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A154146 Numbers k such that 16 plus the k-th triangular number is a perfect square.

%I A154146 #26 Dec 23 2024 14:53:42
%S A154146 0,14,17,87,104,510,609,2975,3552,17342,20705,101079,120680,589134,
%T A154146 703377,3433727,4099584,20013230,23894129,116645655,139265192,
%U A154146 679860702,811697025
%N A154146 Numbers k such that 16 plus the k-th triangular number is a perfect square.
%C A154146 Numbers k such that x=2*k+1 satisfies the Pell-type equation x^2 = 8*y^2 - 127. - _Robert Israel_, Jul 18 2019
%H A154146 Robert Israel, <a href="/A154146/b154146.txt">Table of n, a(n) for n = 0..2608</a>
%H A154146 F. T. Adams-Watters, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2009-October/002504.html">SeqFan Discussion</a>, Oct 2009
%H A154146 Robert Israel, <a href="/A154146/a154146.pdf">Proof of conjectured recurrence</a>
%H A154146 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6,-1,1).
%F A154146 {k: 16+k*(k+1)/2 in A000290}.
%F A154146 Conjectures: (Start)
%F A154146 a(n) = +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).
%F A154146 G.f.: x^2*(-14-3*x+14*x^2+x^3)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).
%F A154146 G.f.: ( 2 + (8+23*x)/(x^2-2*x-1) + 1/(x-1) + (-7+6*x)/(x^2+2*x-1) )/2. (End)
%F A154146 Conjectures confirmed: see link. - _Robert Israel_, Jul 18 2019
%e A154146 0, 14, 17, and 87 are terms:
%e A154146    0* (0+1)/2 + 16 =  4^2,
%e A154146   14*(14+1)/2 + 16 = 11^2,
%e A154146   17*(17+1)/2 + 16 = 13^2,
%e A154146   87*(87+1)/2 + 16 = 62^2.
%p A154146 f:= gfun:-rectoproc({a(n+4)-6*a(n+2)+a(n)=2, a(0)=0, a(1)=14, a(2)=17, a(3)=87}, a(n), remember):
%p A154146 map(f, [$0..40]); # _Robert Israel_, Jul 18 2019
%t A154146 Join[{0}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 16 &]] (* _G. C. Greubel_, Sep 03 2016 *)
%o A154146 (PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 16), print1(n, ", ") ) );}
%Y A154146 Cf. A000217, A000290, A006451.
%K A154146 nonn,less
%O A154146 0,2
%A A154146 _R. J. Mathar_, Oct 18 2009