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 1-2 of 2 results.

A373424 Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(n) is the continued fraction (-1)^n/(~x - 1/(~x - ... 1/(~x - 1)))...) and where '~' is '-' if n is even, and '+' if n is odd, and x appears n times in the expression.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 5, 1, 0, 1, 5, 10, 14, 8, 1, 0, 1, 6, 15, 30, 31, 13, 1, 0, 1, 7, 21, 55, 85, 70, 21, 1, 0, 1, 8, 28, 91, 190, 246, 157, 34, 1, 0, 1, 9, 36, 140, 371, 671, 707, 353, 55, 1, 0, 1, 10, 45, 204, 658, 1547, 2353, 2037, 793, 89, 1, 0
Offset: 0

Views

Author

Peter Luschny, Jun 09 2024

Keywords

Comments

A variant of both A050446 and A050447 which are the main entries. Differs in indexing and adds a first row to the array resp. a diagonal to the triangle.

Examples

			Generating functions of the rows:
   gf0 =  1;
   gf1 = -1/( x-1);
   gf2 =  1/(-x-1/(-x-1));
   gf3 = -1/( x-1/( x-1/( x-1)));
   gf4 =  1/(-x-1/(-x-1/(-x-1/(-x-1))));
   gf5 = -1/( x-1/( x-1/( x-1/( x-1/( x-1)))));
   gf6 =  1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1))))));
   ...
Array A(n, k) starts:
  [0] 1, 0,  0,  0,   0,    0,    0,     0,      0,      0, ...  A000007
  [1] 1, 1,  1,  1,   1,    1,    1,     1,      1,      1, ...  A000012
  [2] 1, 2,  3,  5,   8,   13,   21,    34,     55,     89, ...  A000045
  [3] 1, 3,  6, 14,  31,   70,  157,   353,    793,   1782, ...  A006356
  [4] 1, 4, 10, 30,  85,  246,  707,  2037,   5864,  16886, ...  A006357
  [5] 1, 5, 15, 55, 190,  671, 2353,  8272,  29056, 102091, ...  A006358
  [6] 1, 6, 21, 91, 371, 1547, 6405, 26585, 110254, 457379, ...  A006359
   A000027,A000330,   A085461,     A244881, ...
       A000217, A006322,    A108675, ...
.
Triangle T(n, k) = A(n - k, k) starts:
  [0] 1;
  [1] 1,  0;
  [2] 1,  1,  0;
  [3] 1,  2,  1,  0;
  [4] 1,  3,  3,  1,  0;
  [5] 1,  4,  6,  5,  1,  0;
  [6] 1,  5, 10, 14,  8,  1, 0;

Links

Crossrefs

Cf. A050446, A050447, A276313 (main diagonal), A373353 (row sums of triangle).
Cf. A373423.

Programs

  • Maple
    row := proc(n, len) local x, a, j, ser; if irem(n, 2) = 1 then
a :=  x - 1; for j from 1 to n do a :=  x - 1 / a od: a :=  a - x; else
a := -x - 1; for j from 1 to n do a := -x - 1 / a od: a := -a - x;
fi; ser := series(a, x, len + 2); seq(coeff(ser, x, j), j = 0..len) end:
A := (n, k) -> row(n, 12)[k+1]:      # array form
T := (n, k) -> row(n - k, k+1)[k+1]: # triangular form
  • SageMath
    def Arow(n, len):
    R. = PowerSeriesRing(ZZ, len)
    if n == 0: return [1] + [0]*(len - 1)
    x = -x if n % 2 else x
    a = x + 1
    for _ in range(n):
        a = x - 1 / a
    a = x - a if n % 2 else a - x
    return a.list()
for n in range(7): print(Arow(n, 10))

A373423 Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(0) = 1, cf(1) = -1/(x - 1), and for n > 1 is cf(n) = ~( ~x - 1/(~x - 1/(~x - 1/(~x - 1/(~x - ... 1/(~x + 1))))...) ) where '~' is '-' if n is even, and '+' if n is odd, and x appears n times in the expression.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 1, 1, 0, 1, 4, 3, 1, 1, 0, 1, 5, 6, 5, 1, 1, 0, 1, 6, 10, 14, 8, 1, 1, 0, 1, 7, 15, 30, 31, 13, 1, 1, 0, 1, 8, 21, 55, 85, 70, 21, 1, 1, 0, 1, 9, 28, 91, 190, 246, 157, 34, 1, 1, 0, 1, 10, 36, 140, 371, 671, 707, 353, 55, 1, 1, 0
Offset: 0

Views

Author

Peter Luschny, Jun 09 2024

Keywords

Examples

			Generating functions of row n:
   gf0 = 1;
   gf1 =   - 1/( x-1);
   gf2 = x + 1/(-x+1);
   gf3 = x - 1/( x-1/( x+1));
   gf4 = x + 1/(-x-1/(-x-1/(-x+1)));
   gf5 = x - 1/( x-1/( x-1/( x-1/( x+1))));
   gf6 = x + 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x+1)))));
.
Array begins:
  [0] 1, 0,  0,   0,   0,    0,     0,     0,      0, ...
  [1] 1, 1,  1,   1,   1,    1,     1,     1,      1, ...
  [2] 1, 2,  1,   1,   1,    1,     1,     1,      1, ...  A373565
  [3] 1, 3,  3,   5,   8,   13,    21,    34,     55, ...  A373566
  [4] 1, 4,  6,  14,  31,   70,   157,   353,    793, ...  A373567
  [5] 1, 5, 10,  30,  85,  246,   707,  2037,   5864, ...  A373568
  [6] 1, 6, 15,  55, 190,  671,  2353,  8272,  29056, ...  A373569
       A000217,  A006322,     A108675, ...
            A000330,   A085461,      A244881, ...
.
Triangle starts:
  [0] 1;
  [1] 1, 0;
  [2] 1, 1,  0;
  [3] 1, 2,  1,  0;
  [4] 1, 3,  1,  1,  0;
  [5] 1, 4,  3,  1,  1,  0;
  [6] 1, 5,  6,  5,  1,  1, 0;

Crossrefs

Cf. A373424, A276312 (main diagonal).
Rows include: A373565, A373566, A373567, A373568, A373569.
Columns include: A000217, A000330, A006322, A085461, A108675, A244881.

Programs

  • Maple
    row := proc(n, len) local x, a, j, ser;
if n = 0 then a := -1 elif n = 1 then a := -1/(x - 1) elif irem(n, 2) = 1 then
  a :=  x + 1; for j from 1 to n-1 do a :=  x - 1 / a od: else
  a := -x + 1; for j from 1 to n-1 do a := -x - 1 / a od: fi;
ser := series((-1)^(n-1)*a, x, len + 2); seq(coeff(ser, x, j), j = 0..len) end:
A := (n, k) -> row(n, 12)[k+1]:      # array form
T := (n, k) -> row(n - k, k+1)[k+1]: # triangular form
seq(lprint([n], row(n, 9)), n = 0..9);
  • SageMath
    def Arow(n, len):
    R. = PowerSeriesRing(ZZ, len)
    if n == 0: return [1] + [0]*(len - 1)
    if n == 1: return [1]*(len - 1)
    x = x if n % 2 == 1 else -x
    a = x + 1
    for _ in range(n - 1):
        a = x - 1 / a
    if n % 2 == 0: a = -a
    return a.list()
for n in range(8): print(Arow(n, 9))
Showing 1-2 of 2 results.

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