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 1-7 of 7 results.

A188554 Number of 3 X n binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

Original entry on oeis.org

1, 4, 7, 12, 20, 32, 49, 72, 102, 140, 187, 244, 312, 392, 485, 592, 714, 852, 1007, 1180, 1372, 1584, 1817, 2072, 2350, 2652, 2979, 3332, 3712, 4120, 4557, 5024, 5522, 6052, 6615, 7212, 7844, 8512, 9217, 9960, 10742, 11564, 12427, 13332, 14280, 15272, 16309
Offset: 0

Views

Author

R. H. Hardin, Apr 04 2011

Keywords

Comments

a(n) is the number of words of length n, x(1)x(2)...x(n), on the alphabet {0,1,2,3} such that, for i=2,...,n, x(i)=either x(i-1) or x(i-1)-1.
For the bijection between arrays and words, notice that the i-th column consists of 1's and then 0's, and there are x(i)=0 to 3 1's. - Miquel A. Fiol, Feb 06 2024

Examples

			Some solutions for 3 X 3:
  1 1 0   1 1 1   1 1 1   1 1 1   1 1 1   0 0 0   1 1 1
  0 0 0   1 1 0   1 1 1   1 1 1   1 1 1   0 0 0   1 1 1
  0 0 0   0 0 0   1 0 0   0 0 0   1 1 0   0 0 0   1 1 1
For n=3, the a(3)=12 solutions are 000, 100, 110, 210, 111, 211, 221, 321, 222, 322, 332, 333. Those corresponding to the above arrays are 110, 221, 322, 222, 332, 000, 333 (as mentioned, consider the sums of the columns of each array). - _Miquel A. Fiol_, Feb 06 2024

Links

Crossrefs

Row 3 of A188553.

Formula

Proved (for the number of sequences): a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (From this, the formulas below follow.) - Miquel A. Fiol, Feb 06 2024
a(n) = (1/6)*n^3 + (11/6)*n + 2 for n>=1.
G.f.: -(x^4 - 4*x^3 + 3*x^2 - 1)/(x - 1)^4. - Colin Barker, Mar 18 2012

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 10 2024

A188555 Number of 4 X n binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

Original entry on oeis.org

1, 5, 9, 16, 28, 48, 80, 129, 201, 303, 443, 630, 874, 1186, 1578, 2063, 2655, 3369, 4221, 5228, 6408, 7780, 9364, 11181, 13253, 15603, 18255, 21234, 24566, 28278, 32398, 36955, 41979, 47501, 53553, 60168, 67380, 75224, 83736, 92953, 102913, 113655, 125219
Offset: 0

Views

Author

R. H. Hardin, Apr 04 2011

Keywords

Comments

Row 4 of A188553.
From Miquel A. Fiol, Feb 06 2024: (Start)
a(n) is the number of words of length n, x(1)x(2)...x(n), on the alphabet {0,1,...4}, such that, for i=2,...,n, x(i)=either x(i-1) or x(i-1)-1.
For the bijection between arrays and words, notice that the i-th column consists of 1's and then 0's, and there are x(i)=0 to 4 of 1's.
The number of such words satisfy the recurrence given below and, hence, the empirical/conjectured formulas become true. (End)

Examples

			Some solutions for 4 X 3:
  1 1 1    1 1 1    1 1 1    1 1 1    1 0 0    1 1 1    1 1 0
  1 1 0    1 1 1    1 1 1    1 1 1    0 0 0    1 1 1    0 0 0
  0 0 0    1 1 0    0 0 0    1 1 1    0 0 0    1 1 1    0 0 0
  0 0 0    1 0 0    0 0 0    1 1 0    0 0 0    0 0 0    0 0 0
For these solutions, the corresponding words are 221, 432, 222, 443, 100, 333, 110. - _Miquel A. Fiol_, Feb 06 2024

Links

Crossrefs

Cf. A188553.

