Displaying 1-10 of 63 results found.
Values of n such that A006046(n)/n^theta, where theta=log(3)/log(2), is a local minimum, computed according to Harborth's recurrence.
+20
10
1, 3, 5, 11, 21, 43, 87, 173, 347, 693, 1387, 2775, 5549, 11099, 22197, 44395, 88789, 177579, 355159, 710317, 1420635, 2841269, 5682539, 11365079, 22730157, 45460315, 90920629, 181841259, 363682519, 727365037, 1454730075
Cumulative minima of A006046(n)/n^theta, where theta=log(3)/log(2), is a local minimum.
+20
8
1, 3, 5, 11, 21, 43, 87, 171, 173, 347, 693, 1387, 2775, 5547, 5549, 11099, 22197, 44395, 88789, 177579, 355159, 710315, 710317, 1420635, 2841269, 5682539, 11365079, 22730155, 22730157, 45460315, 90920629, 181841259, 363682519
1, 3, 7, 11, 19, 23, 31, 39, 55, 59, 67, 75, 91, 99, 115, 131, 163, 167, 175, 183, 199, 207, 223, 239, 271, 279, 295, 311, 343, 359, 391, 423, 487, 491, 499, 507, 523, 531, 547, 563, 595, 603, 619, 635, 667, 683, 715, 747, 811, 819, 835, 851, 883, 899, 931, 963
1, 5, 11, 37, 103, 317, 967, 2869, 8639, 25853, 77623, 232997, 698735, 2096461, 6288871, 18867125, 56600351, 169802077, 509408279, 1528220741, 4584666319, 13753990765, 41261980487, 123785957845, 371357840767, 1114073555069
a(n) = b(n+2) + b(n), where b(n) = A006046(n) is the sequence defined by b(0)=0, b(1)=1, b(n) = 2*b((n-1)/2) + b((n+1)/2) for n =3,5,7,... and b(n) = 3*b(n/2) for n =2,4,6,....
+20
1
3, 6, 12, 16, 24, 30, 42, 48, 60, 66, 78, 86, 102, 114, 138, 148, 168, 174, 186, 194, 210, 222, 246, 258, 282, 294, 318, 334, 366, 390, 438, 456, 492, 498, 510, 518, 534, 546, 570, 582, 606, 618, 642, 658, 690, 714, 762, 782, 822, 834, 858, 874, 906, 930, 978
0, 1, 4, 7, 14, 21, 30, 37, 52, 67, 84, 99, 120, 139, 160, 175, 206, 237, 270, 301, 338, 373, 410, 441, 486, 529, 574, 613, 662, 705, 750, 781, 844, 907, 972, 1035, 1104, 1171, 1240, 1303, 1380, 1455, 1532, 1603, 1684, 1759, 1836, 1899, 1992, 2083, 2176, 2263
a(0)=0; for n >= 1, a(n) = f(n) - 2*f(floor((n-1)/2)), where f(n) = A006046(n).
+20
1
0, 1, 3, 3, 7, 5, 9, 9, 17, 11, 15, 15, 23, 19, 27, 27, 43, 29, 33, 33, 41, 37, 45, 45, 61, 49, 57, 57, 73, 65, 81, 81, 113, 83, 87, 87, 95, 91, 99, 99, 115, 103, 111, 111, 127, 119, 135, 135, 167, 139, 147, 147, 163, 155, 171, 171, 203, 179, 195, 195, 227, 211, 243, 243, 307, 245, 249, 249, 257
EXAMPLE
The rows are a subset of the rows of Pascal's triangle A007318 read mod 2 (see A006943). The binary weights of these rows are given by A001316, whose partial sums are A006046, and the formula in the definition follows easily from this.
MAPLE
f:=proc(n) option remember; # A006046
Gould's sequence: a(n) = Sum_{k=0..n} (binomial(n,k) mod 2); number of odd entries in row n of Pascal's triangle ( A007318); a(n) = 2^ A000120(n).
(Formerly M0297 N0109)
+10
199
1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 4, 8, 8, 16, 8, 16, 16, 32, 8, 16, 16, 32, 16, 32, 32, 64, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 4, 8, 8, 16, 8, 16, 16, 32
Number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton.
+10
91
0, 1, 5, 9, 21, 25, 37, 49, 85, 89, 101, 113, 149, 161, 197, 233, 341, 345, 357, 369, 405, 417, 453, 489, 597, 609, 645, 681, 789, 825, 933, 1041, 1365, 1369, 1381, 1393, 1429, 1441, 1477, 1513, 1621, 1633, 1669, 1705, 1813, 1849, 1957, 2065, 2389, 2401, 2437, 2473
Number of entries in n-th row of Pascal's triangle not divisible by 3.
(Formerly M0422)
+10
29
1, 2, 3, 2, 4, 6, 3, 6, 9, 2, 4, 6, 4, 8, 12, 6, 12, 18, 3, 6, 9, 6, 12, 18, 9, 18, 27, 2, 4, 6, 4, 8, 12, 6, 12, 18, 4, 8, 12, 8, 16, 24, 12, 24, 36, 6, 12, 18, 12, 24, 36, 18, 36, 54, 3, 6, 9, 6, 12, 18, 9, 18, 27, 6, 12, 18, 12, 24, 36, 18, 36, 54, 9, 18, 27, 18, 36, 54, 27, 54
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