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 A100252 Least square n-gonal number greater than 1, or 0 if none exists. 8
 36, 4, 9801, 1225, 81, 225, 9, 0, 196, 64, 36, 441, 3025, 16, 17689, 100, 484, 0, 2601, 729, 68121, 225, 25, 7225, 25921, 81, 1225, 203401, 441, 1089, 4761, 196, 15376, 36, 1936, 511225, 784, 576, 55071241, 47089, 1156, 256, 529046001, 2916, 1134225 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS Also, let j be the smallest integer for which 1+(1+1*n)+(1+2*n)+... +(1+j*n)=k^2=s. Then a(n)=s; if no such j exists, then a(n)=0. Basis for sequence is shortest arithmetic series with initial term 1 and difference n that sums to a perfect square. See A100251 and A188898 for the corresponding indices of these terms. Note that a(n) is zero for n = 10, 20, 52 (numbers in A188896). Although the Mathematica program searches only the first 25000 square numbers for n-gonal numbers, the Reduce function can show that there are no square n-gonal numbers (other than 0 and 1) for these n. - T. D. Noe, Apr 19 2011 LINKS Vincenzo Librandi, Table of n, a(n) for n = 3..100 Eric W. Weisstein, MathWorld: Polygonal Number Eric W. Weisstein, MathWorld: Square Number FORMULA 1+(1+1*n)+(1+2*n)+...+(1+A100254(n)*n) = 1+(1+1*n)+(1+2*n)+...+A100253(n) = A100251(n)^2 = a(n). EXAMPLE a(3)=9801 since 1 + 4 + 7 +...+ (1+80*3)= 99^2 = 9801 and no other arithmetic series with initial term 1, difference 3 and fewer terms sums to a perfect square. MATHEMATICA NgonIndex[n_, v_] := (-4 + n + Sqrt[16 - 8*n + n^2 - 16*v + 8*n*v])/(n - 2)/2; Table[k = 2; While[sqr = k^2; i = NgonIndex[n, sqr]; k < 25000 && ! IntegerQ[i], k++]; If[k == 25000, k = sqr = i = 0]; sqr, {n, 3, 64}] (* T. D. Noe, Apr 19 2011 *) CROSSREFS Cf. A000290 (squares), A188891 (similar sequence for triangular numbers). Sequence in context: A159824 A285575 A227168 * A020340 A255868 A289138 Adjacent sequences:  A100249 A100250 A100251 * A100253 A100254 A100255 KEYWORD nonn AUTHOR Charlie Marion, Nov 21 2004 STATUS approved

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