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A036355
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Fibonacci-Pascal triangle read by rows.
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26
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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
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) /* same as in A092566 but use */
steps=[[1, 0], [2, 0], [0, 1], [0, 2]];
(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])))
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CROSSREFS
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Row sums form sequence A002605. T(n, 0) forms the Fibonacci sequence (A000045). T(n, 1) forms sequence A001629.
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KEYWORD
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AUTHOR
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STATUS
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approved
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