%I #32 Jan 02 2023 12:30:47
%S 6,8,47,57,278,336,1623,1961,9462,11432,55151,66633,321446,388368,
%T 1873527,2263577,10919718,13193096,63644783,76895001,370948982,
%U 448176912,2162049111,2612166473,12601345686,15224821928,73446025007,88736765097,428074804358,517195768656
%N Numbers k such that 28 plus the k-th triangular number is a perfect square.
%H F. T. Adams-Watters, <a href="http://list.seqfan.eu/oldermail/seqfan/2009-October/002504.html">SeqFan Discussion</a>, Oct 2009
%H David A. Corneth, <a href="/A154153/a154153.png">Conjectured formula for a(n)</a>
%F {k: 28+k*(k+1)/2 in A000290}.
%F Conjectures: (Start)
%F a(n) = +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).
%F 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)).
%F G.f.: ( 14 + 1/(x-1) + (14+29*x)/(x^2-2*x-1) + (-1-12*x)/(x^2+2*x-1) )/2. (End)
%F See also the Corneth link - _David A. Corneth_, Mar 18 2019
%e 6, 8, 47, and 57 are terms:
%e 6* (6+1)/2 + 28 = 7^2,
%e 8* (8+1)/2 + 28 = 8^2,
%e 47*(47+1)/2 + 28 = 34^2,
%e 57*(57+1)/2 + 28 = 41^2.
%t Join[{6, 8}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 28 &]] (* _G. C. Greubel_, Sep 03 2016 *)
%o (PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 28), print1(n, ", ") ) );}
%Y Cf. A000217, A000290.
%Y 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).
%K nonn
%O 1,1
%A _R. J. Mathar_, Oct 18 2009
%E a(21)-a(30) from _Amiram Eldar_, Mar 18 2019
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