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 1-3 of 3 results.

A152568 Triangle T(n,k) read by rows: T(n,n) = -1, T(n,0) = 2^(n - 1), T(n,k) = -2^(n - k - 1), 1 <= k <= n - 1.

Original entry on oeis.org

-1, 1, -1, 2, -1, -1, 4, -2, -1, -1, 8, -4, -2, -1, -1, 16, -8, -4, -2, -1, -1, 32, -16, -8, -4, -2, -1, -1, 64, -32, -16, -8, -4, -2, -1, -1, 128, -64, -32, -16, -8, -4, -2, -1, -1, 256, -128, -64, -32, -16, -8, -4, -2, -1, -1, 512, -256, -128, -64, -32, -16, -8, -4, -2
Offset: 0

Views

Author

Roger L. Bagula, Dec 08 2008

Keywords

Comments

Except for n = 0, the row sums are zero.

Examples

			Triangle begins:
   -1;
    1,   -1;
    2,   -1,   -1;
    4,   -2,   -1,  -1;
    8,   -4,   -2,  -1,  -1;
   16,   -8,   -4,  -2,  -1,  -1;
   32,  -16,   -8,  -4,  -2,  -1, -1;
   64,  -32,  -16,  -8,  -4,  -2, -1, -1;
  128,  -64,  -32, -16,  -8,  -4, -2, -1, -1;
  256, -128,  -64, -32, -16,  -8, -4, -2, -1, -1;
  512, -256, -128, -64, -32, -16, -8, -4, -2, -1, -1;
  ...

Crossrefs

Cf. A057728, A152570, A152571, A152572.

Programs

  • Mathematica
    b[0] = {-1}; b[1] = {1, -1};
b[n_] := b[n] = Join[{2^(n - 1)}, {-b[n - 1][[1]]}, Table[b[n - 1][[i]], {i, 2, Length[b[n - 1]]}]]
Flatten[Table[b[n], {n, 0, 10}]]
  • Maxima
    T(n, k) := if k = n then -1 else if k = 0 then 2^(n - 1) else -2^(n - k - 1)$
create_list(T(n, k), n, 0, 20, k, 0, n); /* Franck Maminirina Ramaharo, Jan 08 2019 */

Formula

From Franck Maminirina Ramaharo, Jan 08 2019: (Start)
G.f.: -(1 - 3*y + 2*x*y^2)/(1 - (2 + x)*y + 2*x*y^2).
E.g.f.: (exp(2*y) - exp(x*y))*(1 - x)/(2 - x) - 1. (End)

Extensions

Unrelated material removed by the Assoc. Eds. of the OEIS, Jun 07 2010

A152571 Triangle T(n,k) read by rows: T(n,n) = -1, T(n,0) = 4^(n - 1), T(n,k) = -4^(n - k - 1), 1 <= k <= n - 1.

Original entry on oeis.org

-1, 1, -1, 4, -1, -1, 16, -4, -1, -1, 64, -16, -4, -1, -1, 256, -64, -16, -4, -1, -1, 1024, -256, -64, -16, -4, -1, -1, 4096, -1024, -256, -64, -16, -4, -1, -1, 16384, -4096, -1024, -256, -64, -16, -4, -1, -1, 65536, -16384, -4096, -1024, -256, -64, -16, -4, -1, -1
Offset: 0

Views

Author

Roger L. Bagula, Dec 08 2008

Keywords

Examples

			Triangle begins:
      -1;
       1,     -1;
       4,     -1,     -1;
      16,     -4,     -1,    -1;
      64,    -16,     -4,    -1,    -1;
     256,    -64,    -16,    -4,    -1,   -1;
    1024,   -256,    -64,   -16,    -4,   -1,  -1;
    4096,  -1024,   -256,   -64,   -16,   -4,  -1,  -1;
   16384,  -4096,  -1024,  -256,   -64,  -16,  -4,  -1, -1;
   65536, -16384,  -4096, -1024,  -256,  -64, -16,  -4, -1, -1;
  262144, -65536, -16384, -4096, -1024, -256, -64, -16, -4, -1, -1;
     ...

