A154140 Indices k such that 6 plus the k-th triangular number is a perfect square.

2, 4, 19, 29, 114, 172, 667, 1005, 3890, 5860, 22675, 34157, 132162, 199084, 770299, 1160349, 4489634, 6763012, 26167507, 39417725, 152515410, 229743340, 888924955, 1339042317, 5181034322, 7804510564, 30197280979, 45488021069, 176002651554, 265123615852

Offset

1,1

Comments

In general, indices k such that A001109(2j) plus the k-th triangular number is a perfect square may be found as follows:

b(2n-1) = A001652(n+j-1) - A001653(n-j);

b(2n) = A001652(n-j-1) + A001653(n+j);

Indices k such that A001109(2j-1) plus the k-th triangular number is a perfect square may be found as follows:

b(2n-1) = A001652(n+j-1) - A001653(n-j+1);

b(2n) = A001652(n-j) + A001653(n+j). - Charlie Marion, Mar 10 2011

Links

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

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

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

Formula

{k: 6+k*(k+1)/2 in A000290}.

a(2*n-1) = A001652(n) - A001653(n-1).

a(2*n) = A001652(n-2) + A001653(n+1).

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 +2*x +3*x^2 -2*x^3 -3*x^4)/((1-x)* (x^2-2*x-1)* (x^2+2*x-1))

G.f.: ( 6 + (-1 -4*x)/(x^2+2*x-1) + (6 +13*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End)

a(1..4) = (2,4,19,29); a(n) = 6*a(n-2) - a(n-4) + 2, for n > 4. - Ctibor O. Zizka, Nov 10 2009

Example

2*(2+1)/2+6 = 3^2. 4*(4+1)/2+6 = 4^2. 19*(19+1)/2+6 = 14^2. 29*(29+1)/2+6 = 21^2.

Mathematica

LinearRecurrence[{1, 6, -6, -1, 1}, {2, 4, 19, 29, 114}, 40] (* Following first conjecture *) (* Harvey P. Dale, Apr 11 2016 *)
Join[{2}, Select[Range[1, 1010], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 6 &] ] (* G. C. Greubel, Sep 03 2016 *)

Prog

(Magma) [2] cat [n: n in [0..2*10^7] | (Ceiling(Sqrt(n*(n + 1)/2)) )^2 - n*(n + 1)/2 eq  6]; // Vincenzo Librandi, Sep 03 2016

Crossrefs

Cf. A000217, A000290, A006451.

Sequence in context: A171735 A201379 A056727 * A131578 A018273 A047092

Adjacent sequences: A154137 A154138 A154139 * A154141 A154142 A154143

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