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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.

%I A097757 #16 Feb 16 2025 08:32:54
%S A097757 2,3,6,7,8,11,12,15,18,19,22,23,24,27,28,31,32,33,43,44,47,48,60,67,
%T A097757 72,76,92,96,108,112,128,0,1,4,5,9,10,13,14,16,17,20,21,34,35,36,37,
%U A097757 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
%N 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.
%C A097757 Row 0 is A001422.
%C A097757 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
%H A097757 R. Sprague, <a href="https://dx.doi.org/10.1007/BF01181594">Über Zerlegungen in ungleiche Quadratzahlen</a>, Math. Z. 51, (1948), 289-290.
%H A097757 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number.</a>
%e A097757 Table begins:
%e A097757 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;
%e A097757 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;
%e A097757 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;
%e A097757 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;
%e A097757 Row 4: 65, 74, 78, 101, 105, 106, 111, 115, 117, 120, 121, 122, 127, 141, 153, 160, 164, 168, 193;
%e A097757 Row 5: 94, 125, 129, 131, 143, 145, 149, 156, 161, 163, 167, 173, 197, 213;
%e A097757 Row 6: 90, 114, 134, 135, 138, 139, 147, 180, 181, 208, 212, 217, 228;
%e A097757 Row 7: 110, 142, 151, 154, 158, 169, 184, 189, 204, 224;
%e A097757 Row 8: 155, 159, 162, 165, 166, 182, 187, 196, 201, 202, 203, 216, 229, 233, 240, 252, 253;
%e A097757 Row 9: 126, 130, 146, 150, 171, 178, 179, 183, 185, 200, 209, 236, 237, 241, 288;
%e A097757 Row 10: 191, 205, 218, 232, 249, 257;
%e A097757 Row 11: 170, 175, 198, 207, 220, 221, 227, 245, 272, 277, 293;
%e A097757 Row 12: 186, 214, 225, 244, 248, 268, 297;
%e A097757 Row 13: 174, 199, 223, 256, 265, 292;
%e A097757 Row 14: 190, 194, 206, 215, 261, 269, 273, 281, 313, 317;
%e A097757 Row 15: 211, 219, 242, 262, 301;
%e A097757 Row 16: 195, 222, 239, 243, 276, 278, 289, 333;
%e A097757 Row 17: 226, 230, 238, 264, 266, 284;
%e A097757 Row 18: 210, 258, 263, 267, 285, 304, 308, 337;
%e A097757 Row 19: 231, 246, 254, 260, 357;
%e A097757 Row 20: 234, 247, 251, 282, 305, 309, 353;
%e A097757 Row 21: 235, 250, 280, 298, 321, 329;
%e A097757 Row 22: 332.
%e A097757 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
%o A097757 (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
%Y A097757 Cf. A001422, A003995, A033461, A097563, A097758, A097759, A097760.
%K A097757 nonn,tabf
%O A097757 0,1
%A A097757 _Ray Chandler_, Sep 06 2004