A154147 Indices k such that 19 plus the k-th triangular number is a perfect square.

3, 9, 30, 60, 179, 353, 1046, 2060, 6099, 12009, 35550, 69996, 207203, 407969, 1207670, 2377820, 7038819, 13858953, 41025246, 80775900, 239112659, 470796449, 1393650710, 2744002796, 8122791603, 15993220329, 47343098910, 93215319180, 275935801859

Offset

1,1

Links

Colin Barker, Table of n, a(n) for n = 1..500

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

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

Formula

{k: 19+k*(k+1)/2 in A000290}.

a(n)= +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).

G.f.: x*(3 +6*x +3*x^2 -6*x^3 -4*x^4)/((1-x) * (x^2-2*x-1) * (x^2+2*x-1)).

Example

3*(3+1)/2+19 = 5^2. 9*(9+1)/2+19 = 8^2. 30*(30+1)/2+19 = 22^2. 60*(60+1)/2+19 = 43^2.

Mathematica

Join[{3, 9}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 19 &]] (* or *) LinearRecurrence[{1, 6, -6, -1, 1}, {3, 9, 30, 60, 179}, 25] (* G. C. Greubel, Sep 03 2016 *)

Prog

(PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 19), print1(n, ", ") ) ); }

Crossrefs

Cf. A000217, A000290, A006451.

Sequence in context: A292756 A151451 A138938 * A179545 A163129 A074003

Adjacent sequences: A154144 A154145 A154146 * A154148 A154149 A154150

Keyword

nonn,less

Author

R. J. Mathar, Oct 18 2009

Status

approved