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%I #48 Aug 26 2020 02:17:40
%S 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,
%T 13,21,71,149,207,207,149,71,21,34,130,304,478,556,478,304,130,34,55,
%U 235,604,1060,1390,1390,1060,604,235,55,89,420,1177,2272,3310,3736,3310
%N Fibonacci-Pascal triangle read by rows.
%C 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
%C For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - _Boris Putievskiy_, Aug 18 2013
%C For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 09 2013
%H Reinhard Zumkeller, <a href="/A036355/b036355.txt">Rows n = 0..120 of triangle, flattened</a>
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F 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.
%F G.f.: 1/(1-(1+y)*x-(1+y^2)*x^2). - _Vladeta Jovovic_, Oct 11 2003
%e Triangle begins
%e 1;
%e 1, 1;
%e 2, 2, 2;
%e 3, 5, 5, 3;
%e 5, 10, 14, 10, 5;
%e 8, 20, 32, 32, 20, 8;
%e 13, 38, 71, 84, 71, 38, 13;
%e 21, 71, 149, 207, 207, 149, 71, 21;
%e 34, 130, 304, 478, 556, 478, 304, 130, 34;
%e 55, 235, 604, 1060, 1390, 1390, 1060, 604, 235, 55;
%e with indices
%e T(0,0);
%e T(1,0), T(1,1);
%e T(2,0), T(2,1), T(2,2);
%e T(3,0), T(3,1), T(3,2), T(3,3);
%e T(4,0), T(4,1), T(4,2), T(4,3), T(4,4);
%e 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
%t 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 *)
%o (PARI) /* same as in A092566 but use */
%o steps=[[1,0], [2,0], [0,1], [0,2]];
%o /* _Joerg Arndt_, Jun 30 2011 */
%o (Haskell)
%o a036355 n k = a036355_tabl !! n !! k
%o a036355_row n = a036355_tabl !! n
%o a036355_tabl = [1] : f [1] [1,1] where
%o f us vs = vs : f vs (zipWith (+)
%o (zipWith (+) ([0,0] ++ us) (us ++ [0,0]))
%o (zipWith (+) ([0] ++ vs) (vs ++ [0])))
%o -- _Reinhard Zumkeller_, Apr 23 2013
%Y Row sums form sequence A002605. T(n, 0) forms the Fibonacci sequence (A000045). T(n, 1) forms sequence A001629.
%Y Derived sequences: A036681, A036682, A036683, A036684, A036692 (central terms).
%Y Cf. A007318, A051159, A228196, A228576.
%Y Some other Fibonacci-Pascal triangles: A027926, A037027, A074829, A105809, A109906, A111006, A114197, A162741, A228074.
%K nonn,tabl,easy,nice
%O 0,4
%A _Floor van Lamoen_, Dec 28 1998
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