A154153 Numbers k such that 28 plus the k-th triangular number is a perfect square.

6, 8, 47, 57, 278, 336, 1623, 1961, 9462, 11432, 55151, 66633, 321446, 388368, 1873527, 2263577, 10919718, 13193096, 63644783, 76895001, 370948982, 448176912, 2162049111, 2612166473, 12601345686, 15224821928, 73446025007, 88736765097, 428074804358, 517195768656

Offset

1,1

Links

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

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

David A. Corneth, Conjectured formula for a(n)

Formula

{k: 28+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*(-6-2*x-3*x^2+2*x^3+7*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).

G.f.: ( 14 + 1/(x-1) + (14+29*x)/(x^2-2*x-1) + (-1-12*x)/(x^2+2*x-1) )/2. (End)

See also the Corneth link - David A. Corneth, Mar 18 2019

Example

6, 8, 47, and 57 are terms:
   6* (6+1)/2 + 28 =  7^2,
   8* (8+1)/2 + 28 =  8^2,
  47*(47+1)/2 + 28 = 34^2,
  57*(57+1)/2 + 28 = 41^2.

Mathematica

Join[{6, 8}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 28 &]] (* G. C. Greubel, Sep 03 2016 *)

Prog

(PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 28), print1(n, ", ") ) ); }

Crossrefs

Cf. A000217, A000290.

Cf. A001108 (0), A006451 (1), A154138 (3), A154139 (4), A154140 (6), A154141 (8), A154142 (9), A154143 (10), A154144 (13), A154145 (15), A154146 (16), A154147 (19), A154148 (21), A154149 (22), A154150(24), A154151 (25), A154151 (26), this sequence (28), A154154 (30).

Sequence in context: A054102 A261063 A000380 * A164640 A223852 A227167

Adjacent sequences: A154150 A154151 A154152 * A154154 A154155 A154156

Keyword

nonn

Author

R. J. Mathar, Oct 18 2009

Extensions

a(21)-a(30) from Amiram Eldar, Mar 18 2019

Status

approved