cp's OEIS Frontend

A154153 Numbers k such that 28 plus the k-th triangular number is a perfect square.

%I A154153 #33 Dec 23 2024 14:53:42
%S A154153 6,8,47,57,278,336,1623,1961,9462,11432,55151,66633,321446,388368,
%T A154153 1873527,2263577,10919718,13193096,63644783,76895001,370948982,
%U A154153 448176912,2162049111,2612166473,12601345686,15224821928,73446025007,88736765097,428074804358,517195768656
%N A154153 Numbers k such that 28 plus the k-th triangular number is a perfect square.
%H A154153 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 A154153 David A. Corneth, <a href="/A154153/a154153.png">Conjectured formula for a(n)</a>
%F A154153 {k: 28+k*(k+1)/2 in A000290}.
%F A154153 Conjectures: (Start)
%F A154153 a(n) = +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).
%F A154153 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 A154153 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 A154153 See also the Corneth link - _David A. Corneth_, Mar 18 2019
%e A154153 6, 8, 47, and 57 are terms:
%e A154153    6* (6+1)/2 + 28 =  7^2,
%e A154153    8* (8+1)/2 + 28 =  8^2,
%e A154153   47*(47+1)/2 + 28 = 34^2,
%e A154153   57*(57+1)/2 + 28 = 41^2.
%t A154153 Join[{6, 8}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 28 &]] (* _G. C. Greubel_, Sep 03 2016 *)
%o A154153 (PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 28), print1(n, ", ") ) );}
%Y A154153 Cf. A000217, A000290.
%Y A154153 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 A154153 nonn
%O A154153 1,1
%A A154153 _R. J. Mathar_, Oct 18 2009
%E A154153 a(21)-a(30) from _Amiram Eldar_, Mar 18 2019