A036692 -id:A036692 - cp's OEIS frontend

A036355 Fibonacci-Pascal triangle read by rows.

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, 2272, 1177, 420, 89

Offset: 0

Floor van Lamoen, Dec 28 1998

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

			Triangle begins
   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;
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

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Row sums form sequence A002605. T(n, 0) forms the Fibonacci sequence (A000045). T(n, 1) forms sequence A001629.
Derived sequences: A036681, A036682, A036683, A036684, A036692 (central terms).
Cf. A007318, A051159, A091533, A228196, A228576.
Some other Fibonacci-Pascal triangles: A027926, A037027, A074829, A105809, A109906, A111006, A114197, A162741, A228074.

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

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 *)

/* same as in A092566 but use */
steps=[[1,0], [2,0], [0,1], [0,2]];
/* Joerg Arndt, Jun 30 2011 */

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

A122680 Number of possible basketball games that end in a tie (n:n). Equivalently, the number of walks on the square lattice from (0,0) to (n,n) where the allowed steps are {(1,0),(2,0),(3,0), (0,1),(0,2),(0,3)}.

Original entry on oeis.org

1, 2, 14, 106, 784, 6040, 47332, 375196, 3001966, 24190148, 196034522, 1596030740, 13044459766, 106961525744, 879512777006, 7249483605580, 59881171431050, 495545064567260, 4107666916668414, 34099685718629264, 283454909832384416, 2359069189033880228

Offset: 0

Doron Zeilberger, Oct 20 2006

Diagonal of the rational function 1 / (1 - x - y - x^2 - y^2 - x^3 - y^3). - Ilya Gutkovskiy, Apr 23 2025

			a(1) = 2 because the number of ways of getting the score to be (1,1) is
(0,0) -> (1,0) -> (1,1),
(0,0) -> (0,1) -> (1,1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Moa Apagodu and Doron Zeilberger, FIVE Applications of Wilf-Zeilberger Theory to Enumeration and Probability; Local copy [Pdf file only, no active links]
M. Apagodu and D. Zeilberger, Maple package to generate recurrence; Local copy
M. Apagodu and D. Zeilberger, 6th order recurrence; Local copy

b:= proc(x, y) option remember; `if`(x<0 or y<0, 0,
      `if`([x, y]=[0, 0], 1, add(b(x-i, y) +b(x, y-i), i=1..3)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 24 2011

b[x_, y_] := b[x, y] = If[x < 0 || y < 0, 0, If[{x, y} == {0, 0}, 1, Sum[b[x - i, y] + b[x, y - i], {i, 3}]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, May 13 2020, after Alois P. Heinz *)

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A036355 Fibonacci-Pascal triangle read by rows.

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