OFFSET
0,1
COMMENTS
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
KEYWORD
nonn,tabl
AUTHOR
Charles L. Hohn, Aug 20 2019
STATUS
approved