + 838848)*a(n-2) + 4*(32625*n^7 - 514170*n^6 + 3412449*n^5 - 12326760*n^4 + 26068504*n^3 - 32079108*n^2 + 21061664*n - 5595312)*a(n-3) - 3*(n-3)*(66555*n^6 - 918894*n^5 + 5135607*n^4 - 14853260*n^3 + 23457868*n^2 - 19205728*n + 6392000)*a(n-4) + 16*(n-4)*(n-3)*(6525*n^5 - 73080*n^4 + 309312*n^3 - 634788*n^2 + 644096*n - 264112)*a(n-5) - 4*(n-5)*(n-4)*(n-3)*(1305*n^4 - 8874*n^3 + 22869*n^2 - 26724*n + 12172)*a(n-6). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ sqrt(58+26*sqrt(5)) * (9+4*sqrt(5))^n/(20*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 13 2013
Equivalently, a(n) ~ phi^(6*n + 7/2) / (10*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021
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MATHEMATICA
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Table[Sum[Binomial[n, k]^2Fibonacci[2n-k+1]^2, {k, 0, n}], {n, 0, 20}]
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PROG
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(Maxima) makelist(sum(binomial(n, k)^2*fib(2*n-k+1)^2, k, 0, n), n, 0, 20);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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