

A100252


Least square ngonal number greater than 1, or 0 if none exists.


12



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
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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 ngonal numbers, the Reduce function can show that there are no square ngonal 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



