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