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

1,2

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)

...

Links

Table of n, a(n) for n=1..37.

Formula

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

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 *)

Crossrefs

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

Sequence in context: A137713 A326490 A191385 * A335093 A143642 A192685

Adjacent sequences: A345666 A345667 A345668 * A345670 A345671 A345672

Keyword

nonn

Author

Christoph B. Kassir, Jun 21 2021

Status

approved