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A309853
Array read by antidiagonals: ((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k for columns k >= 0 and rows n >= 0, where x = 4*n+1 and y = ceiling(sqrt(x)) and z = y+1-(y mod 2).
1
2, 1, 2, 1, 3, 2, 1, 7, 3, 2, 1, 18, 9, 5, 2, 1, 47, 27, 19, 5, 2, 1, 123, 81, 80, 21, 5, 2, 1, 322, 243, 343, 95, 23, 5, 2, 1, 843, 729, 1475, 433, 110, 25, 7, 2, 1, 2207, 2187, 6346, 1975, 527, 125, 39, 7, 2, 1, 5778, 6561, 27305, 9009, 2525, 625, 238, 41, 7, 2
OFFSET
0,1
COMMENTS
One of 4 related arrays (the others being A191347, A191348, and A309852) where the two halves of the main formula approach the integers shown and 0 respectively, and also with A309852 where rows represent various Fibonacci series a(n) = a(n-2)*b + a(n-1)*c where b and c are integers >= 0.
EXAMPLE
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 3, 7, 18, 47, 123, 322, 843, 2207, 5778, ...
2, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, ...
2, 5, 19, 80, 343, 1475, 6346, 27305, 117487, 505520, ...
2, 5, 21, 95, 433, 1975, 9009, 41095, 187457, 855095, ...
2, 5, 23, 110, 527, 2525, 12098, 57965, 277727, 1330670, ...
2, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, ...
2, 7, 39, 238, 1471, 9107, 56394, 349223, 2162591, 13392022, ...
2, 7, 41, 259, 1649, 10507, 66953, 426643, 2718689, 17324251, ...
2, 7, 43, 280, 1831, 11977, 78346, 512491, 3352399, 21929320, ...
2, 7, 45, 301, 2017, 13517, 90585, 607061, 4068257, 27263677, ...
...
PROG
(PARI) T(n, k) = my(x = 4*n+1, y = ceil(sqrt(x)), z = y+1-(y % 2)); round(((z+sqrt(x))/2)^k + ((z-sqrt(x))/2)^k);
matrix(9, 9, n, k, T(n-1, k-1)) \\ Michel Marcus, Aug 22 2019
CROSSREFS
Row 2 is A005248, row 3 (except the first term) is A000244, row 4 is A228569, row 5 is A159289, row 6 is A003501, row 7 (except the first term) is A000351.
Sequence in context: A375577 A372704 A203301 * A107456 A334864 A165112
KEYWORD
nonn,tabl
AUTHOR
Charles L. Hohn, Aug 20 2019
EXTENSIONS
Revised orientation of n and k to customary T(n, k), by Charles L. Hohn, Sep 27 2024
STATUS
approved