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A154143 Indices k such that 10 plus the k-th triangular number is a perfect square. 4
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 (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: 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

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Last modified December 9 23:05 EST 2022. Contains 358710 sequences. (Running on oeis4.)