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.

A349819 Central column (ignoring the zeros) of A349813, or leading entries in rows of A349814.

Original entry on oeis.org

3, 1, 3, 4, 20, 31, 182, 304, 1932, 3364, 22407, 40044, 275132, 500522, 3515564, 6478784, 46260604, 86094668, 622636764, 1167752848, 8531618640, 16100882359, 118615471982, 225001039696, 1669094344820, 3179713880524, 23725950404610, 45364387834120, 340192536782760
Offset: 0

Views

Author

N. J. A. Sloane, Dec 24 2021

Keywords

Links

Crossrefs

Cf. A349813, A349814.

Programs

  • PARI
    a(n) = if(n==0, 3, polcoef((-3 - x + x^2 + 3*x^3)*(1 + x + x^2 + x^3)^(n-1), 3*n\2+1)) \\ Andrew Howroyd, Feb 28 2023

Formula

a(n) = [x^floor(3*n/2+1)](-3 - x + x^2 + 3*x^3)*(1 + x + x^2 + x^3)^(n-1) for n > 0. - Andrew Howroyd, Feb 28 2023

Extensions

Offset corrected and terms a(21) and beyond from Andrew Howroyd, Feb 28 2023

A349814 Irregular triangle read by rows: the right-hand side of the triangle in A349813.

Original entry on oeis.org

3, 1, 3, 3, 4, 3, 4, 10, 10, 7, 3, 20, 31, 30, 20, 10, 3, 31, 81, 101, 91, 63, 33, 13, 3, 182, 304, 336, 288, 200, 112, 49, 16, 3, 304, 822, 1110, 1128, 936, 649, 377, 180, 68, 19, 3, 1932, 3364, 3996, 3823, 3090, 2142, 1274, 644, 270, 90, 22, 3, 3364, 9292, 13115, 14273, 13051, 10329, 7150, 4330, 2278, 1026, 385, 115, 25, 3, 22407, 40044, 49731, 50768, 44803, 34860, 24087, 14784, 8019, 3804, 1551, 528, 143, 28, 3
Offset: 0

Views

Author

N. J. A. Sloane, Dec 23 2021

Keywords

Comments

It seems more symmetrical to omit the central column of zeros in A349813.

Examples

			Triangle begins:
     3;
     1,    3;
     3,    4,    3;
     4,   10,   10,    7,    3;
    20,   31,   30,   20,   10,    3;
    31,   81,  101,   91,   63,   33,   13,   3;
   182,  304,  336,  288,  200,  112,   49,  16,   3;
   304,  822, 1110, 1128,  936,  649,  377, 180,  68, 19,  3;
  1932, 3364, 3996, 3823, 3090, 2142, 1274, 644, 270, 90, 22, 3;
  ...

Crossrefs

Cf. A348813, A349819.

A349812 Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1/x + x)*(1/x + 1 + x)^(n-1) in order of increasing powers of x.

Original entry on oeis.org

1, -1, 0, 1, -1, -1, 0, 1, 1, -1, -2, -2, 0, 2, 2, 1, -1, -3, -5, -4, 0, 4, 5, 3, 1, -1, -4, -9, -12, -9, 0, 9, 12, 9, 4, 1, -1, -5, -14, -25, -30, -21, 0, 21, 30, 25, 14, 5, 1, -1, -6, -20, -44, -69, -76, -51, 0, 51, 76, 69, 44, 20, 6, 1, -1, -7, -27, -70, -133, -189, -196, -127, 0, 127, 196, 189, 133, 70, 27, 7, 1
Offset: 0

Views

Author

N. J. A. Sloane, Dec 23 2021

Keywords

Comments

The rule for constructing this triangle (ignoring row 0) is the same as that for A027907: each number is the sum of the three numbers immediately above it in the previous row. Here row 1 is [-1, 0, 1] instead of [1, 1, 1].

Examples

			Triangle begins:
   1;
  -1,  0,   1;
  -1, -1,   0,   1,    1;
  -1, -2,  -2,   0,    2,    2,    1;
  -1, -3,  -5,  -4,    0,    4,    5,    3,  1;
  -1, -4,  -9, -12,   -9,    0,    9,   12,  9,   4,   1;
  -1, -5, -14, -25,  -30,  -21,    0,   21, 30,  25,  14,   5,   1;
  -1, -6, -20, -44,  -69,  -76,  -51,    0, 51,  76,  69,  44,  20,  6,  1;
  -1, -7, -27, -70, -133, -189, -196, -127,  0, 127, 196, 189, 133, 70, 27, 7, 1;
  ...

Links

  • Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]

Crossrefs

Cf. A007318, A027907, A112467, A349813, A348815.
The left half of the triangle is A026300, the right half is A064189 (or A122896). The central (nonzero) column gives the Motzkin numbers A001006.

Programs

  • Maple
    t1:=-1/x+x; m:=1/x+1+x;
lprint([1]);
for n from 1 to 12 do
w1:=expand(t1*m^(n-1));
w3:=expand(x^n*w1);
w4:=series(w3,x,2*n+1);
w5:=seriestolist(w4);
lprint(w5);
od:

A349815 Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1 - x + x^2 + x^3)*(1 + x + x^2 + x^3)^(n-1) in order of increasing powers of x.

Original entry on oeis.org

1, -1, -1, 1, 1, -1, -2, -1, 0, 1, 2, 1, -1, -3, -4, -4, -2, 2, 4, 4, 3, 1, -1, -4, -8, -12, -13, -8, 0, 8, 13, 12, 8, 4, 1, -1, -5, -13, -25, -37, -41, -33, -13, 13, 33, 41, 37, 25, 13, 5, 1, -1, -6, -19, -44, -80, -116, -136, -124, -74, 0, 74, 124, 136, 116, 80, 44, 19, 6, 1, -1, -7, -26, -70, -149, -259, -376, -456, -450, -334, -124, 124, 334, 450, 456, 376, 259, 149, 70, 26, 7, 1
Offset: 0

Views

Author

N. J. A. Sloane, Dec 23 2021

Keywords

Comments

The row polynomials can be further factorized, since -3 - x + x^2 + 3*x^3 = -(1-x)*(1+x)^2 and 1 + x + x^2 + x^3 = (1+x)*(1+x^2).
The rule for constructing this triangle (ignoring row 0) is the same as that for A008287: each number is the sum of the four numbers immediately above it in the previous row. Here row 1 is [-1, -1, 1, 3] instead of [1, 1, 1, 1].
This is a companion to A008287 and A349813.

Examples

			Triangle begins:
   1;
  -1, -1,   1,   1;
  -1, -2,  -1,   0,   1,   2,   1;
  -1, -3,  -4,  -4,  -2,   2,   4,   4,  3,  1;
  -1, -4,  -8, -12, -13,  -8,   0,   8, 13, 12,  8,  4,  1;
  -1, -5, -13, -25, -37, -41, -33, -13, 13, 33, 41, 37, 25, 13, 5, 1;
  ...

Links

  • Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]

Crossrefs

Cf. A007318, A008287, A349812, A349813.
The right half of the triangle gives A349816. For the central nonzero entries see A349818.

Programs

  • Maple
    t1:=-1-x+x^2+x^3;
m:=1+x+x^2+x^3;
lprint([3]);
for n from 1 to 12 do
w1:=expand(t1*m^(n-1));
w4:=series(w1,x,3*n+1);
w5:=seriestolist(w4);
lprint(w5);
od:
Showing 1-4 of 4 results.

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

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