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).

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.

Links

Table of n, a(n) for n=0..65.

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.

Cf. A191347, A191348, and A309853.

Sequence in context: A218828 A075117 A279387 * A029810 A321601 A366461

Adjacent sequences: A309849 A309850 A309851 * A309853 A309854 A309855

Keyword

nonn,tabl

Author

Charles L. Hohn, Aug 20 2019

Status

approved