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A059216 Variation of Boustrophedon transform applied to all-1's sequence (see Comments for details). 22
1, 2, 5, 14, 45, 169, 740, 3721, 21142, 133850, 933770, 7114115, 58758459, 522892624, 4987285553, 50751731950, 548839590949, 6285265061237, 75985249771496, 967047685739501, 12923640789599709, 180945893711983990, 2648725169100050894 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Variation of Boustrophedon transform applied to all-1's sequence. Fill an array by diagonals in alternating directions - 'up' and 'down'. The first element of each diagonal is 1. When 'going up', add to the previous element the elements of the row the new element is in. When 'going down', add to the previous element the elements of the column the new element is in. The final element of the n-th diagonal is a(n).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..480

Index entries for sequences related to boustrophedon transform

EXAMPLE

The array begins

   1  2  1 14  1 ...

   1  3 10 15 ...

   5  6 26 ...

   1 37 ...

  45 ...

MAPLE

# To get the array used to produce this sequence:

aaa := proc(m, n) option remember; local i, j, r, s, t1; if m=0 and n=0 then RETURN(1); fi; if n = 0 and m mod 2 = 1 then RETURN(1); fi; if m = 0 and n mod 2 = 0 then RETURN(1); fi; s := m+n; if s mod 2 = 1 then t1 := aaa(m+1, n-1); for j from 0 to n-1 do t1 := t1+aaa(m, j); od: else t1 := aaa(m-1, n+1); for j from 0 to m-1 do t1 := t1+aaa(j, n); od: fi; RETURN(t1); end; # the n-th antidiagonal in the up direction is aaa(n, 0), aaa(n-1, 1), aaa(n-2, 2), ..., aaa(0, n)

# To get the array formed when the transformation is applied to an arbitrary input sequence b = [b[1], b[2], ..., b[N]]:

aab := proc(b, N, m, n) local i, j, r, s, t1; option remember; if m>N or n>N then error "asking for too many terms"; fi; if m = 0 and n mod 2 = 0 then RETURN(b[n+1]) end if; if n = 0 and m mod 2 = 1 then RETURN(b[m+1]) end if; s := m + n; if s mod 2 = 1 then t1 := aab(b, N, m + 1, n - 1); for j from 0 to n - 1 do t1 := t1 + aab(b, N, m, j) end do else t1 := aab(b, N, m - 1, n + 1); for j from 0 to m - 1 do t1 := t1 + aab(b, N, j, n) end do end if; RETURN(t1) end proc;

# To get the output sequence when the transformation is applied to an arbitrary input sequence b = [b[1], b[2], ..., b[N]]:

ff := proc(b) local N, t1, i; N := min(35, nops(b)); t1 := []; for i from 0 to N-1 do if i mod 2 = 0 then t1 := [op(t1), aab(b, N, i, 0)]; else t1 := [op(t1), aab(b, N, 0, i)]; fi; od: t1; end;

MATHEMATICA

max = 22; t[0, 0] = 1; t[0, _?EvenQ] = 1; t[_?OddQ, 0] = 1; t[n_, k_] /; OddQ[n + k](*up*):= t[n, k] = t[n + 1, k - 1] + Sum[t[n, j], {j, 0, k - 1}]; t[n_, k_] /; EvenQ[n + k](*down*):= t[n, k] = t[n - 1, k + 1] + Sum[t[j, k], {j, 0, n - 1}]; tnk = Table[t[n, k], {n, 0, max}, {k, 0, max - n}]; Join[{1}, Rest[Union[tnk[[1]], tnk[[All, 1]]]]] (* Jean-Fran├žois Alcover, Jun 15 2012 *)

CROSSREFS

Cf. A000667, A059217, A059219, A059220, A059718.

Sequence in context: A119429 A290615 A174300 * A259871 A285201 A007823

Adjacent sequences:  A059213 A059214 A059215 * A059217 A059218 A059219

KEYWORD

easy,nonn,nice

AUTHOR

Floor van Lamoen, Jan 18 2001

EXTENSIONS

More terms from N. J. A. Sloane and Larry Reeves (larryr(AT)acm.org), Jan 23 2001

STATUS

approved

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Last modified August 11 23:45 EDT 2020. Contains 336434 sequences. (Running on oeis4.)