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 1-4 of 4 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

A152570 Triangle T(n,k) read by rows: T(n,n) = -1, T(n,0) = 3^(n - 1), T(n,k) = -3^(n - k - 1), 1 <= k <= n - 1.

Original entry on oeis.org

-1, 1, -1, 3, -1, -1, 9, -3, -1, -1, 27, -9, -3, -1, -1, 81, -27, -9, -3, -1, -1, 243, -81, -27, -9, -3, -1, -1, 729, -243, -81, -27, -9, -3, -1, -1, 2187, -729, -243, -81, -27, -9, -3, -1, -1, 6561, -2187, -729, -243, -81, -27, -9, -3, -1, -1, 19683, -6561, -2187
Offset: 0

Views

Author

Roger L. Bagula, Dec 08 2008

Keywords

Examples

			Triangle begins:
     -1;
      1,    -1;
      3,    -1,    -1;
      9,    -3,    -1,   -1;
     27,    -9,    -3,   -1,   -1;
     81,   -27,    -9,   -3,   -1,  -1;
    243,   -81,   -27,   -9,   -3,  -1,  -1;
    729,  -243,   -81,  -27,   -9,  -3,  -1, -1;
   2187,  -729,  -243,  -81,  -27,  -9,  -3, -1, -1;
   6561, -2187,  -729, -243,  -81, -27,  -9, -3, -1, -1;
  19683, -6561, -2187, -729, -243, -81, -27, -9, -3, -1, -1;
    ...

Crossrefs

Row sums (except row 0): A003462.
Cf. A057728, A152568, A152571, A152572.

Programs

  • Mathematica
    b[0] = {-1}; b[1] = {1, -1};
b[n_] := b[n] = Join[{3^(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 3^(n - 1) else -3^(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 - 4*y + 2*x*y^2)/(1 - (3 + x)*y + 3*x*y^2).
E.g.f.: -(6 - 2*x - (3 - 2*x)*exp(3*y) + (6 - 3*x)*exp(x*y))/(9 - 3*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

A262616 Triangle read by rows: T(n,k) = 4^(n-k), n>=0, 0<=k<=n.

Original entry on oeis.org

1, 4, 1, 16, 4, 1, 64, 16, 4, 1, 256, 64, 16, 4, 1, 1024, 256, 64, 16, 4, 1, 4096, 1024, 256, 64, 16, 4, 1, 16384, 4096, 1024, 256, 64, 16, 4, 1, 65536, 16384, 4096, 1024, 256, 64, 16, 4, 1, 262144, 65536, 16384, 4096, 1024, 256, 64, 16, 4, 1, 1048576, 262144, 65536, 16384, 4096, 1024, 256, 64, 16, 4, 1
Offset: 0

Views

Author

Omar E. Pol, Nov 23 2015

Keywords

Comments

A triangle of the same family of A130321 and A140303, with the same offset.
T(n,k) is also the number of hidden crosses of size k+1 formed by squares and rectangles in the toothpick structure of A139250 after 2^(n+2) stages. The last term in every row represents the central cross of the toothpick structure.

Examples

			Triangle begins:
1;
4,       1;
16,      4,       1;
64,      16,      4,      1;
256,     64,      16,     4,     1;
1024,    256,     64,     16,    4,     1;
4096,    1024,    256,    64,    16,    4,    1;
16384,   4096,    1024,   256,   64,    16,   4,    1;
65536,   16384,   4096,   1024,  256,   64,   16,   4,   1;
262144,  65536,   16384,  4096,  1024,  256,  64,   16,  4,  1;
1048576, 262144,  65536,  16384, 4096,  1024, 256,  64,  16, 4,  1;
4194304, 1048576, 262144, 65536, 16384, 4096, 1024, 256, 64, 16, 4, 1;
...

Links

Crossrefs

Column k gives A000302.
Row sums give the positive terms of A002450.
Alternating row sums give the positive terms of A015521.
Cf. A130321, A139250, A140303, A152571, A152716.

Programs

  • Mathematica
    Table[4^(n - k), {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 17 2016 *)

Formula

T(n,k) = A000302(n-k).
Showing 1-4 of 4 results.

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

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