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

1,1

Links

Table of n, a(n) for n=1..30.

F. T. Adams-Watters, SeqFan Discussion, Oct 2009

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

Example

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.

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 *)

Prog

(PARI) isok(n) = issquare(10 + n*(n+1)/2); \\ Michel Marcus, Sep 03 2016
(Magma) [n: n in [0..2*10^7] | IsSquare(10+n*(n+1)/2)]; /* or */ [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

Crossrefs

Cf. A000217, A000290, A006451.

Sequence in context: A327468 A140127 A226318 * A101611 A268409 A182030

Adjacent sequences: A154140 A154141 A154142 * A154144 A154145 A154146

Keyword

nonn

Author

R. J. Mathar, Oct 18 2009

Extensions

a(17)-a(24) from Donovan Johnson, Nov 01 2010

a(25)-a(30) from Lars Blomberg, Jul 07 2015

Status

approved