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A154140
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Indices k such that 6 plus the k-th triangular number is a perfect square.
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4
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2, 4, 19, 29, 114, 172, 667, 1005, 3890, 5860, 22675, 34157, 132162, 199084, 770299, 1160349, 4489634, 6763012, 26167507, 39417725, 152515410, 229743340, 888924955, 1339042317, 5181034322, 7804510564, 30197280979, 45488021069, 176002651554, 265123615852
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OFFSET
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1,1
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COMMENTS
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In general, indices k such that A001109(2j) plus the k-th triangular number is a perfect square may be found as follows:
Indices k such that A001109(2j-1) plus the k-th triangular number is a perfect square may be found as follows:
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LINKS
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FORMULA
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Conjectures: (Start)
a(n) = +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).
G.f.: x*(2 +2*x +3*x^2 -2*x^3 -3*x^4)/((1-x)* (x^2-2*x-1)* (x^2+2*x-1))
G.f.: ( 6 + (-1 -4*x)/(x^2+2*x-1) + (6 +13*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End)
a(1..4) = (2,4,19,29); a(n) = 6*a(n-2) - a(n-4) + 2, for n > 4. - Ctibor O. Zizka, Nov 10 2009
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EXAMPLE
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2*(2+1)/2+6 = 3^2. 4*(4+1)/2+6 = 4^2. 19*(19+1)/2+6 = 14^2. 29*(29+1)/2+6 = 21^2.
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MATHEMATICA
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LinearRecurrence[{1, 6, -6, -1, 1}, {2, 4, 19, 29, 114}, 40] (* Following first conjecture *) (* Harvey P. Dale, Apr 11 2016 *)
Join[{2}, Select[Range[1, 1010], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 6 &] ] (* G. C. Greubel, Sep 03 2016 *)
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PROG
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(Magma) [2] cat [n: n in [0..2*10^7] | (Ceiling(Sqrt(n*(n + 1)/2)) )^2 - n*(n + 1)/2 eq 6]; // Vincenzo Librandi, Sep 03 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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