A154143 Indices k such that 10 plus the k-th triangular number is a perfect square.
3, 5, 26, 36, 155, 213, 906, 1244, 5283, 7253, 30794, 42276, 179483, 246405, 1046106, 1436156, 6097155, 8370533, 35536826, 48787044, 207123803, 284351733, 1207205994, 1657323356, 7036112163, 9659588405, 41009466986, 56300207076, 239020689755, 328141654053
Offset: 1
Keywords
Examples
3*(3+1)/2+10 = 4^2. 5*(5+1)/2+10 = 5^2. 26*(26+1)/2+10 = 19^2. 36*(36+1)/2+10 = 26^2.
Links
- F. T. Adams-Watters, SeqFan Discussion, Oct 2009
Programs
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Magma
[n: n in [0..2*10^7] | IsSquare(10+n*(n+1)/2)]; // Vincenzo Librandi, Sep 03 2016
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Magma
[3,5] cat [n: n in [0..2*10^7] | (Ceiling(Sqrt(n*(n+ 1)/2)))^2-n*(n+1)/2 eq 10]; // Vincenzo Librandi, Sep 03 2016
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Mathematica
Join[{3, 5}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 10 &]] (* G. C. Greubel, Sep 03 2016 *) Select[Range[0, 2 10^7], IntegerQ[Sqrt[10 + # (# + 1) / 2]] &] (* Vincenzo Librandi, Sep 03 2016 *) -
PARI
isok(n) = issquare(10 + n*(n+1)/2); \\ Michel Marcus, Sep 03 2016
Formula
{k: 10+k*(k+1)/2 in A000290}.
Conjectures: (Start)
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - a(n-4) + a(n-5).
G.f.: x*(3 +2*x +3*x^2 -2*x^3 -4*x^4)/((1-x) * (x^2-2*x-1) * (x^2+2*x-1))
G.f.: ( 8 + (-1-6*x)/(x^2+2*x-1) + (8+17*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End)
a(1..4) = (3,5,26,36); a(n) = 6*a(n-2) - a(n-4) + 2, for n > 4. - Ctibor O. Zizka, Nov 10 2009
Extensions
a(17)-a(24) from Donovan Johnson, Nov 01 2010
a(25)-a(30) from Lars Blomberg, Jul 07 2015