A345669 Antidiagonal sums of array containing i-bonacci sequences nac(i,n), where nac(i,n) is the n-th i-bonacci number with nac(i,1..i) = 1 (see comments).

1, 2, 3, 5, 7, 12, 18, 31, 51, 89, 153, 273, 483, 870, 1571, 2860, 5225, 9603, 17711, 32805, 60967, 113685, 212610, 398723, 749615, 1412585, 2667549, 5047345, 9567527, 18166272, 34546857, 65793343, 125471295, 239584610, 458028439, 876628109, 1679581899

Offset: 1

Author

Christoph B. Kassir, Jun 21 2021

Keywords

• nonn

Comments

Antidiagonal sum of below array:

1, 1, 1, 1, 1, 1, ... (1-bonacci numbers)

1, 1, 2, 3, 5, 8, ... (2-bonacci or Fibonacci numbers)

1, 1, 1, 3, 5, 9, ... (3-bonacci or tribonacci numbers)

1, 1, 1, 1, 4, 7, ... (4-bonacci or tetranacci numbers)

...

Crossrefs

Cf. A000045, A000213, A000288, A000322, A000383, A060455, A061451.

Programs

Maple

b:= proc(i, n) option remember; `if`(n=0, 0,
      `if`(n<=i, 1, add(b(i, n-j), j=1..i)))
    end:
a:= n-> add(b(i+1, n-i), i=0..n):
seq(a(n), n=1..37);  # Alois P. Heinz, Jun 21 2021

Mathematica

b[i_, n_] := b[i, n] = If[n == 0, 0, If[n <= i, 1, Sum[b[i, n - j], {j, 1, i}]]];
a[n_] := Sum[b[i + 1, n - i], {i, 0, n}];
Table[a[n], {n, 1, 37}] (* Jean-François Alcover, Dec 27 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{i=1..n} of nac(i,n-i+1) = Sum_{i=1..n} of nac(n-i+1,i).