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A259561
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Numbers k such that k^2+2 is the product of a Fibonacci number and a Lucas number.
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1
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0, 1, 2, 3, 4, 10, 11, 29, 40, 76, 199, 521, 1364, 3571, 9349, 24476, 64079, 167761, 439204, 1149851, 3010349, 7881196, 20633239, 54018521, 141422324, 370248451, 969323029, 2537720636, 6643838879, 17393796001, 45537549124, 119218851371, 312119004989, 817138163596
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OFFSET
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1,3
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COMMENTS
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Except for a(6)=10 and a(9)=40, it seems that a(n) is a Lucas number.
Except for a(3)=2, a(6)=10 and a(9)=40, it seems that a(n)^2+2 is a Lucas number.
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LINKS
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FORMULA
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a(n) = 3*a(n-1)-a(n-2) for n>11. - Colin Barker, Jun 30 2015
G.f.: x^2*(11*x^9-15*x^8-36*x^7+6*x^6-15*x^5+x^4-3*x^3-2*x^2-x+1) / (x^2-3*x+1). - Colin Barker, Jun 30 2015
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EXAMPLE
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0^2+2 = 2 = 1*2 = F(1)*L(0);
1^2+2 = 3 = 1*3 = F(1)*L(2);
2^2+2 = 6 = 2*3 = F(3)*L(2);
3^2+2 = 11 = 1*11 = F(1)*L(5);
4^2+2 = 18 = 1*18 = F(1)*L(6);
10^2+2 = 102 = 34*3 = F(9)*L(2).
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MAPLE
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with(combinat, fibonacci):nn:=100:lst:={}:
a:=n->2*fibonacci(n-1)+fibonacci(n):
for i from 0 to nn do:
for j from 0 to nn do:
x:=sqrt(a(i)*fibonacci(j)-2):
if x=floor(x) then lst:=lst union {x}:
else fi:
od:
od:
print(lst):
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PROG
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(PARI) concat(0, Vec(x^2*(11*x^9-15*x^8-36*x^7+6*x^6-15*x^5+x^4-3*x^3-2*x^2-x+1)/(x^2-3*x+1) + O(x^50))) \\ Colin Barker, Jun 30 2015
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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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