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approved

editing

proposed

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

proposed

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editing

proposed

Array read by antidiagonals: ((z+sqrt(x))/2)^n k + ((z-sqrt(x))/2)^n k for columns n k >= 0 and rows k n >= 0, where x = 4*kn+1 and y = ceiling(sqrt(x)) and z = y+1-(y mod 2).

(PARI) T(k, n, k) = my(x = 4*kn+1, y = ceil(sqrt(x)), z = y+1-(y % 2)); round(((z+sqrt(x))/2)^n k + ((z-sqrt(x))/2)^nk);

matrix(9, 9, k, n, k, T(kn-1, n k-1)) \\ _Michel Marcus_, , Aug 22 2019

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

approved

editing

proposed

approved

editing

proposed

2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

2, 3, 7, 18, 47, 123, 322, 843, 2207, 5778, 15127, ...

2, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, ...

2, 5, 19, 80, 343, 1475, 6346, 27305, 117487, 505520, 2175139, ...

2, 5, 21, 95, 433, 1975, 9009, 41095, 187457, 855095, 3900561, ...

2, 5, 23, 110, 527, 2525, 12098, 57965, 277727, 1330670, 6375623, ...

2, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, ...

2, 7, 39, 238, 1471, 9107, 56394, 349223, 2162591, 13392022, 82931199, ...

2, 7, 41, 259, 1649, 10507, 66953, 426643, 2718689, 17324251, 110395001, ...

2, 7, 43, 280, 1831, 11977, 78346, 512491, 3352399, 21929320, 143448043, ...

2, 7, 45, 301, 2017, 13517, 90585, 607061, 4068257, 27263677, 182709225, ...

proposed

editing

editing

proposed

(PARI) T(k, n) = my(x = 4*k+1, y = ceil(sqrt(x)), z = y+1-(y % 2)); round(((z+sqrt(x))/2)^n + ((z-sqrt(x))/2)^n);

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

proposed

editing