%I #23 Oct 22 2024 23:42:27

%S 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,

%T 322,243,343,95,23,5,2,1,843,729,1475,433,110,25,7,2,1,2207,2187,6346,

%U 1975,527,125,39,7,2,1,5778,6561,27305,9009,2525,625,238,41,7,2

%N 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).

%C 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.

%e 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 2, 3, 7, 18, 47, 123, 322, 843, 2207, 5778, ...

%e 2, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, ...

%e 2, 5, 19, 80, 343, 1475, 6346, 27305, 117487, 505520, ...

%e 2, 5, 21, 95, 433, 1975, 9009, 41095, 187457, 855095, ...

%e 2, 5, 23, 110, 527, 2525, 12098, 57965, 277727, 1330670, ...

%e 2, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, ...

%e 2, 7, 39, 238, 1471, 9107, 56394, 349223, 2162591, 13392022, ...

%e 2, 7, 41, 259, 1649, 10507, 66953, 426643, 2718689, 17324251, ...

%e 2, 7, 43, 280, 1831, 11977, 78346, 512491, 3352399, 21929320, ...

%e 2, 7, 45, 301, 2017, 13517, 90585, 607061, 4068257, 27263677, ...

%e ...

%o (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);

%o matrix(9, 9, n, k, T(n-1, k-1)) \\ Michel Marcus, Aug 22 2019

%Y 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.

%Y Cf. A191347, A191348, and A309852.

%K nonn,tabl

%O 0,1

%A _Charles L. Hohn_, Aug 20 2019

%E Revised orientation of n and k to customary T(n, k), by _Charles L. Hohn_, Sep 27 2024