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A154143 Indices k such that 10 plus the k-th triangular number is a perfect square.

%I A154143 #27 Jan 07 2025 12:45:07
%S A154143 3,5,26,36,155,213,906,1244,5283,7253,30794,42276,179483,246405,
%T A154143 1046106,1436156,6097155,8370533,35536826,48787044,207123803,
%U A154143 284351733,1207205994,1657323356,7036112163,9659588405,41009466986,56300207076,239020689755,328141654053
%N A154143 Indices k such that 10 plus the k-th triangular number is a perfect square.
%H A154143 F. T. Adams-Watters, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2009-October/002506.html">SeqFan Discussion</a>, Oct 2009
%F A154143 {k: 10+k*(k+1)/2 in A000290}.
%F A154143 Conjectures: (Start)
%F A154143 a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - a(n-4) + a(n-5).
%F A154143 G.f.: x*(3 +2*x +3*x^2 -2*x^3 -4*x^4)/((1-x) * (x^2-2*x-1) * (x^2+2*x-1))
%F A154143 G.f.: ( 8 + (-1-6*x)/(x^2+2*x-1) + (8+17*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End)
%F A154143 a(1..4) = (3,5,26,36); a(n) = 6*a(n-2) - a(n-4) + 2, for n > 4. - _Ctibor O. Zizka_, Nov 10 2009
%e A154143 3*(3+1)/2+10 = 4^2. 5*(5+1)/2+10 = 5^2. 26*(26+1)/2+10 = 19^2. 36*(36+1)/2+10 = 26^2.
%t A154143 Join[{3, 5}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 10 &]] (* _G. C. Greubel_, Sep 03 2016 *)
%t A154143 Select[Range[0, 2 10^7], IntegerQ[Sqrt[10 + # (# + 1) / 2]] &] (* _Vincenzo Librandi_, Sep 03 2016 *)
%o A154143 (PARI) isok(n) = issquare(10 + n*(n+1)/2); \\ _Michel Marcus_, Sep 03 2016
%o A154143 (Magma) [n: n in [0..2*10^7] | IsSquare(10+n*(n+1)/2)]; // _Vincenzo Librandi_, Sep 03 2016
%o A154143 (Magma) [3,5] cat [n: n in [0..2*10^7] | (Ceiling(Sqrt(n*(n+ 1)/2)))^2-n*(n+1)/2 eq 10]; // _Vincenzo Librandi_, Sep 03 2016
%Y A154143 Cf. A000217, A000290, A006451.
%K A154143 nonn
%O A154143 1,1
%A A154143 _R. J. Mathar_, Oct 18 2009
%E A154143 a(17)-a(24) from _Donovan Johnson_, Nov 01 2010
%E A154143 a(25)-a(30) from _Lars Blomberg_, Jul 07 2015