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A191347
Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.
3
1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 4, 2, 1, 0, 16, 17, 10, 8, 2, 1, 0, 32, 41, 28, 32, 9, 2, 1, 0, 64, 99, 76, 128, 38, 10, 2, 1, 0, 128, 239, 208, 512, 161, 44, 11, 2, 1, 0, 256, 577, 568, 2048, 682, 196, 50, 12, 3, 1
OFFSET
0,8
FORMULA
For each row n>=0 let T(n,0)=1 and T(n,1)=floor(sqrt(n)), then for each column k>=2: T(n,k)=T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 22 2019
T(n, k) = Sum_{i=0..floor((k+1)/2)} binomial(k, 2*i)*floor(sqrt(n))^(k-2*i)*n^i for n > 0, with T(0, 0) = 1 and T(0, k) = 0 for k > 0. - Michel Marcus, Aug 23 2019
EXAMPLE
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...
1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, ...
1, 1, 4, 10, 28, 76, 208, 568, 1552, 4240, 11584, ...
1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, ...
1, 2, 9, 38, 161, 682, 2889, 12238, 51841, 219602, 930249, ...
1, 2, 10, 44, 196, 872, 3880, 17264, 76816, 341792, 1520800, ...
1, 2, 11, 50, 233, 1082, 5027, 23354, 108497, 504050, 2341691, ...
1, 2, 12, 56, 272, 1312, 6336, 30592, 147712, 713216, 3443712, ...
1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, ...
1, 3, 19, 117, 721, 4443, 27379, 168717, 1039681, 6406803, 39480499, ...
1, 3, 20, 126, 796, 5028, 31760, 200616, 1267216, 8004528, 50561600, ...
1, 3, 21, 135, 873, 5643, 36477, 235791, 1524177, 9852435, 63687141, ...
1, 3, 22, 144, 952, 6288, 41536, 274368, 1812352, 11971584, 79078912, ...
1, 3, 23, 153, 1033, 6963, 46943, 316473, 2133553, 14383683, 96969863, ...
...
PROG
(PARI) T(n, k) = if (n==0, k==0, my(x=sqrtint(n)); sum(i=0, (k+1)\2, binomial(k, 2*i)*x^(k-2*i)*n^i));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Michel Marcus, Aug 22 2019
(PARI) T(n, k) = if (k==0, 1, if (k==1, sqrtint(n), T(n, k-2)*(n-T(n, 1)^2) + T(n, k-1)*T(n, 1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 22 2019
CROSSREFS
Row 1 is A000007, row 2 is A011782, row 3 is A001333, row 4 is A026150, row 5 is A081294, row 6 is A001077, row 7 is A084059, row 8 is A108851, row 9 is A084128, row 10 is A081341, row 11 is A005667, row 13 is A141041.
Row 3*2 is A002203, row 4*2 is A080040, row 5*2 is A155543, row 6*2 is A014448, row 8*2 is A080042, row 9*2 is A170931, row 11*2 is A085447.
Cf. A191348 which uses ceiling() in place of floor().
Sequence in context: A318686 A214546 A255704 * A106234 A238125 A062507
KEYWORD
nonn,tabl
AUTHOR
Charles L. Hohn, May 31 2011
STATUS
approved