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A154144 Indices k such that 13 plus the k-th triangular number is a perfect square. 2
2, 8, 23, 53, 138, 312, 807, 1821, 4706, 10616, 27431, 61877, 159882, 360648, 931863, 2102013, 5431298, 12251432, 31655927, 71406581, 184504266, 416188056, 1075369671, 2425721757, 6267713762, 14138142488, 36530912903, 82403133173, 212917763658, 480280656552 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

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

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

FORMULA

{k: 13+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 +6*x +3*x^2 -6*x^3 -3*x^4)/((1-x) * (x^2-2*x-1) * (x^2+2*x-1))

G.f.: ( 6 + (-3-2*x)/(x^2+2*x-1) + 1/(x-1) + (8+19*x)/(x^2-2*x-1) )/2 . (End)

a(1..4) = (2,8,23,53); a(n) = 6*a(n-2) - a(n-4) + 2, for n>2. - Ctibor O. Zizka, Nov 10 2009

EXAMPLE

2*(2+1)/2+13 = 4^2. 8*(8+1)/2+13 = 7^2. 23*(23+1)/2+13 = 17^2. 53*(53+1)/2+13 = 38^2.

MATHEMATICA

With[{nn=25000}, Transpose[Select[Thread[{Range[nn], Accumulate[ Range[nn]]}], IntegerQ[Sqrt[#[[2]]+13]]&]][[1]]] (* Harvey P. Dale, Jan 13 2012 *)

Join[{2, 8}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 13 &]] (* G. C. Greubel, Sep 03 2016 *)

CROSSREFS

Cf. A000217, A000290, A006451.

Sequence in context: A190021 A014285 A079460 * A255942 A180664 A294959

Adjacent sequences:  A154141 A154142 A154143 * A154145 A154146 A154147

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

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Last modified August 18 08:57 EDT 2019. Contains 326077 sequences. (Running on oeis4.)