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

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..2351

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

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.

a(2*k+1) = (8 * A001109(k) -7 * A001541(k) - 1)/2.

a(2*k) = (8 * A001109(k) -9 * A001541(k) - 1)/2. (End)

Example

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.

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

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 *)

Prog

(Magma) [n: n in [0..2*10^7] | IsSquare(9 + n*(n+1)/2)];
/* or */ [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

Crossrefs

Cf. A000217, A000290, A006451, A001109, A001541.

Sequence in context: A062370 A069960 A219715 * A219804 A108761 A219455

Adjacent sequences: A154139 A154140 A154141 * A154143 A154144 A154145

Keyword

nonn

Author

R. J. Mathar, Oct 18 2009

Extensions

a(16)-a(24) from Donovan Johnson, Nov 01 2010

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

Status

approved