Displaying 11-20 of 231 results found.
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a(n) is the least squared distance between 2 points of an n X n grid not occurring between two points of an (n-1) X (n-1) grid.
+0
2
1, 4, 9, 16, 26, 36, 49, 64, 81, 101, 121, 144, 173, 196, 226, 256, 293, 324, 361, 401, 441, 484, 529, 576, 626, 677, 729, 784, 842, 904, 961, 1024, 1089, 1172, 1226, 1296, 1373, 1444, 1522, 1601, 1697, 1764, 1849, 1936, 2026, 2116, 2209, 2304, 2401, 2504, 2602, 2708
Numbers that are the sum of two 4th powers in more than one way.
+0
10
635318657, 3262811042, 8657437697, 10165098512, 51460811217, 52204976672, 68899596497, 86409838577, 138519003152, 160961094577, 162641576192, 264287694402, 397074160625, 701252453457, 823372979472, 835279626752
Intersection of A001481 and A002479: N = a^2 + b^2 = c^2 + 2d^2 for some integers a,b,c,d.
+0
2
0, 1, 2, 4, 8, 9, 16, 17, 18, 25, 32, 34, 36, 41, 49, 50, 64, 68, 72, 73, 81, 82, 89, 97, 98, 100, 113, 121, 128, 136, 137, 144, 146, 153, 162, 164, 169, 178, 193, 194, 196, 200, 225, 226, 233, 241, 242, 256, 257, 272, 274, 281, 288, 289, 292, 306, 313, 324, 328
a(n) is the smallest number such that there are precisely n squares between a(n) and 2*a(n) inclusive.
+0
3
1, 8, 25, 61, 98, 162, 221, 288, 392, 481, 613, 722, 841, 1013, 1152, 1352, 1513, 1741, 1922, 2113, 2381, 2592, 2888, 3121, 3362, 3698, 3961, 4325, 4608, 5000, 5305, 5618, 6050, 6385, 6845, 7200, 7565, 8065, 8450, 8978, 9385, 9800, 10368, 10805, 11401, 11858
COMMENTS
If a(n) is even, then 2*a(n)-1 is not a square by considering mod 4. Then 2*a(n) must be square, so a(n) is itself twice a square. Next, if a(n) is odd, then 2*a(n) is not a square. So 2*a(n)-1 is a square, and then a(n) is a sum of consecutive squares. This also shows that a(n) is expressible as a sum of two squares, and so is a subsequence of A001481. - Elvar Wang Atlason, Jul 25 2021
Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).
(Formerly M3218)
+0
123
1, 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8, 0, 0, 0, 0, 12, 8, 0, 0, 8, 0, 0, 4, 0, 8, 0, 4, 8, 0, 0, 8, 8, 0, 0, 0, 8, 0, 0, 0, 4, 12, 0, 8, 8, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 4, 16, 0, 0, 8, 0, 0, 0, 4, 8, 8, 0, 0, 0, 0, 0, 8, 4, 8, 0, 0, 16, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 8, 4, 0, 12, 8
a(n) is the number of distinct rectangles with area n whose vertices lie on points of a unit square grid.
+0
1
1, 2, 1, 3, 2, 3, 1, 4, 2, 5, 1, 5, 2, 3, 3, 5, 2, 5, 1, 8, 2, 3, 1, 7, 3, 5, 2, 5, 2, 9, 1, 6, 2, 5, 3, 8, 2, 3, 3, 11, 2, 6, 1, 5, 5, 3, 1, 9, 2, 8, 3, 8, 2, 6, 3, 7, 2, 5, 1, 15, 2, 3, 3, 7, 5, 6, 1, 8, 2, 9, 1, 11, 2, 5, 5, 5, 2, 9, 1, 14, 3, 5, 1, 10, 5, 3
Numbers that are not the sum of a prime and a nonzero triangular number.
+0
1
1, 2, 7, 36, 61, 105, 171, 210, 211, 216, 325, 351, 406, 528, 561, 630, 741, 780, 990, 1081, 1176, 1275, 1596, 1711, 1830, 1953, 2016, 2145, 2346, 2628, 2775, 3003, 3081, 3240, 3321, 3655, 3741, 3916, 4278, 4371, 4465, 4560, 4851, 5253, 5460, 5565, 5886
CROSSREFS
Cf. A064233, A000217, A076768, A020756, A072386, A014089, A014090, A000404, A001481, A002654, A100570, A101181.
Number of ways of writing n as a sum of at most two nonzero squares, where order matters; also (number of divisors of n of form 4m+1) - (number of divisors of form 4m+3).
(Formerly M0012 N0001)
+0
105
1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 0, 2, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 3, 2, 0, 0, 2, 0
Triangle with T(n,k) = n^2 + k^2.
+0
7
0, 1, 2, 4, 5, 8, 9, 10, 13, 18, 16, 17, 20, 25, 32, 25, 26, 29, 34, 41, 50, 36, 37, 40, 45, 52, 61, 72, 49, 50, 53, 58, 65, 74, 85, 98, 64, 65, 68, 73, 80, 89, 100, 113, 128, 81, 82, 85, 90, 97, 106, 117, 130, 145, 162, 100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200
CROSSREFS
Cf. A001481 for terms in this sequence, A000161 for number of times each term appears, A048147 for square array.
Determinant of the matrix [Jacobi(i^2+5*i*j+5*j^2,2*n+1)]_{1<i,j<2*n}, where Jacobi(a,m) denotes the Jacobi symbol (a/m).
+0
1
0, 0, 0, 33, 0, 0, 0, -77539, 1811939328, -405798912, 0, 0, 649564705105200, -2787119627540625, 86463597248512, 0, 0, 0, 353143905335474188320, -66016543975248459410178048, 0, 23092056382629010556862857216, 0, 0, 0, 0, -5310136941067623723354761986048
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