OFFSET
1,2
COMMENTS
LINKS
Clark Kimberling, Antidiagonals n = 1..60
FORMULA
T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = n + x - (n - 1)*x and g(x) = (1 - x - x^2)*(1 - x)^2.
T(n, k) = Fibonacci(k+4) + n*Fibonacci(k+3) - 2*(n+k) - 3. - G. C. Greubel, Jul 08 2019
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1...4....10...21...40....72
2...7....16...32...59....104
3...10...22...43...78....136
4...13...28...54...97....168
5...16...34...65...116...200
6...19...40...76...135...232
MATHEMATICA
(* First program *)
b[n_]:= Fibonacci[n+1]; c[n_]:= n;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213584 *)
r[n_]:= Table[T[n, k], {k, 40}] (* columns of antidiagonal triangle *)
d = Table[T[n, n], {n, 1, 40}] (* A213585 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213586 *)
(* Second program *)
Table[Fibonacci[n-k+5] + k*Fibonacci[n-k+4] -2*n-5, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)
PROG
(PARI) t(n, k) = fibonacci(n-k+5) + k*fibonacci(n-k+4) -(2*n+5);
for(n=1, 12, for(k=1, n, print1(t(n, k), ", "))) \\ G. C. Greubel, Jul 08 2019
(Magma) [[Fibonacci(n-k+5) + k*Fibonacci(n-k+4) -(2*n+5): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019
(Sage) [[fibonacci(n-k+5) + k*fibonacci(n-k+4) -(2*n+5) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019
(GAP) Flat(List([1..12], n-> List([1..n], k-> Fibonacci(n-k+5) + k*Fibonacci(n-k+4) -(2*n+5)))) # G. C. Greubel, Jul 08 2019
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jun 18 2012
STATUS
approved