A061451 -id:A061451 - cp's OEIS frontend

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

Christoph B. Kassir, Jun 21 2021

nonn

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)
...

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

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

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 *)

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

A345372 a(n) = Sum_{i=1..n} nac(i,n) where nac(i,n) is the n-th i-bonacci number. The n-th i-bonacci number here is equal to 1 for the first i terms, with subsequent terms equaling the sum of the previous n terms.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 60, 114, 217, 411, 780, 1481, 2820, 5379, 10288, 19720, 37884, 72924, 140640, 271695, 525698, 1018611, 1976276, 3838889, 7465191, 14531683, 28313776, 55214993, 107762464, 210477611, 411387724, 804609206, 1574671586, 3083549861, 6041628460

Offset: 1

Christoph B. Kassir, Jun 16 2021

nonn

a(n) is the sum of the first n elements of the n-th column of the following array:
  1, 1, 1, 1, 1, ... (1-bonacci numbers)
  1, 1, 2, 3, 5, ... (2-bonacci or Fibonacci numbers)
  1, 1, 1, 3, 5, ... (3-bonacci or tribonacci numbers)
  1, 1, 1, 1, 4, ... (4-bonacci or tetranacci numbers)
  ...
For n >= 3, this sequence is 2 + antidiagonal sums of A061451.

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

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, n), i=1..n):
seq(a(n), n=1..36);  # Alois P. Heinz, Jun 16 2021

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, n], {i, 1, n}];
Table[a[n], {n, 1, 36}] (* Jean-François Alcover, May 29 2022, after Alois P. Heinz *)

a(n) = Sum_{i=1..n} nac(i,n) where nac(i,n) = 1 if 1 <= n <= i, Sum_{k=1..i} nac(i,n-k) if n > i.

cp's OEIS Frontend

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).

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