cp's OEIS Frontend

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).

%I A309853 #23 Oct 22 2024 23:42:27
%S A309853 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 A309853 322,243,343,95,23,5,2,1,843,729,1475,433,110,25,7,2,1,2207,2187,6346,
%U A309853 1975,527,125,39,7,2,1,5778,6561,27305,9009,2525,625,238,41,7,2
%N 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).
%C A309853 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 A309853 2, 1,  1,   1,    1,     1,     1,      1,       1,        1, ...
%e A309853 2, 3,  7,  18,   47,   123,   322,    843,    2207,     5778, ...
%e A309853 2, 3,  9,  27,   81,   243,   729,   2187,    6561,    19683, ...
%e A309853 2, 5, 19,  80,  343,  1475,  6346,  27305,  117487,   505520, ...
%e A309853 2, 5, 21,  95,  433,  1975,  9009,  41095,  187457,   855095, ...
%e A309853 2, 5, 23, 110,  527,  2525, 12098,  57965,  277727,  1330670, ...
%e A309853 2, 5, 25, 125,  625,  3125, 15625,  78125,  390625,  1953125, ...
%e A309853 2, 7, 39, 238, 1471,  9107, 56394, 349223, 2162591, 13392022, ...
%e A309853 2, 7, 41, 259, 1649, 10507, 66953, 426643, 2718689, 17324251, ...
%e A309853 2, 7, 43, 280, 1831, 11977, 78346, 512491, 3352399, 21929320, ...
%e A309853 2, 7, 45, 301, 2017, 13517, 90585, 607061, 4068257, 27263677, ...
%e A309853 ...
%o A309853 (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 A309853 matrix(9, 9, n, k, T(n-1, k-1)) \\ Michel Marcus, Aug 22 2019
%Y A309853 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 A309853 Cf. A191347, A191348, and A309852.
%K A309853 nonn,tabl
%O A309853 0,1
%A A309853 _Charles L. Hohn_, Aug 20 2019
%E A309853 Revised orientation of n and k to customary T(n, k), by _Charles L. Hohn_, Sep 27 2024