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 A296619 The number of nonnegative walks of n steps with step sizes 1 and 2, starting at 0 and ending at 2. 3
 0, 1, 1, 6, 13, 52, 152, 550, 1813, 6453, 22427, 80330, 286895, 1038931, 3772801, 13807294, 50726893, 187332517, 694364517, 2583714636, 9644852364, 36115537269, 135607526865, 510496492338, 1926284451923, 7284476707597, 27602839227883, 104791979218326 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS a(n) is the number of 2-D walks with n steps of type {(1,-2), (1,-1), (1,1), or (1,2)} starting at (0,0), ending at (n,2), and not dropping below the x-axis. The sequence corresponds to element (1,3) of the matrix B(n)^n (see Maple script). Furthermore, element (1,1) of the matrix is A187430, the element (1,2) of these matrix is A055113. LINKS Robert Israel, Table of n, a(n) for n = 0..1667 FORMULA a(n) = A185286(n,2). - Robert Israel, Dec 19 2017 EXAMPLE There are 6 walks of length 3:         __        |  |         __      __|  |_     __|  |_     __    _     |           |           |  |__|    _|          _|          _|     2+2-2=2     2+1-1=2     2-1+1=2                     __      __    _       |  |_           _     |  |  |      __|         __   |    _|  |__|    _|          _|  |__|     2-2+2=2     1+2-1=2     1-1+2=2 MAPLE B := n -> LinearAlgebra:-ToeplitzMatrix([0, 1, 1, seq(0, k=0..n-2)], symmetric): seq((B(n)^n)(1, 3), n=0..27); # alternative: T:= proc(n, k) option remember;   if k < 0 or k > 2*n then return 0 fi;   procname(n-1, k-2)+procname(n-1, k-1)+procname(n-1, k+1)+procname(n-1, k+2) end proc: T(0, 0):= 1: seq(T(n, 2), n=0..40); # Robert Israel, Dec 19 2017 MATHEMATICA b[n_] := ToeplitzMatrix[Join[{0, 1, 1}, ConstantArray[0, n-1]]]; Prepend[Table[MatrixPower[b[n], n][[1, 3]], {n, 20}], 0] (* Andrey Zabolotskiy, Dec 19 2017 *) PROG (PARI) Next(v)={vector(#v+2, i, if(i<3||i>#v-2, 0, v[i-2]+v[i-1]+v[i+1]+v[i+2]))} my(v=vector(7, i, i==3)); for(n=1, 50, print1(v[5], ", "); v=Next(v)) \\ Andrew Howroyd, Dec 18 2017 CROSSREFS Cf. A055113, A185286, A187430. Sequence in context: A003757 A187985 A320043 * A064521 A330283 A262238 Adjacent sequences:  A296616 A296617 A296618 * A296620 A296621 A296622 KEYWORD nonn,walk AUTHOR Feng Jishe, Dec 17 2017 STATUS approved

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Last modified July 30 19:33 EDT 2021. Contains 346359 sequences. (Running on oeis4.)