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.

Previous Showing 11-14 of 14 results.

A318775 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 2 * T(n-5,k-1) for k = 0..floor(n/5); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12, 4, 1, 14, 12, 1, 16, 24, 1, 18, 40, 1, 20, 60, 1, 22, 84, 8, 1, 24, 112, 32, 1, 26, 144, 80, 1, 28, 180, 160, 1, 30, 220, 280, 1, 32, 264, 448, 16, 1, 34, 312, 672, 80, 1, 36, 364, 960, 240, 1, 38, 420, 1320, 560, 1, 40, 480, 1760, 1120
Offset: 0

Views

Author

Zagros Lalo, Sep 04 2018

Keywords

Comments

The numbers in rows of the triangle are along a "fourth layer" skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along a "fourth layer" skew diagonals pointing top-left in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-x-2*x^5) are given by the sequence generated by the row sums.
The row sums give A318777.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.4510850920547191..., when n approaches infinity.

Examples

			Triangle begins:
  1;
  1;
  1;
  1;
  1;
  1,  2;
  1,  4;
  1,  6;
  1,  8;
  1, 10;
  1, 12,   4;
  1, 14,  12;
  1, 16,  24;
  1, 18,  40;
  1, 20,  60;
  1, 22,  84,    8;
  1, 24, 112,   32;
  1, 26, 144,   80;
  1, 28, 180,  160;
  1, 30, 220,  280;
  1, 32, 264,  448,  16;
  1, 34, 312,  672,  80;
  1, 36, 364,  960, 240;
  1, 38, 420, 1320, 560;
  ...

References

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

Links

Crossrefs

Row sums give A318777.
Cf. A013609, A038207, A128099, A207538.

Programs

  • Mathematica
    t[n_, k_] := t[n, k] = 2^k/((n - 5 k)! k!) (n - 4 k)!; Table[t[n, k], {n, 0, 24}, {k, 0, Floor[n/5]} ] // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 5, k - 1]]; Table[t[n, k], {n, 0, 24}, {k, 0, Floor[n/5]}] // Flatten

Formula

T(n,k) = 2^k / ((n - 5*k)! k!) * (n - 4*k)! where n >= 0 and 0 <= k <= floor(n/5).

A318776 Triangle read by rows: T(0,0) = 1; T(n,k) = 2*T(n-1,k) + T(n-5,k-1) for k = 0..floor(n/5); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 1, 64, 4, 128, 12, 256, 32, 512, 80, 1024, 192, 1, 2048, 448, 6, 4096, 1024, 24, 8192, 2304, 80, 16384, 5120, 240, 32768, 11264, 672, 1, 65536, 24576, 1792, 8, 131072, 53248, 4608, 40, 262144, 114688, 11520, 160, 524288, 245760, 28160, 560, 1048576, 524288, 67584, 1792, 1, 2097152, 1114112, 159744, 5376, 10
Offset: 0

Views

Author

Zagros Lalo, Sep 04 2018

Keywords

Comments

The numbers in rows of the triangle are along a "fourth layer" skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and along a "fourth layer" skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-2*x-x^5) are given by the sequence generated by the row sums.
The row sums give A098588.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.0559673967128..., when n approaches infinity.

Examples

			Triangle begins:
        1;
        2;
        4;
        8;
       16;
       32,       1;
       64,       4;
      128,      12;
      256,      32;
      512,      80;
     1024,     192,      1;
     2048,     448,      6;
     4096,    1024,     24;
     8192,    2304,     80;
    16384,    5120,    240;
    32768,   11264,    672,    1;
    65536,   24576,   1792,    8;
   131072,   53248,   4608,   40;
   262144,  114688,  11520,  160;
   524288,  245760,  28160,  560;
  1048576,  524288,  67584, 1792,  1;
  2097152, 1114112, 159744, 5376, 10;
  ...

