cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A059511 Main diagonal of the array A059217.

Original entry on oeis.org

1, 3, 26, 471, 13953, 627421, 39873948, 3406344237, 376591685322, 52319828460127, 8922977163384085, 1832831375530080986, 446306812127457163805, 127130280682997045232005, 41881054710966221044222595, 15800074329058136531647698779
Offset: 1

Views

Author

Floor van Lamoen, Jan 21 2001

Keywords

Links

Programs

  • Mathematica
    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}]; Table[t[n, n], {n,0,50}] (* G. C. Greubel, Sep 10 2017 *)

Extensions

Terms a(11) onward added by G. C. Greubel, Sep 10 2017

A059216 Variation of Boustrophedon transform applied to all-1's sequence (see Comments for details).

Original entry on oeis.org

1, 2, 5, 14, 45, 169, 740, 3721, 21142, 133850, 933770, 7114115, 58758459, 522892624, 4987285553, 50751731950, 548839590949, 6285265061237, 75985249771496, 967047685739501, 12923640789599709, 180945893711983990, 2648725169100050894
Offset: 1

Views

Author

Floor van Lamoen, Jan 18 2001

Keywords

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).

Examples

			The array begins
   1  2  1 14  1 ...
   1  3 10 15 ...
   5  6 26 ...
   1 37 ...
  45 ...

Links

Crossrefs

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

Programs

  • 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-François Alcover, Jun 15 2012 *)

Extensions

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

A059219 Variation of Boustrophedon transform applied to sequence 1,0,0,0,...: fill an array by diagonals in alternating directions - 'up' and 'down'. The first element of each diagonal after the first is 0. 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).

Original entry on oeis.org

1, 1, 2, 5, 15, 55, 239, 1199, 6810, 43108, 300731, 2291162, 18923688, 168402163, 1606199354, 16345042652, 176758631046, 2024225038882, 24471719797265, 311446235344127, 4162172487402027, 58275220793611957, 853045299274146032
Offset: 0

Views

Author

N. J. A. Sloane, Jan 18 2001

Keywords

Examples

			The array begins
1 1 0 5 0 55 0 ...
0 1 3 5 48 55 ...
2 2 8 39 103 ...
0 12 27 152 ...
15 15 190 ...
0 221 ...

Links

Crossrefs

Cf. A000667, A059216, A059217, A059220, A059237, A059720, A059718.

Programs

  • Maple
    aaa := proc(m,n) option remember; local j,s,t1; if m=0 and n=0 then RETURN(1); fi; if n = 0 and m mod 2 = 1 then RETURN(0); fi; if m = 0 and n mod 2 = 0 then RETURN(0); 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)
  • Mathematica
    max = 22; t[0, 0] = 1; t[0, ?EvenQ] = 0; t[?OddQ, 0] = 0; 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-François Alcover, May 16 2012 *)

Extensions

More terms from Floor van Lamoen, Jan 19 2001; and from N. J. A. Sloane Jan 20 2001.

A059220 The array in A059219 read by antidiagonals in 'up' direction.

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 0, 2, 3, 5, 15, 12, 8, 5, 0, 0, 15, 27, 39, 48, 55, 239, 221, 190, 152, 103, 55, 0, 0, 239, 460, 680, 871, 1025, 1137, 1199, 6810, 6553, 6062, 5374, 4493, 3471, 2336, 1199, 0, 0, 6810, 13363, 19903, 25958, 31351, 35884, 39399, 41847
Offset: 1

Views

Author

N. J. A. Sloane, Jan 18 2001

Keywords

Examples

			The array begins
   1   1   0   5   0  55   0 ...
   0   1   3   5  48  55   ...
   2   2   8  39 103  ...
   0  12  27 152 ...
  15  15 190 ...
   0 221 ...

Links

Crossrefs

Cf. A000667, A059216, A059219, A059217, A059235.

Programs

  • Maple
    See A059219 for Maple code.
  • Mathematica
    max = 9; t[0, 0] = 1; t[0, ?EvenQ] = 0; t[?OddQ, 0] = 0; 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}]; Table[t[n - k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 19 2013 *)

Extensions

More terms from Floor van Lamoen, Jan 19 2001

A059235 The array in A059219 read by antidiagonals in the direction in which it was constructed.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 3, 5, 0, 5, 8, 12, 15, 0, 15, 27, 39, 48, 55, 0, 55, 103, 152, 190, 221, 239, 0, 239, 460, 680, 871, 1025, 1137, 1199, 0, 1199, 2336, 3471, 4493, 5374, 6062, 6553, 6810, 0, 6810, 13363, 19903, 25958, 31351, 35884, 39399, 41847, 43108, 0
Offset: 1

Views

Author

N. J. A. Sloane, Jan 18 2001

Keywords

Examples

			The array begins
1 1 0 5 0 55 0 ...
0 1 3 5 48 55 ...
2 2 8 39 103 ...
0 12 27 152 ...
15 15 190 ...
0 221 ...

Links

Crossrefs

Cf. A000667, A059216, A059219, A059217, A059220.

Programs

  • Mathematica
    max = 10; t[0, 0] = 1; t[0, ?EvenQ] = 0; t[?OddQ, 0] = 0; 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}]; Table[tn = Table[t[n-k, k], {k, 0, n}]; If[OddQ[n], tn, tn // Reverse] , {n, 0, max}] // Flatten (* Jean-François Alcover, Nov 20 2012 *)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jan 24 2001

A059234 The array in A059216 read by antidiagonals in the direction in which it was constructed.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 6, 10, 14, 1, 15, 26, 37, 45, 1, 46, 84, 121, 150, 169, 1, 170, 321, 471, 592, 686, 740, 1, 741, 1428, 2111, 2704, 3183, 3532, 3721, 1, 3722, 7255, 10777, 13953, 16685, 18826, 20347, 21142, 1, 21143, 41491, 61798, 80598, 97345, 111419
Offset: 1

Views

Author

Floor van Lamoen, Jan 18 2001

Keywords

Examples

			The array begins
   1  2  1 14  1 ...
   1  3 10 15 ...
   5  6 26 ...
   1 37 ...
  45 ...

Crossrefs

Cf. A000667, A059216, A059219, A059220, A059217.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jan 23 2001

A059237 Variation of Boustrophedon transform described in A059219 applied to sequence 0,1,0,0,0,....

Original entry on oeis.org

0, 1, 2, 5, 16, 59, 258, 1296, 7362, 46609, 325147, 2477212, 20460278, 182076531, 1736623109, 17672266151, 191111489038, 2188592796698, 26458831601847, 336735773968857, 4500142285227330, 63007188219787855, 922312862937555109
Offset: 0

Views

Author

N. J. A. Sloane, Jan 20 2001

Keywords

Links

Crossrefs

Cf. A000667, A059216, A059217, A059220, A059219.

Programs

  • Maple
    aaa := proc(m,n) option remember; local i,j,r,s,t1; if m=0 and n=0 then RETURN(0); fi; if m=1 and n=0 then RETURN(1); fi; if n = 0 and m mod 2 = 1 then RETURN(0); fi; if m = 0 and n mod 2 = 0 then RETURN(0); 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)
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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