Formula

Empirical: a(n) = (1/24)*n^4 - (1/12)*n^3 + (23/24)*n^2 + (13/12)*n + 3.
Conjectures from Colin Barker, Apr 27 2018: (Start)
G.f.: -(2*x^5 - 7*x^4 + 11*x^3 - 6*x^2 + 1)/(x - 1)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 10 2024

A188556 Number of 5 X n binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

Original entry on oeis.org

6, 11, 20, 36, 64, 112, 192, 321, 522, 825, 1268, 1898, 2772, 3958, 5536, 7599, 10254, 13623, 17844, 23072, 29480, 37260, 46624, 57805, 71058, 86661, 104916, 126150, 150716, 178994, 211392, 248347, 290326, 337827, 391380, 451548, 518928, 594152
Offset: 1

Views

Author

R. H. Hardin, Apr 04 2011

Keywords

Comments

Row 5 of A188553.

Examples

			Some solutions for 5 X 3:
..1..1..0....1..1..1....1..1..1....0..0..0....1..1..1....1..1..1....1..1..1
..0..0..0....1..1..1....0..0..0....0..0..0....1..1..0....1..1..1....1..1..1
..0..0..0....1..1..1....0..0..0....0..0..0....1..0..0....1..1..1....1..1..1
..0..0..0....1..1..0....0..0..0....0..0..0....0..0..0....1..1..0....1..1..1
..0..0..0....1..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0

Links

Crossrefs

Cf. A188553.

Formula

Empirical: a(n) = (1/120)*n^5 - (1/24)*n^4 + (3/8)*n^3 + (1/24)*n^2 + (157/60)*n + 3.
Conjectures from Colin Barker, Apr 27 2018: (Start)
G.f.: x*(6 - 25*x + 44*x^2 - 39*x^3 + 18*x^4 - 3*x^5) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)

A188557 Number of 6 X n binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

Original entry on oeis.org

7, 13, 24, 44, 80, 144, 256, 448, 769, 1291, 2116, 3384, 5282, 8054, 12012, 17548, 25147, 35401, 49024, 66868, 89940, 119420, 156680, 203304, 261109, 332167, 418828, 523744, 649894, 800610, 979604, 1190996, 1439343, 1729669, 2067496, 2458876
Offset: 1

Views

Author

R. H. Hardin, Apr 04 2011

Keywords

Comments

Row 6 of A188553.

Examples

			Some solutions for 6 X 3:
..1..1..1....1..1..1....1..1..1....1..1..1....1..1..1....1..1..1....0..0..0
..1..1..1....1..1..1....1..1..0....1..1..1....1..1..1....1..1..1....0..0..0
..1..1..1....1..1..1....0..0..0....1..1..1....1..1..1....1..1..1....0..0..0
..1..1..0....1..1..0....0..0..0....1..1..1....1..1..1....1..1..1....0..0..0
..0..0..0....1..0..0....0..0..0....1..1..0....1..1..1....1..1..0....0..0..0
..0..0..0....0..0..0....0..0..0....1..0..0....1..1..1....0..0..0....0..0..0

Links

Crossrefs

Cf. A188553.

Formula

Empirical: a(n) = (1/720)*n^6 - (1/80)*n^5 + (17/144)*n^4 - (3/16)*n^3 + (497/360)*n^2 + (17/10)*n + 4.
Conjectures from Colin Barker, Apr 28 2018: (Start)
G.f.: x*(7 - 36*x + 80*x^2 - 96*x^3 + 66*x^4 - 24*x^5 + 4*x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A188558 Number of 7 X n binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

Original entry on oeis.org

8, 15, 28, 52, 96, 176, 320, 576, 1024, 1793, 3084, 5200, 8584, 13866, 21920, 33932, 51480, 76627, 112028, 161052, 227920, 317860, 437280, 593960, 797264, 1058373, 1390540, 1809368, 2333112, 2983006, 3783616, 4763220, 5954216, 7393559, 9123228
Offset: 1

Views

Author

R. H. Hardin, Apr 04 2011

Keywords

Comments

Row 7 of A188553.

