|
| |
|
|
A100252
|
|
Least square n-gonal number greater than 1, or 0 if none exists.
|
|
7
| |
|
|
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; 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
| 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: A156645 A037935 A159824 * A020340 A181759 A097488
Adjacent sequences: A100249 A100250 A100251 * A100253 A100254 A100255
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Charlie Marion (charliemath(AT)optonline.net), Nov 21 2004
|
| |
|
|
