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).

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

Author

Charles L. Hohn, Aug 20 2019

Keywords

• nonn
• tabl

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.

Examples

			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, ...
...
		

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.

Cf. A191347, A191348, and A309852.

Programs

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

Extensions

Revised orientation of n and k to customary T(n, k), by Charles L. Hohn, Sep 27 2024