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.

A304213 Triangle read by rows: T(0,0) = 1; T(n,k) = 2*T(n-1,k) - 2*T(n-2,k-1) + T(n-3,k-2) for k = 0..floor(2*n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, -2, 8, -8, 1, 16, -24, 8, 32, -64, 36, -4, 64, -160, 128, -32, 1, 128, -384, 400, -160, 18, 256, -896, 1152, -640, 136, -6, 512, -2048, 3136, -2240, 720, -80, 1, 1024, -4608, 8192, -7168, 3120, -592, 32, 2048, -10240, 20736, -21504, 11872, -3264, 360, -8, 4096, -22528, 51200, -61440, 41216, -15008, 2624, -160, 1
Offset: 0

Views

Author

Shara Lalo, May 08 2018

Keywords

Comments

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A304209.
The coefficients in the expansion of 1/(1-2*x+2*x^2-x^3) are given by the sequence generated by the row sums.

Examples

			Triangle begins:
      1;
      2;
      4,      -2;
      8,      -8,      1;
     16,     -24,      8;
     32,     -64,     36,      -4;
     64,    -160,    128,     -32,      1;
    128,    -384,    400,    -160,     18;
    256,    -896,   1152,    -640,    136,      -6;
    512,   -2048,   3136,   -2240,    720,     -80,     1;
   1024,   -4608,   8192,   -7168,   3120,    -592,    32;
   2048,  -10240,  20736,  -21504,  11872,   -3264,   360,     -8;
   4096,  -22528,  51200,  -61440,  41216,  -15008,  2624,   -160,   1;
   8192,  -49152, 123904, -168960, 133632,  -60928, 14896,  -1632,  50;
  16384, -106496, 294912, -450560, 410880, -225792, 71680, -11776, 780, -10;
  ...

References

  • Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 204, 205.

Links

Crossrefs

Row sums is similar to A021823.
Cf. A304209.

Programs

  • PARI
    T(n,k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 2*T(n-1,k)-2*T(n-2,k-1)+T(n-3,k-2)));
tabf(nn) = for (n=0, nn, for (k=0, 2*n\3, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 10 2018

A304223 Triangle read by rows: T(0,0)=1; T(n,k) = T(n-1,k)-2*T(n-2,k-1)+2*T(n-3,k-2) for k = 0..floor(2*n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 1, -2, 1, -4, 2, 1, -6, 8, 1, -8, 18, -8, 1, -10, 32, -32, 4, 1, -12, 50, -80, 36, 1, -14, 72, -160, 136, -24, 1, -16, 98, -280, 360, -160, 8, 1, -18, 128, -448, 780, -592, 128, 1, -20, 162, -672, 1484, -1632, 720, -64
Offset: 0

Views

Author

Shara Lalo, May 08 2018

Keywords

Comments

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A304209.
The coefficients in the expansion of 1/(1-x+2*x^2-2*x^3) are given by the sequence generated by the row sums.

Examples

			Triangle begins:
  1;
  1;
  1,  -2;
  1,  -4,   2;
  1,  -6,   8;
  1,  -8,  18,   -8;
  1, -10,  32,  -32,    4;
  1, -12,  50,  -80,   36;
  1, -14,  72, -160,  136,   -24;
  1, -16,  98, -280,  360,  -160,    8;
  1, -18, 128, -448,  780,  -592,  128;
  1, -20, 162, -672, 1484, -1632,  720,  -64;
  1, -22, 200, -960, 2576, -3752, 2624, -640,  16;
  ...

References

  • Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 204, 205.

Links

Crossrefs

Row sums is A077953.
Cf. A304209.

Programs

  • PARI
    T(n,k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1,k)-2*T(n-2,k-1)+2*T(n-3,k-2)));
tabf(nn) = for (n=0, nn, for (k=0, 2*n\3, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 10 2018
Showing 1-2 of 2 results.

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