A354267 A Fibonacci-Pascal triangle read by rows: T(n, n) = 1, T(n, n-1) = n - 1, T(n, 0) = T(n-1, 1) and T(n, k) = T(n-1, k-1) + T(n-1, k) for 0 < k < n-1.

1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 4, 3, 1, 3, 5, 7, 7, 4, 1, 5, 8, 12, 14, 11, 5, 1, 8, 13, 20, 26, 25, 16, 6, 1, 13, 21, 33, 46, 51, 41, 22, 7, 1, 21, 34, 54, 79, 97, 92, 63, 29, 8, 1, 34, 55, 88, 133, 176, 189, 155, 92, 37, 9, 1, 55, 89, 143, 221, 309, 365, 344, 247, 129, 46, 10, 1

Offset

0,8

Links

Table of n, a(n) for n=0..77.

Index entries for triangles and arrays related to Pascal's triangle

Formula

T(n, 0) = Fibonacci(n - 1).

Example

[0]  1;
[1]  0,  1;
[2]  1,  1,  1;
[3]  1,  2,  2,  1;
[4]  2,  3,  4,  3,  1;
[5]  3,  5,  7,  7,  4,  1;
[6]  5,  8, 12, 14, 11,  5,  1;
[7]  8, 13, 20, 26, 25, 16,  6,  1;
[8] 13, 21, 33, 46, 51, 41, 22,  7, 1;
[9] 21, 34, 54, 79, 97, 92, 63, 29, 8, 1;

Maple

T := proc(n, k) option remember;
if n = k then 1 elif k = n-1 then n-1 elif k = 0 then T(n-1, 1) else
T(n-1, k) + T(n-1, k-1) fi end: seq(seq(T(n, k), k = 0..n), n = 0..11);

Mathematica

T[n_, k_] := Which[n == k, 1, k == n-1, n-1, k == 0, T[n-1, 1], True, T[n-1, k] + T[n-1, k-1]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023 *)

Prog

(Python) from functools import cache
@cache
def A354267row(n):
    if n == 0: return [1]
    if n == 1: return [0, 1]
    row = A354267row(n - 1) + [1]
    s = row[1]
    for k in range(n-1, 0, -1):
        row[k] += row[k - 1]
    row[0] = s
    return row
for n in range(10): print(A354267row(n))

Crossrefs

Cf. A212804 (first column, which is also row 0 of A352744), A099036 (row sums), A228074 (subtriangle), A000045 (Fibonacci).

Sequence in context: A352889 A230507 A352888 * A366804 A259910 A072549

Adjacent sequences: A354264 A354265 A354266 * A354268 A354269 A354270

Keyword

nonn,tabl

Author

Peter Luschny, May 31 2022

Status

approved