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A154146
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Numbers k such that 16 plus the k-th triangular number is a perfect square.
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4
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0, 14, 17, 87, 104, 510, 609, 2975, 3552, 17342, 20705, 101079, 120680, 589134, 703377, 3433727, 4099584, 20013230, 23894129, 116645655, 139265192, 679860702, 811697025
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OFFSET
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0,2
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COMMENTS
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Numbers k such that x=2*k+1 satisfies the Pell-type equation x^2 = 8*y^2 - 127. - Robert Israel, Jul 18 2019
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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*(-14-3*x+14*x^2+x^3)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).
G.f.: ( 2 + (8+23*x)/(x^2-2*x-1) + 1/(x-1) + (-7+6*x)/(x^2+2*x-1) )/2. (End)
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EXAMPLE
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0, 14, 17, and 87 are terms:
0* (0+1)/2 + 16 = 4^2,
14*(14+1)/2 + 16 = 11^2,
17*(17+1)/2 + 16 = 13^2,
87*(87+1)/2 + 16 = 62^2.
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MAPLE
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f:= gfun:-rectoproc({a(n+4)-6*a(n+2)+a(n)=2, a(0)=0, a(1)=14, a(2)=17, a(3)=87}, a(n), remember):
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MATHEMATICA
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Join[{0}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 16 &]] (* G. C. Greubel, Sep 03 2016 *)
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PROG
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(PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 16), print1(n, ", ") ) ); }
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CROSSREFS
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KEYWORD
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nonn,less
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AUTHOR
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STATUS
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approved
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