References

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

Links

Crossrefs

Row sums give A098588.
Cf. A013609, A038207, A128099, A207538.
Cf. also A000079 (column 0), A001787 (column 1), A001788 (column 2), A001789 (column 3)

Programs

  • Mathematica
    t[n_, k_] := t[n, k] = 2^(n - 5 k)/((n - 5 k)! k!) (n - 4 k)!; Table[t[n, k], {n, 0, 21}, {k, 0, Floor[n/5]} ] // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 5, k - 1]]; Table[t[n, k], {n, 0, 21}, {k, 0, Floor[n/5]}] // Flatten

Formula

T(n,k) = 2^(n - 5*k) / ((n - 5*k)! k!) * (n - 4*k)! where n >= 0 and 0 <= k <= floor(n/5).

A317500 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 2 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 4, 1, 12, 12, 1, 14, 24, 1, 16, 40, 1, 18, 60, 8, 1, 20, 84, 32, 1, 22, 112, 80, 1, 24, 144, 160, 1, 26, 180, 280, 16, 1, 28, 220, 448, 80, 1, 30, 264, 672, 240, 1, 32, 312, 960, 560, 1, 34, 364, 1320, 1120, 32
Offset: 0

Views

Author

Zagros Lalo, Sep 03 2018

Keywords

Comments

The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-x-2*x^4) are given by the sequence generated by the row sums.
The row sums give A052942.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.543689012692076... (A256099: Decimal expansion of the real root of a cubic used by Omar Khayyám in a geometrical problem), when n approaches infinity.

Examples

			Triangle begins:
  1;
  1;
  1;
  1;
  1,  2;
  1,  4;
  1,  6;
  1,  8;
  1, 10,   4;
  1, 12,  12;
  1, 14,  24;
  1, 16,  40;
  1, 18,  60,   8;
  1, 20,  84,  32;
  1, 22, 112,  80;
  1, 24, 144, 160;
  1, 26, 180, 280,  16;
  1, 28, 220, 448,  80;
  1, 30, 264, 672, 240;
...

References

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

Links

Crossrefs

Row sums give A052942.
Cf. A013609, A038207, A128099, A207538, A256099.

Programs

  • Mathematica
    t[n_, k_] := t[n, k] = 2^k/((n - 4 k)! k!) (n - 3 k)!; Table[t[n, k], {n, 0, 20}, {k, 0, Floor[n/4]} ] // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 20}, {k, 0, Floor[n/4]}] // Flatten

Formula

T(n,k) = 2^k / ((n - 4*k)! k!) * (n - 3*k)! where n >= 0 and 0 <= k <= floor(n/4).

A317501 Triangle read by rows: T(0,0) = 1; T(n,k) = 2*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, 16, 1, 32, 4, 64, 12, 128, 32, 256, 80, 1, 512, 192, 6, 1024, 448, 24, 2048, 1024, 80, 4096, 2304, 240, 1, 8192, 5120, 672, 8, 16384, 11264, 1792, 40, 32768, 24576, 4608, 160, 65536, 53248, 11520, 560, 1, 131072, 114688, 28160, 1792, 10, 262144, 245760, 67584, 5376, 60
Offset: 0

Views

Author

Zagros Lalo, Sep 03 2018

Keywords

Comments

Unsigned version of the triangle in A317506.
The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-2*x-x^4) are given by the sequence generated by the row sums.
The row sums give A008999.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.106919340376..., when n approaches infinity.

Examples

			Triangle begins:
       1;
       2;
       4;
       8;
      16,      1;
      32,      4;
      64,     12;
     128,     32;
     256,     80,     1;
     512,    192,     6;
    1024,    448,    24;
    2048,   1024,    80;
    4096,   2304,   240,    1;
    8192,   5120,   672,    8;
   16384,  11264,  1792,   40;
   32768,  24576,  4608,  160;
   65536,  53248, 11520,  560,  1;
  131072, 114688, 28160, 1792, 10;
  262144, 245760, 67584, 5376, 60;

References

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

Links

Crossrefs

Row sums give A008999.
Cf. A013609, A038207, A128099, A207538, A317506.

Programs

  • Mathematica
    t[n_, k_] := t[n, k] = 2^(n - 4 k)/((n - 4 k)! k!) (n - 3 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/4]} ] // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/4]}] // Flatten

Formula

T(n,k) = 2^(n - 4*k) / ((n - 4*k)! k!) * (n - 3*k)! where n >= 0 and 0 <= k <= floor(n/4).
Previous Showing 11-14 of 14 results.

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