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A154151 Indices k such that 25 plus the k-th triangular number is a perfect square. 2
0, 18, 21, 111, 128, 650, 749, 3791, 4368, 22098, 25461, 128799, 148400, 750698, 864941, 4375391, 5041248, 25501650, 29382549, 148634511, 171254048, 866305418, 998141741, 5049197999, 5817596400, 29428882578, 33907436661, 171524097471, 197627023568 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-1,1).

FORMULA

{k: 25+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*(-18-3*x+18*x^2+x^3)/( (x-1) * (x^2+2*x-1) * (x^2-2*x-1)).

G.f.: ( 2 + 1/(x-1) + (10+29*x)/(x^2-2*x-1) + (-9+8*x)/(x^2+2*x-1) )/2. (End)

The first conjecture is true for the first 1000 terms of the sequence. - Harvey P. Dale, Jun 15 2013

EXAMPLE

0*(0+1)/2+25 = 5^2. 18*(18+1)/2+25 = 14^2. 21*(21+1)/2+25 = 16^2. 111*(111+1)/2+25 = 79^2.

MATHEMATICA

Join[{0}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 25 &]] (* or *) LinearRecurrence[{1, 6, -6, -1, 1}, {0, 18, 21, 111, 128}, 25] (* G. C. Greubel_, Sep 03 2016 *)

PROG

(PARI) for(n=1, 10^10, if(issquare(25+n*(n+1)/2), print1(n, ", ")))

CROSSREFS

Cf. A000217, A000290, A006451.

Sequence in context: A090891 A303298 A121851 * A296008 A049734 A096282

Adjacent sequences:  A154148 A154149 A154150 * A154152 A154153 A154154

KEYWORD

nonn,less,easy

AUTHOR

R. J. Mathar, Oct 18 2009

STATUS

approved

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Last modified March 26 18:36 EDT 2019. Contains 321511 sequences. (Running on oeis4.)