cp's OEIS Frontend

A176542 Numbers n such that there are only a finite nonzero number of sets of n consecutive triangular numbers that sum to a square.

%I A176542 #27 Apr 18 2023 05:26:42
%S A176542 32,50,98,128,200,242,338,392,512,578,722,800,968,1058,1250,1352,1568,
%T A176542 1682,1922,2048,2312,2450,2738,2888,3200,3362,3698,3872,4232,4418,
%U A176542 4802,5000,5408,5618,6050,6272,6728,6962,7442,7688,8192,8450,8978,9248,9800
%N A176542 Numbers n such that there are only a finite nonzero number of sets of n consecutive triangular numbers that sum to a square.
%C A176542 Members of A176541, for which there are only a finite number of solutions.
%C A176542 Integer n is in this sequence if n=2*m^2 and the equation (2*x-m*y)*(2*x+m*y)=A077415(n)/2 has integer solutions with y>=n. - _Max Alekseyev_, May 10 2010
%C A176542 It seems that a(n) = 2*A001651(n+2)^2. - _Colin Barker_, Sep 25 2015
%F A176542 Conjectures from _Colin Barker_, Sep 24 2015: (Start)
%F A176542 a(n) = (9*n^2+24*n+16)/2 for n even.
%F A176542 a(n) = (9*n^2+30*n+25)/2 for n odd.
%F A176542 a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5.
%F A176542 G.f.: -2*x*(4*x^4-3*x^3-8*x^2+9*x+16) / ((x-1)^3*(x+1)^2).
%F A176542 (End)
%e A176542 32 is in this sequence because there is only one set of 32 consecutive triangular numbers that sum to a square (namely, A000217(26) thru A000217(57), which sum to 29584 = 172^2).
%e A176542 3 is NOT in this sequence, because there are infinitely many sets of 3 consecutive triangular numbers that sum to a square (cf. A165517).
%e A176542 4 is NOT in this sequence, because there are infinitely many sets of 4 consecutive triangular numbers that sum to a square (cf. A202391).
%e A176542 5 is NOT in this sequence, because there are NO sets of 5 consecutive triangular numbers that sum to a square.
%e A176542 11 is NOT in this sequence, since there are infinitely many sets of 11 consecutive triangular numbers that sum to a square (cf. A116476).
%Y A176542 Cf. A176541, A000217, A000292, A077415.
%K A176542 nonn
%O A176542 1,1
%A A176542 _Andrew Weimholt_, Apr 20 2010
%E A176542 Terms a(6) onward from _Max Alekseyev_, May 10 2010