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A309852
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 = floor(sqrt(x)) and z = y-1+(y mod 2).
1
2, 1, 2, 1, 1, 2, 1, 3, 3, 2, 1, 4, 9, 3, 2, 1, 7, 27, 11, 3, 2, 1, 11, 81, 36, 13, 3, 2, 1, 18, 243, 119, 45, 15, 5, 2, 1, 29, 729, 393, 161, 54, 25, 5, 2, 1, 47, 2187, 1298, 573, 207, 125, 27, 5, 2, 1, 76, 6561, 4287, 2041, 783, 625, 140, 29, 5, 2
OFFSET
0,1
COMMENTS
One of 4 related arrays (the others being A191347, A191348, and A309853) where the two halves of the main formula approach the integers shown and 0 respectively, and also with A309853 where rows represent various Fibonacci series a(n) = a(n-2)*b + a(n-1)*c where b and c are integers >= 0.
FORMULA
For each row n>=0 let x = 4*n+1, y = floor(sqrt(x)), T(n,0)=2, and T(n,1)=y-1+(y % 2)), then for each column k>=2: T(n, k-2)*((x-T(n, 1)^2)/4) + T(n, k-1)*T(n, 1). - Charles L. Hohn, Aug 23 2019
EXAMPLE
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...
2, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, ...
2, 3, 11, 36, 119, 393, 1298, 4287, 14159, 46764, 154451, ...
2, 3, 13, 45, 161, 573, 2041, 7269, 25889, 92205, 328393, ...
2, 3, 15, 54, 207, 783, 2970, 11259, 42687, 161838, 613575, ...
2, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, ...
2, 5, 27, 140, 727, 3775, 19602, 101785, 528527, 2744420, 14250627, ...
2, 5, 29, 155, 833, 4475, 24041, 129155, 693857, 3727595, 20025689, ...
2, 5, 31, 170, 943, 5225, 28954, 160445, 889087, 4926770, 27301111, ...
2, 5, 33, 185, 1057, 6025, 34353, 195865, 1116737, 6367145, 36302673, ...
...
PROG
(PARI) T(n, k) = my(x = 4*n+1, y = sqrtint(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
(PARI) T(n, k) = my(x = 4*n+1, y = sqrtint(x), z=y-1+(y % 2)); v=if(k==0, 2, k==1, z, mapget(m2, n)*((x-z^2)/4) + mapget(m1, n)*z); mapput(m2, n, if(mapisdefined(m1, n), mapget(m1, n), 0)); mapput(m1, n, v); v;
m1=Map(); m2=Map(); matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 26 2019
CROSSREFS
Row 2 is A000032, row 3 (except the first term) is A000244, row 4 is A006497, row 5 is A206776, row 6 is A172012, row 7 (except the first term) is A000351, row 8 is A087130.
Sequence in context: A218828 A075117 A279387 * A029810 A321601 A366461
KEYWORD
nonn,tabl
AUTHOR
Charles L. Hohn, Aug 20 2019
STATUS
approved