A377290
For each row n in array A374602(n, k), the period size, as a count of terms, that divides the row into congruent subsequences.
Original entry on oeis.org
1, 2, 2, 6, 4, 4, 14, 10, 8, 10, 6, 8, 14, 16, 34, 10, 8, 34, 8, 12, 22, 22, 32, 18, 18, 30, 14, 18, 16, 12, 38, 22, 28, 26, 42, 20, 74, 36, 14, 54, 12, 16, 34, 38, 54, 26, 58, 50, 24, 36, 102, 46, 32, 78, 14, 22, 38, 46, 118, 22, 30, 68, 36, 32, 130, 74, 34
Offset: 1
Given formula sqrt((d-c)*b^2 + c*(b+1)^2) from A374602, for n=5, the first few terms of A374602(5, k) are:
sqrt((7-3)*1^2 + 3*(1+1)^2) = 4,
sqrt((7-6)*1^2 + 6*(1+1)^2) = 5,
sqrt((7-1)*4^2 + 1*(4+1)^2) = 11,
sqrt((7-4)*10^2 + 4*(10+1)^2) = 28,
sqrt((7-3)*23^2 + 3*(23+1)^2) = 62,
sqrt((7-6)*29^2 + 6*(29+1)^2) = 79,
sqrt((7-1)*66^2 + 1*(66+1)^2) = 175,
sqrt((7-4)*168^2 + 4*(168+1)^2) = 446,
producing the repeating pattern of c values {3, 6, 1, 4}, of length 4 -> a(5).
A377291
For each row n in array A374602(n, k), the asymptotic geometric growth factor of every A377290(n) terms, represented by its nearest integer.
Original entry on oeis.org
6, 14, 7, 98, 16, 34, 1442, 398, 194, 119, 30, 62, 4354, 1154, 115598, 322, 23, 155234, 48, 98, 10402, 2702, 64514, 727, 482, 3040, 1154, 2114, 70, 142, 21314, 5474, 2498, 1442, 16793602, 674, 48497294, 158402, 47, 48670, 96, 194, 39202, 9998, 1684802, 2599
Offset: 1
For n = 5, the first few terms of A374602(5, k) are {4, 5, 11, 28, 62, 79, 175, 446, 988} and the period size is A377290(5) = 4, giving A374602(5, 1+4)/A374602(5, 1) = 62/4 = 15.5, 79/5 = 15.8, 175/11 = 15.909..., 446/28 = 15.928..., 988/62 = 15.935..., ..., to limit 15.937... -> g(5), from which g(5)+(1/g(5)) = 16 -> a(5).
A375336
For n>=4, irregular triangular array of successive integer solutions to sqrt((d-c)*b^2 + c*(b+1)^2) for square integers d = n^2, where b and c are positive integers and c < d, read by rows.
Original entry on oeis.org
5, 7, 7, 8, 9, 13, 17, 27, 8, 10, 11, 13, 16, 19, 10, 11, 13, 14, 19, 21, 25, 31, 59, 61, 12, 15, 22, 23, 29, 34, 39, 42, 11, 13, 14, 16, 17, 19, 25, 33, 37, 41, 49, 103, 107, 125, 13, 14, 16, 17, 19, 20, 23, 27, 28, 32, 37, 40, 46, 53, 82, 83, 15, 18, 21, 26
Offset: 4
4: {5, 7}
5: {7, 8}
6: {9, 13, 17, 27}
7: {8, 10, 11, 13, 16, 19}
8: {10, 11, 13, 14, 19, 21, 25, 31, 59, 61}
9: {12, 15, 22, 23, 29, 34, 39, 42}
10: {11, 13, 14, 16, 17, 19, 25, 33, 37, 41, 49, 103, 107, 125}
11: {13, 14, 16, 17, 19, 20, 23, 27, 28, 32, 37, 40, 46, 53, 82, 83}
12: {15, 18, 21, 26, 29, 31, 34, 41, 43, 51, 54, 57, 61, 71, 159, 165, 209, 211}
...
sqrt((2^2-1)*1^2 + 1*(1+1)^2) = sqrt(7) -> not an integer so not included.
sqrt((4^2-1)*1^2 + 1*(1+1)^2) = sqrt(19) -> not an integer so not included.
sqrt((4^2-3)*1^2 + 3*(1+1)^2) = 5 -> T(4,1).
sqrt((4^2-11)*1^2 + 11*(1+1)^2) = 7 -> T(4,2).
sqrt((5^2-8)*1^2 + 8*(1+1)^2) = 7 -> T(5,1).
sqrt((6^2-5)*2^2 + 5*(2+1)^2) = 13 -> T(6,2).
Showing 1-3 of 3 results.
Comments