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A154142 Indices k such that 9 plus the k-th triangular number is a perfect square. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 19 23:35 EDT 2017. Contains 290821 sequences.