Crossrefs

Row sums (except row 0): A020988.
Cf. A057728, A152568, A152570, A152572.

Programs

  • Mathematica
    b[0] = {-1}; b[1] = {1, -1};
b[n_] := b[n] = Join[{4^(n - 1)}, {-b[n - 1][[1]]}, Table[b[n - 1][[i]], {i, 2, Length[b[n - 1]]}]];
Flatten[Table[b[n], {n, 0, 10}]]
  • Maxima
    T(n, k) := if k = n then -1 else if k = 0 then 4^(n - 1) else -4^(n - k - 1)$
create_list(T(n, k), n, 0, 20, k, 0, n); /* Franck Maminirina Ramaharo, Jan 08 2019 */

Formula

From Franck Maminirina Ramaharo, Jan 08 2019: (Start)
G.f.: -(1 - 5*y + 2*x*y^2)/(1 - (4 + x)*y + 4*x*y^2).
E.g.f.: -(4 - x - (2 - x)*exp(4*y) + (6 - 2*x)*exp(x*y))/(8 - 2*x). (End)

Extensions

Edited by Franck Maminirina Ramaharo, Jan 08 2019

A152572 Triangle T(n,k) read by rows: T(n,n) = -1, T(n,0) = 5^(n - 1), T(n,k) = -5^(n - k - 1), 1 <= k <= n - 1.

Original entry on oeis.org

-1, 1, -1, 5, -1, -1, 25, -5, -1, -1, 125, -25, -5, -1, -1, 625, -125, -25, -5, -1, -1, 3125, -625, -125, -25, -5, -1, -1, 15625, -3125, -625, -125, -25, -5, -1, -1, 78125, -15625, -3125, -625, -125, -25, -5, -1, -1, 390625, -78125, -15625, -3125, -625, -125, -25, -5, -1, -1
Offset: 0

Views

Author

Roger L. Bagula, Dec 08 2008

Keywords

Examples

			Triangle begins:
       -1;
        1,      -1;
        5,      -1,     -1;
       25,      -5,     -1,     -1;
      125,     -25,     -5,     -1,    -1;
      625,    -125,    -25,     -5,    -1,   -1;
     3125,    -625,   -125,    -25,    -5,   -1,   -1;
    15625,   -3125,   -625,   -125,   -25,   -5,   -1,  -1;
    78125,  -15625,  -3125,   -625,  -125,  -25,   -5,  -1, -1;
   390625,  -78125, -15625,  -3125,  -625, -125,  -25,  -5, -1, -1;
  1953125, -390625, -78125, -15625, -3125, -625, -125, -25, -5, -1, -1;
      ...

Crossrefs

Row sums (except row 0): A125833.
Cf. A057728, A152568, A152570, A152571.

Programs

  • Mathematica
    b[0] = {-1}; b[1] = {1, -1};
b[n_] := b[n] = Join[{5^(n - 1)}, {-b[n - 1][[1]]}, Table[b[n - 1][[i]], {i, 2, Length[b[n - 1]]}]];
Flatten[Table[b[n], {n, 0, 10}]]
  • Maxima
    T(n, k) := if k = n then -1 else if k = 0 then 5^(n - 1) else -5^(n - k - 1);
create_list(T(n, k), n, 0, 20, k, 0, n); /* Franck Maminirina Ramaharo, Jan 08 2019 */

Formula

From Franck Maminirina Ramaharo, Jan 08 2019: (Start)
G.f.: -(1 - 6*y + 2*x*y^2)/(1 - (5 + x)*y + 5*x*y^2).
E.g.f.: -(10 - 2*x - (5 - 2*x)*exp(5*y) + (20 - 5*x)*exp(x*y))/(25 - 5*x). (End)

Extensions

Edited by Franck Maminirina Ramaharo, Jan 08 2019
Showing 1-3 of 3 results.

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