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 A176542 Numbers n such that there are only a finite nonzero number of sets of n consecutive triangular numbers that sum to a square. 9
 32, 50, 98, 128, 200, 242, 338, 392, 512, 578, 722, 800, 968, 1058, 1250, 1352, 1568, 1682, 1922, 2048, 2312, 2450, 2738, 2888, 3200, 3362, 3698, 3872, 4232, 4418, 4802, 5000, 5408, 5618, 6050, 6272, 6728, 6962, 7442, 7688, 8192, 8450, 8978, 9248, 9800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Members of A176541, for which there are only a finite number of solutions. It seems that a(n) = 2*A001651(n+2)^2. - Colin Barker, Sep 25 2015 LINKS FORMULA 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 Conjectures from Colin Barker, Sep 24 2015: (Start) a(n) = (9*n^2+24*n+16)/2 for n even. a(n) = (9*n^2+30*n+25)/2 for n odd. a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5. G.f.: -2*x*(4*x^4-3*x^3-8*x^2+9*x+16) / ((x-1)^3*(x+1)^2). (End) EXAMPLE 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). 3 is NOT in this sequence, because there is an infinite number of sets of 3 consecutive triangular numbers that sum to a square (cf. A165517). 4 is NOT in this sequence, because there is an infinite number of sets of 4 consecutive triangular numbers that sum to a square (cf. A202391). 5 is NOT is this sequence, because there are NO sets of 5 consecutive triangular numbers that sum to a square. 11 is NOT in this sequence, since there is an infinite number of sets of 11 consecutive triangular numbers that sum to a square (cf. A116476). CROSSREFS Cf. A176541, A000217, A000292. Sequence in context: A256521 A037008 A099048 * A048734 A277870 A077534 Adjacent sequences:  A176539 A176540 A176541 * A176543 A176544 A176545 KEYWORD nonn AUTHOR Andrew Weimholt, Apr 20 2010 EXTENSIONS Terms a(6) onward from Max Alekseyev, May 10 2010 STATUS approved

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