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 1-6 of 6 results.

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

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

A152722 Triangle read by rows: T(n,0) = prime(n+2), T(n,1) = 1 - T(n,0), T(n,k) = T(n-1,k-1), T(1,0) = 1 T(n,n) = -1.

Original entry on oeis.org

-1, 1, -1, 7, -6, -1, 11, -10, -6, -1, 13, -12, -10, -6, -1, 17, -16, -12, -10, -6, -1, 19, -18, -16, -12, -10, -6, -1, 23, -22, -18, -16, -12, -10, -6, -1, 29, -28, -22, -18, -16, -12, -10, -6, -1, 31, -30, -28, -22, -18, -16, -12, -10, -6, -1, 37, -36, -30, -28, -22, -18, -16, -12, -10, -6, -1
Offset: 0

Views

Author

Roger L. Bagula and Alexander R. Povolotsky, Dec 11 2008

Keywords

Examples

			Triangle begins as:
  -1;
   1,  -1;
   7,  -6,  -1;
  11, -10,  -6,  -1;
  13, -12, -10,  -6,  -1;
  17, -16, -12, -10,  -6,  -1;
  19, -18, -16, -12, -10,  -6,  -1;
  23, -22, -18, -16, -12, -10,  -6, -1;
  29, -28, -22, -18, -16, -12, -10, -6, -1;

Links

Crossrefs

Cf. A152568, A027293.

Programs

  • Mathematica
    T[n_, n_]:= -1; T[1, 0]:= 1; T[n_, 0]:= Prime[n+2]; T[n_, 1]:= 1 - Prime[n+2]; T[n_, k_]:= T[n-1, k-1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 07 2019 *)
  • PARI
    {T(n,k) = if(k==n, -1, if(n==1 && k==0, 1, if(k==0, prime(n+2), if(k==1, 1-prime(n+2), T(n-1,k-1) ))))}; \\ G. C. Greubel, Apr 07 2019
  • Sage
    @CachedFunction
def T(n,k):
   if k==n: return -1
   elif n==1 and k==0: return 1
   elif k==0: return nth_prime(n+2)
   elif k==1: return 1 - nth_prime(n+2)
   else: return T(n-1,k-1)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 07 2019

Formula

T(n,n) = -1, T(1,0) = 1, T(n,0) = prime(n+2), T(n,1) = 1 - prime(n+2), T(n,k) = T(n-1,k-1). - G. C. Greubel, Apr 07 2019

Extensions

Edited by G. C. Greubel, Apr 07 2019

A152720 A prime-based vector recursion: a(n)={Prime[n],Prime[n-1],-Prime[n-2],...,-1,-1}.

Original entry on oeis.org

-1, 1, -1, 3, -1, -1, 5, -3, -1, -1, 7, -5, -3, -1, -1, 11, -7, -5, -3, -1, -1, 13, -11, -7, -5, -3, -1, -1, 17, -13, -11, -7, -5, -3, -1, -1, 19, -17, -13, -11, -7, -5, -3, -1, -1, 23, -19, -17, -13, -11, -7, -5, -3, -1, -1, 29, -23, -19, -17, -13, -11, -7, -5, -3, -1, -1
Offset: 0

Views

Author

Roger L. Bagula, Dec 11 2008

Keywords

Comments

Row sums are: {-1, 0, 1, 0, -3, -6, -15, -24, -39, -54, -71,...}

Examples

			{-1},
{1, -1},
{3, -1, -1},
{5, -3, -1, -1},
{7, -5, -3, -1, -1},
{11, -7, -5, -3, -1, -1},
{13, -11, -7, -5, -3, -1, -1},
{17, -13, -11, -7, -5, -3, -1, -1},
{19, -17, -13, -11, -7, -5, -3, -1, -1},
{23, -19, -17, -13, -11, -7, -5, -3, -1, -1},
{29, -23, -19, -17, -13, -11, -7, -5, -3, -1, -1}

Crossrefs

A152568

Programs

  • Mathematica
    b[0] = {-1}; b[1] = {1, -1};
b[n_] := b[n] = Join[{Prime[n ]}, {-b[n - 1][[ 1]]}, Table[b[n - 1][[i]], {i, 2, Length[b[n - 1]]}]];
Table[b[n], {n, 0, 10}]; Flatten[%]

Formula

a(n)={Prime[n],Prime[n-1],-Prime[n-2],...,-1,-1}.

A152721 A prime based vector recursion: a(n)={Prime[n+1],Prime[n],Prime[n-1],-Prime[n-2],...,-1,-1}.

Original entry on oeis.org

-1, 1, -1, 5, -1, -1, 7, -5, -1, -1, 11, -7, -5, -1, -1, 13, -11, -7, -5, -1, -1, 17, -13, -11, -7, -5, -1, -1, 19, -17, -13, -11, -7, -5, -1, -1, 23, -19, -17, -13, -11, -7, -5, -1, -1, 29, -23, -19, -17, -13, -11, -7, -5, -1, -1, 31, -29, -23, -19, -17, -13, -11, -7, -5
Offset: 0

Views

Author

Roger L. Bagula, Dec 11 2008

Keywords

Comments

Row sums are:
{-1, 0, 3, 0, -3, -12, -21, -36, -51, -68, -95,...}

Examples

			{-1},
{1, -1},
{5, -1, -1},
{7, -5, -1, -1},
{11, -7, -5, -1, -1},
{13, -11, -7, -5, -1, -1},
{17, -13, -11, -7, -5, -1, -1},
{19, -17, -13, -11, -7, -5, -1, -1},
{23, -19, -17, -13, -11, -7, -5, -1, -1},
{29, -23, -19, -17, -13, -11, -7, -5, -1, -1},
{31, -29, -23, -19, -17, -13, -11, -7, -5, -1, -1}

Crossrefs

A152568, A027293

Programs

  • Mathematica
    b[0] = {-1}; b[1] = {1, -1};
b[n_] := b[n] = Join[{Prime[n + 1 ]}, {-b[n - 1][[1]]}, Table[b[n - 1][[i]], {i, 2, Length[b[n - 1]]}]];
Table[b[n], {n, 0, 10}]; Flatten[%]

Formula

a(n)={Prime[n+1],Prime[n],Prime[n-1],-Prime[n-2],...,-1,-1}.
Showing 1-6 of 6 results.

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