OFFSET
0,12
LINKS
G. C. Greubel, Antidiagonals n = 0..100, flattened
FORMULA
T(n, k) = ( ((1+sqrt(1+4*n))/2)^k - ((1-sqrt(1+4*n))/2)^k )/sqrt(1+4*n).
EXAMPLE
Square array begins as:
0, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 2, 3, 5, 8, ...
0, 1, 1, 3, 5, 11, 21, ...
0, 1, 1, 4, 7, 19, 40, ...
0, 1, 1, 5, 9, 29, 65, ...
0, 1, 1, 6, 11, 41, 96, ...
MAPLE
seq(seq( round((((1+sqrt(1+4*k))/2)^(n-k) - ((1-sqrt(1+4*k))/2)^(n-k) )/sqrt(1+4*k)), k=0..n), n=0..12); # G. C. Greubel, Jan 15 2020
MATHEMATICA
T[n_, k_]:= If[n==k==0, 0, Round[(((1+Sqrt[1+4n])/2)^k - ((1-Sqrt[1+4n])/2)^k)/Sqrt[1+4n]]]; Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 15 2020 *)
PROG
(PARI) T(n, k) = ( ((1+sqrt(1+4*n))/2)^k - ((1-sqrt(1+4*n))/2)^k )/sqrt(1+4*n);
for(n=0, 12, for(k=0, n, print1( round(T(k, n-k)), ", "))) \\ G. C. Greubel, Jan 15 2020
(Magma) [Round( (((1+Sqrt(1+4*k))/2)^(n-k) - ((1-Sqrt(1+4*k))/2)^(n-k) )/Sqrt(1+4*k) ): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 15 2020
(Sage) [[ round( (((1+sqrt(1+4*k))/2)^(n-k) - ((1-sqrt(1+4*k))/2)^(n-k) )/sqrt(1+4*k) ) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 15 2020
(GAP) Flat(List([0..12], n-> List([0..n], k-> (((1+Sqrt(1+4*k))/2)^(n-k) - ((1-Sqrt(1+4*k))/2)^(n-k))/Sqrt(1+4*k) ))); # G. C. Greubel, Jan 15 2020
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, May 10 2001
STATUS
approved