A036683 -id:A036683

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A036355

Fibonacci-Pascal triangle read by rows.

+0
26

1, 1, 1, 2, 2, 2, 3, 5, 5, 3, 5, 10, 14, 10, 5, 8, 20, 32, 32, 20, 8, 13, 38, 71, 84, 71, 38, 13, 21, 71, 149, 207, 207, 149, 71, 21, 34, 130, 304, 478, 556, 478, 304, 130, 34, 55, 235, 604, 1060, 1390, 1390, 1060, 604, 235, 55, 89, 420, 1177, 2272, 3310, 3736, 3310

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

T(n,k) is the number of lattice paths from (0,0) to (n-k,k) using steps (1,0),(2,0),(0,1),(0,2). - Joerg Arndt, Jun 30 2011, corrected by Greg Dresden, Aug 25 2020

For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 18 2013

For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

LINKS

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

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

FORMULA

T(n, m) = T'(n-1, m-1)+T'(n-2, m-2)+T'(n-1, m)+T'(n-2, m), where T'(n, m) = T(n, m) if 0<=m<=n and n >= 0 and T'(n, m)=0 otherwise. Initial term T(0, 0)=1.

G.f.: 1/(1-(1+y)*x-(1+y^2)*x^2). - Vladeta Jovovic, Oct 11 2003

EXAMPLE

Triangle begins

1, 1;

2, 2, 2;

3, 5, 5, 3;

5, 10, 14, 10, 5;

8, 20, 32, 32, 20, 8;

13, 38, 71, 84, 71, 38, 13;

21, 71, 149, 207, 207, 149, 71, 21;

34, 130, 304, 478, 556, 478, 304, 130, 34;

55, 235, 604, 1060, 1390, 1390, 1060, 604, 235, 55;

with indices

T(0,0);

T(1,0), T(1,1);

T(2,0), T(2,1), T(2,2);

T(3,0), T(3,1), T(3,2), T(3,3);

T(4,0), T(4,1), T(4,2), T(4,3), T(4,4);

For example, T(4,2) = 14 and there are 14 lattice paths from (0,0) to (4-2,2) = (2,2) using steps (1,0),(2,0),(0,1),(0,2). - Greg Dresden, Aug 25 2020

MATHEMATICA

nmax = 11; t[n_, m_] := t[n, m] = tp[n-1, m-1] + tp[n-2, m-2] + tp[n-1, m] + tp[n-2, m]; tp[n_, m_] /; 0 <= m <= n && n >= 0 := t[n, m]; tp[n_, m_] = 0; t[0, 0] = 1; Flatten[ Table[t[n, m], {n, 0, nmax}, {m, 0, n}]] (* Jean-François Alcover, Nov 09 2011, after formula *)

PROG

(PARI) /* same as in A092566 but use */

steps=[[1, 0], [2, 0], [0, 1], [0, 2]];

/* Joerg Arndt, Jun 30 2011 */

(Haskell)

a036355 n k = a036355_tabl !! n !! k

a036355_row n = a036355_tabl !! n

a036355_tabl = [1] : f [1] [1, 1] where

f us vs = vs : f vs (zipWith (+)

(zipWith (+) ([0, 0] ++ us) (us ++ [0, 0]))

(zipWith (+) ([0] ++ vs) (vs ++ [0])))

-- Reinhard Zumkeller, Apr 23 2013