A154142 Indices k such that 9 plus the k-th triangular number is a perfect square.
0, 10, 13, 63, 80, 370, 469, 2159, 2736, 12586, 15949, 73359, 92960, 427570, 541813, 2492063, 3157920, 14524810, 18405709, 84656799, 107276336, 493415986, 625252309, 2875839119, 3644237520, 16761618730, 21240172813, 97693873263, 123796799360, 569401620850
Offset: 1
Keywords
Examples
0*(0+1)/2+9 = 3^2. 10*(10+1)/2+9 = 8^2. 13*(13+1)/2+9 = 10^2. 63*(63+1)/2+9 = 45^2.
Links
- Robert Israel, Table of n, a(n) for n = 1..2351
- F. T. Adams-Watters, SeqFan Discussion, Oct 2009
Programs
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Magma
[n: n in [0..2*10^7] | IsSquare(9 + n*(n+1)/2)]; // Vincenzo Librandi, Sep 03 2016
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Magma
[0] cat [n: n in [0..2*10^7] | (Ceiling(Sqrt(n*(n+ 1)/2)))^2-n*(n+1)/2 eq 9]; // Vincenzo Librandi, Sep 03 2016
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Maple
seq(seq((8*orthopoly[U](k+j,3) - (8 - (-1)^j)*orthopoly[T](k+j,3)-1)/2, j=0..1),k=0..20); # Robert Israel, Jul 07 2015
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Mathematica
Join[{0}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 9 &]] (* G. C. Greubel, Sep 03 2016 *) Select[Range[0, 2 10^7], IntegerQ[Sqrt[9 + # (# + 1) / 2]] &] (* Vincenzo Librandi, Sep 03 2016 *) (Sqrt[8#+1]-1)/2&/@Select[Accumulate[Range[0,5*10^6]],IntegerQ[Sqrt[#+9]]&] (* The program generates the first 17 terms of the sequence. *) (* Harvey P. Dale, Oct 21 2024 *)
Formula
{k: 9+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^2*(10 +3*x -10*x^2 -x^3)/((1-x) * (x^2-2*x-1) * (x^2+2*x-1))
G.f.: ( 2 + (-5+4*x)/(x^2+2*x-1) + (6+17*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End)
a(1..4) = (0,10,13,63); a(n) = 6*a(n-2) - a(n-4) + 2, for n > 4. - Ctibor O. Zizka, Nov 10 2009
From Robert Israel, Jul 07 2015: (Start)
These conjectures follow from the theory of Pell-like equations.
Extensions
a(16)-a(24) from Donovan Johnson, Nov 01 2010
a(25)-a(30) from Lars Blomberg, Jul 07 2015