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
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
Programs
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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
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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
Comments