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Number of sequences {s(i): i=0..n} such that |s(i)-s(i-1)|=1, i=1..n and s(i)=0 at four values of i, one of which is i=0.
1

%I #14 Oct 30 2021 13:44:36

%S 0,0,0,0,0,8,16,40,80,168,336,672,1344,2640,5280,10296,20592,40040,

%T 80080,155584,311168,604656,1209312,2351440,4702880,9152528,18305056,

%U 35659200,71318400,139070880,278141760,542911320,1085822640,2121460200,4242920400,8297266560

%N Number of sequences {s(i): i=0..n} such that |s(i)-s(i-1)|=1, i=1..n and s(i)=0 at four values of i, one of which is i=0.

%C Calculation suggests that {a(2n)/8} is A002054 (shifted).

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a052/A052207.java">Java program</a> (github)

%e a(6) = 8: [0,-1,0,-1,0,-1,0], [0,-1,0,-1,0,1,0], [0,-1,0,1,0,-1,0], [0,-1,0,1,0,1,0], [0,1,0,-1,0,-1,0], [0,1,0,-1,0,1,0], [0,1,0,1,0,-1,0], [0,1,0,1,0,1,0].

%p a:= n-> max(0, (m-> 8*(d+1)*binomial(2*m+1, m-1))(iquo(n-4, 2, 'd'))):

%p seq(a(n), n=1..36); # _Alois P. Heinz_, Oct 30 2021

%Y Cf. A002054.

%K nonn

%O 1,6

%A _John W. Layman_, Jan 26 2000

%E More terms from _Sean A. Irvine_, Oct 29 2021