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%I #16 Mar 11 2022 12:00:44
%S 1,2,5,13,40,145,616,3017,16752,103973,713040,5352729,43645848,
%T 384059537,3626960272,36585357429,392545057280,4463791225145,
%U 53622168102640,678508544425721,9020035443775264,125684948107190045,1831698736650660952,27866044704218390113
%N Di-Boustrophedon transform of all 1's sequence: Fill in an array by diagonals alternating in the 'up' and 'down' directions. Each diagonal starts with a 1. When going in the 'up' direction the next element is the sum of the previous element of the diagonal and the previous two elements of the row the new element is in. When going in the 'down' direction the next element is the sum of the previous element of the diagonal and the previous two elements of the column the new element is in. The final element of the n-th diagonal is a(n).
%H Alois P. Heinz, <a href="/A062704/b062704.txt">Table of n, a(n) for n = 1..200</a>
%e The array begins:
%e 1 2 1 13 1
%e 1 3 10 14
%e 5 6 25
%e 1 34
%e 40
%p T:= proc(n, k) option remember;
%p if n<1 or k<1 then 0
%p elif n=1 and irem(k, 2)=1 or k=1 and irem(n, 2)=0 then 1
%p elif irem(n+k, 2)=0 then T(n-1, k+1)+T(n-1, k)+T(n-2, k)
%p else T(n+1, k-1)+T(n, k-1)+T(n, k-2)
%p fi
%p end:
%p a:= n-> `if`(irem (n, 2)=0, T(1, n), T(n, 1)):
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Feb 08 2011
%t T[n_, k_] := T[n, k] = Which[n < 1 || k < 1, 0
%t , n == 1 && Mod[k, 2] == 1 || k == 1 && Mod[n, 2] == 0, 1
%t , Mod[n + k, 2] == 0, T[n - 1, k + 1] + T[n - 1, k] + T[n - 2, k]
%t , True, T[n + 1, k - 1] + T[n, k - 1] + T[n, k - 2]];
%t a[n_] := If[Mod [n, 2] == 0, T[1, n], T[n, 1]];
%t Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Mar 11 2022, after _Alois P. Heinz_ *)
%Y Cf. A000667, A059216, A063179.
%K easy,nonn
%O 1,2
%A _Floor van Lamoen_, Jul 11 2001
%E More terms from _Alois P. Heinz_, Feb 08 2011
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