A309853 Array read by antidiagonals: ((z+sqrt(x))/2)^n + ((z-sqrt(x))/2)^n for columns n >= 0 and rows k >= 0, where x = 4*k+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

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.

Links

Table of n, a(n) for n=0..64.

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(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

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.

Sequence in context: A205696 A029635 A203301 * A107456 A334864 A165112

Adjacent sequences: A309850 A309851 A309852 * A309854 A309855 A309856

Keyword

nonn,tabl

Author

Charles L. Hohn, Aug 20 2019

Status

approved