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.

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A063179 Di-Boustrophedon transform of (1,0,0,0,...): Fill in an array by diagonals alternating in the 'up' and 'down' directions. The n-th diagonal starts with the n-th element of (1,0,0,0,...). 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).

Original entry on oeis.org

1, 1, 2, 4, 12, 42, 178, 870, 4830, 29976, 205572, 1543210, 12583242, 110725638, 1045664646, 10547679660, 113172039256, 1286925785286, 15459448549274, 195616259182162, 2600506074185090, 36235386548738016, 528084808585261568, 8033872923106040478
Offset: 1

Views

Author

Floor van Lamoen, Jul 09 2001

Keywords

Examples

			Array begins:
   1  1  0  4  0 42 ...
   0  1  3  4 38 ...
   2  2  7 31 ...
   0 10 22 ...
  12 12 ...
   0 ...

Crossrefs

Cf. A000667, A059216, A062704, A063180, A063181.

Extensions

More terms from Sean A. Irvine, Apr 21 2023

A059513 Variation of Boustrophedon transform applied to 1,1,1,1,... Fill an array by diagonals, in alternating directions. The first entry is 1 each time. For the next element of a diagonal, add to the previous element the elements of the row and the column the new element is in. The final element of each diagonal gives a(n).

Original entry on oeis.org

1, 2, 6, 23, 116, 736, 5659, 50796, 521040, 6006587, 76874524, 1081439062, 16586149365, 275442822510, 4924040788654
Offset: 1

Views

Author

Floor van Lamoen, Jan 23 2001

Keywords

Examples

			The array begins
1 ....2 ...1 ..23 ..1 ...
1 ....4 ..19 ..48 ...
6 ...13 ..87 ...
1 ..107 ...
116 ...
1 ...

Crossrefs

Cf. A000667, A035002, A059216, A059219, A059575, A059574.

A059574 The array described in A059513 read by antidiagonals in the 'up' direction.

Original entry on oeis.org

1, 1, 2, 6, 4, 1, 1, 13, 19, 23, 116, 107, 87, 48, 1, 1, 243, 458, 635, 708, 736, 5659, 5533, 5163, 4239, 2967, 1517, 1, 1, 11562, 22824, 33291, 41772, 47733, 50031, 50796, 521040, 515254, 497789, 452016, 385422, 301161, 204598, 103125, 1
Offset: 1

Views

Author

Floor van Lamoen, Jan 23 2001

Keywords

Crossrefs

Cf. A000667, A035002, A059216, A059219, A059575.

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

A059505 Transform of A059502 applied to sequence 2,3,4,...

Original entry on oeis.org

2, 5, 14, 40, 114, 323, 910, 2551, 7120, 19796, 54852, 151525, 417434, 1147145, 3145394, 8606848, 23507190, 64093031, 174474790, 474261691, 1287398452, 3490267820, 9451319304, 25565098825, 69080289074
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The second row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502.

Programs

  • Mathematica
    LinearRecurrence[{6,-11,6,-1},{2,5,14,40}, 50] (* or *) Rest[CoefficientList[Series[x*(2 - 7*x + 6*x^2 - x^3)/(1 - 3*x + x^2)^2, {x,0,50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    x='x+O('x^50); Vec(x*(2-7*x+6*x^2-x^3)/(1-3*x+x^2)^2) \\ G. C. Greubel, Sep 10 2017

Formula

G.f.: x*(2 - 7*x + 6*x^2 - x^3)/(1 - 3*x + x^2)^2.
From G. C. Greubel, Sep 10 2017: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
a(n) = ((3 - n)*Fibonacci(2*n) + (5 + 3*n)*Fibonacci(2*n - 1))/5. (End)

A059506 Transform of A059502 applied to sequence 3,4,5,...

Original entry on oeis.org

3, 7, 19, 53, 148, 412, 1143, 3161, 8717, 23977, 65798, 180182, 492459, 1343563, 3659623, 9953117, 27031768, 73320496, 198632607, 537507677, 1452978593, 3923762257, 10586222474, 28536313898, 76859031123
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The third row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502.

Programs

  • Mathematica
    LinearRecurrence[{6,-11,6,-1},{3,7,19,53},30] (* Harvey P. Dale, Jul 30 2015 *)
Rest[CoefficientList[Series[x*(1 - x)*(2*x^2 - 8*x + 3)/(x^2 - 3*x + 1)^2, {x,0,50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    Vec(x*(1-x)*(2*x^2-8*x+3)/(x^2-3*x+1)^2 + O(x^30)) \\ Michel Marcus, Sep 09 2017

Formula

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(2*x^2-8*x+3)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (10 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017

A059507 Transform of A059502 applied to sequence 4,5,6,...

Original entry on oeis.org

4, 9, 24, 66, 182, 501, 1376, 3771, 10314, 28158, 76744, 208839, 567484, 1539981, 4173852, 11299386, 30556346, 82547961, 222790424, 600753663, 1618558734, 4357256694, 11721125644, 31507528971, 84637773172
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The fourth row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502.

Programs

  • Mathematica
    Rest[CoefficientList[Series[x*(1 - x)*(3*x^2 - 11*x + 4)/(x^2 - 3*x + 1)^2, {x, 0, 50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    Vec(x*(1-x)*(3*x^2-11*x+4)/(x^2-3*x+1)^2 + O(x^40)) \\ Michel Marcus, Sep 09 2017

Formula

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(3*x^2-11*x+4)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (15 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017

A059508 Transform of A059502 applied to sequence 5,6,7,...

Original entry on oeis.org

5, 11, 29, 79, 216, 590, 1609, 4381, 11911, 32339, 87690, 237496, 642509, 1736399, 4688081, 12645655, 34080924, 91775426, 246948241, 663999649, 1784138875, 4790751131, 12856028814, 34478744044, 92416515221
Offset: 1

Views

Author

Floor van Lamoen, Jan 19 2001

Keywords

Comments

The fifth row of the array A059503.

Links

Crossrefs

Cf. A000667, A059216, A059219, A059502, A059503.

Programs

  • Mathematica
    Rest[CoefficientList[Series[x*(1-x)*(4*x^2 - 14*x + 5)/(x^2 - 3*x + 1)^2, {x, 0, 50}], x]] (* G. C. Greubel, Sep 10 2017 *)
  • PARI
    Vec(-x*(x-1)*(4*x^2-14*x+5)/(x^2-3*x+1)^2 + O(x^40)) \\ Michel Marcus, Sep 09 2017

Formula

From Colin Barker, Nov 30 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: x*(1-x)*(4*x^2-14*x+5)/(x^2-3*x+1)^2. (End)
a(n) = ((3 - n)*Fibonacci(2*n) + (20 + 3*n)*Fibonacci(2*n - 1))/5. - G. C. Greubel, Sep 10 2017

A059575 The array described in A059513 read by antidiagonals in the direction of construction.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 13, 19, 23, 1, 48, 87, 107, 116, 1, 243, 458, 635, 708, 736, 1, 1517, 2967, 4239, 5163, 5533, 5659, 1, 11562, 22824, 33291, 41772, 47733, 50031, 50796, 1, 103125, 204598, 301161, 385422, 452016, 497789, 515254, 521040
Offset: 1

Views

Author

Floor van Lamoen, Jan 23 2001

Keywords

Crossrefs

Cf. A000667, A035002, A059216, A059219, A059574.

Extensions

Sequence contained two errors corrected by N. J. A. Sloane, Jun 14 2005

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

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