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

Author

Charles L. Hohn, Aug 20 2019

Keywords

• nonn
• tabl

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.

Examples

			Array begins:
  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, ...
  ...
		

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.

Programs

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

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