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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).

%I A345669 #18 Dec 27 2022 10:58:51
%S A345669 1,2,3,5,7,12,18,31,51,89,153,273,483,870,1571,2860,5225,9603,17711,
%T A345669 32805,60967,113685,212610,398723,749615,1412585,2667549,5047345,
%U A345669 9567527,18166272,34546857,65793343,125471295,239584610,458028439,876628109,1679581899
%N 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).
%C A345669 Antidiagonal sum of below array:
%C A345669 1, 1, 1, 1, 1, 1, ... (1-bonacci numbers)
%C A345669 1, 1, 2, 3, 5, 8, ... (2-bonacci or Fibonacci numbers)
%C A345669 1, 1, 1, 3, 5, 9, ... (3-bonacci or tribonacci numbers)
%C A345669 1, 1, 1, 1, 4, 7, ... (4-bonacci or tetranacci numbers)
%C A345669 ...
%F A345669 a(n) = Sum_{i=1..n} of nac(i,n-i+1) = Sum_{i=1..n} of nac(n-i+1,i).
%p A345669 b:= proc(i, n) option remember; `if`(n=0, 0,
%p A345669       `if`(n<=i, 1, add(b(i, n-j), j=1..i)))
%p A345669     end:
%p A345669 a:= n-> add(b(i+1, n-i), i=0..n):
%p A345669 seq(a(n), n=1..37);  # _Alois P. Heinz_, Jun 21 2021
%t A345669 b[i_, n_] := b[i, n] = If[n == 0, 0, If[n <= i, 1, Sum[b[i, n - j], {j, 1, i}]]];
%t A345669 a[n_] := Sum[b[i + 1, n - i], {i, 0, n}];
%t A345669 Table[a[n], {n, 1, 37}] (* _Jean-François Alcover_, Dec 27 2022, after _Alois P. Heinz_ *)
%Y A345669 Cf. A000045, A000213, A000288, A000322, A000383, A060455, A061451.
%K A345669 nonn
%O A345669 1,2
%A A345669 _Christoph B. Kassir_, Jun 21 2021