A097757 Table read by rows where row n consists of integers that can be expressed as the sum of distinct squares in exactly n ways.

2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128, 0, 1, 4, 5, 9, 10, 13, 14, 16, 17, 20, 21, 34, 35, 36, 37, 38, 39, 40, 42, 51, 52, 55, 56, 57, 58, 59, 63, 64, 68, 71, 73, 80, 83, 88, 97, 124, 132, 25, 26, 29

Offset: 0

Author

Ray Chandler, Sep 06 2004

Keywords

• nonn
• tabf

Comments

Row 0 is A001422.

Only positive squares are allowed, not 0. The inclusion of 0 in row 1 is for the empty sum, not for a sum with a single 0. - Franklin T. Adams-Watters, Sep 20 2009

Examples

			Table begins:
Row 0: 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128;
Row 1: 0, 1, 4, 5, 9, 10, 13, 14, 16, 17, 20, 21, 34, 35, 36, 37, 38, 39, 40, 42, 51, 52, 55, 56, 57, 58, 59, 63, 64, 68, 71, 73, 80, 83, 88, 97, 124, 132;
Row 2: 25, 26, 29, 30, 41, 45, 46, 49, 53, 54, 61, 69, 70, 77, 79, 82, 84, 87, 93, 103, 107, 133, 144, 148, 188;
Row 3: 50, 62, 66, 75, 81, 85, 86, 89, 91, 95, 98, 99, 100, 102, 104, 109, 113, 116, 118, 119, 123, 136, 137, 140, 152, 157, 172, 176, 177, 192;
Row 4: 65, 74, 78, 101, 105, 106, 111, 115, 117, 120, 121, 122, 127, 141, 153, 160, 164, 168, 193;
Row 5: 94, 125, 129, 131, 143, 145, 149, 156, 161, 163, 167, 173, 197, 213;
Row 6: 90, 114, 134, 135, 138, 139, 147, 180, 181, 208, 212, 217, 228;
Row 7: 110, 142, 151, 154, 158, 169, 184, 189, 204, 224;
Row 8: 155, 159, 162, 165, 166, 182, 187, 196, 201, 202, 203, 216, 229, 233, 240, 252, 253;
Row 9: 126, 130, 146, 150, 171, 178, 179, 183, 185, 200, 209, 236, 237, 241, 288;
Row 10: 191, 205, 218, 232, 249, 257;
Row 11: 170, 175, 198, 207, 220, 221, 227, 245, 272, 277, 293;
Row 12: 186, 214, 225, 244, 248, 268, 297;
Row 13: 174, 199, 223, 256, 265, 292;
Row 14: 190, 194, 206, 215, 261, 269, 273, 281, 313, 317;
Row 15: 211, 219, 242, 262, 301;
Row 16: 195, 222, 239, 243, 276, 278, 289, 333;
Row 17: 226, 230, 238, 264, 266, 284;
Row 18: 210, 258, 263, 267, 285, 304, 308, 337;
Row 19: 231, 246, 254, 260, 357;
Row 20: 234, 247, 251, 282, 305, 309, 353;
Row 21: 235, 250, 280, 298, 321, 329;
Row 22: 332.
Denoting r(n) the index of the row of the number n, among n = {350, ..., 1000} the only r-values below 28 are a(357) = 19, a(353) = 20, a(373) = 25, a(397) = 26, a(362) = 27, and all n >= 400 have r(n) > 30. So the above rows appear to be complete, and rows 23 & 24 would be empty, as would be rows 39 & 40 unless they have elements > 1000. - _M. F. Hasler_, May 26 2020
		

Links

• R. Sprague, Über Zerlegungen in ungleiche Quadratzahlen, Math. Z. 51, (1948), 289-290.
• Eric Weisstein's World of Mathematics, Square Number.

Crossrefs

Cf. A001422, A003995, A033461, A097563, A097758, A097759, A097760.

Programs

PARI

{r(n,m=n)=sum(x=1,min(sqrtint(n),m),r(n-x^2,x-1),!n)} \\ Gives index of the row in which number n is listed. - M. F. Hasler, May 26 2020