Examples

			Some solutions for 7 X 3:
..1..1..1....1..1..1....1..1..1....1..1..1....1..1..1....1..1..1....1..1..1
..1..1..1....1..1..1....1..1..1....1..1..1....1..1..1....1..1..1....1..1..1
..1..1..1....1..1..0....1..1..1....1..1..1....1..1..1....1..1..1....1..1..1
..1..1..1....1..0..0....1..1..1....1..1..1....1..1..0....1..1..1....1..1..1
..1..1..1....0..0..0....1..1..0....1..1..1....0..0..0....1..1..1....1..1..1
..1..1..1....0..0..0....1..0..0....1..0..0....0..0..0....1..1..1....1..1..0
..0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....1..0..0....1..0..0

Links

Crossrefs

Cf. A188553.

Formula

Empirical: a(n) = (1/5040)*n^7 - (1/360)*n^6 + (11/360)*n^5 - (1/9)*n^4 + (427/720)*n^3 + (41/360)*n^2 + (709/210)*n + 4.
Conjectures from Colin Barker, Apr 28 2018: (Start)
G.f.: x*(8 - 49*x + 132*x^2 - 200*x^3 + 184*x^4 - 102*x^5 + 32*x^6 - 4*x^7) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.
(End)

A188559 Number of 8 X n binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

Original entry on oeis.org

9, 17, 32, 60, 112, 208, 384, 704, 1280, 2304, 4097, 7181, 12381, 20965, 34831, 56751, 90683, 142163, 218790, 330818, 491870, 719790, 1037650, 1474930, 2068890, 2866154, 3924527, 5315067, 7124435, 9457547, 12440553, 16224169, 20987389
Offset: 1

Views

Author

R. H. Hardin, Apr 04 2011

Keywords

Comments

Row 8 of A188553.

Examples

			Some solutions for 8 X 3:
..1..1..1....1..1..1....1..1..1....1..1..1....1..1..1....1..1..1....1..1..1
..1..1..1....1..1..0....1..1..1....1..1..1....1..1..1....1..1..1....1..1..1
..1..1..1....1..0..0....1..1..1....1..1..1....1..1..1....1..1..1....1..1..1
..1..1..1....0..0..0....1..1..1....1..1..1....1..1..1....1..1..0....1..1..1
..1..1..1....0..0..0....1..1..1....1..1..1....1..1..1....1..0..0....1..1..0
..1..1..0....0..0..0....1..1..1....1..0..0....1..1..1....0..0..0....1..0..0
..0..0..0....0..0..0....1..1..1....0..0..0....1..1..0....0..0..0....0..0..0
..0..0..0....0..0..0....1..0..0....0..0..0....0..0..0....0..0..0....0..0..0

Links

Crossrefs

Cf. A188553.

Formula

Empirical: a(n) = (1/40320)*n^8 - (1/2016)*n^7 + (19/2880)*n^6 - (7/180)*n^5 + (1247/5760)*n^4 - (85/288)*n^3 + (17911/10080)*n^2 + (1961/840)*n + 5.
Conjectures from Colin Barker, Apr 28 2018: (Start)
G.f.: x*(9 - 64*x + 203*x^2 - 372*x^3 + 430*x^4 - 320*x^5 + 150*x^6 - 40*x^7 + 5*x^8) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>9.
(End)

A210489 Array read by ascending antidiagonals where row n contains the second partial sums of row n of Pascal's triangle.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 8, 7, 5, 1, 6, 12, 12, 9, 6, 1, 7, 17, 20, 16, 11, 7, 1, 8, 23, 32, 28, 20, 13, 8, 1, 9, 30, 49, 48, 36, 24, 15, 9, 1, 10, 38, 72, 80, 64, 44, 28, 17, 10, 1, 11, 47, 102, 129, 112, 80, 52, 32, 19, 11, 1, 12, 57, 140, 201, 192, 144, 96, 60, 36, 21, 12
Offset: 0

Views

Author

Jakub Jaroslaw Ciaston, Jan 23 2013

Keywords

Comments

Appears to be a transposed version of A188553 with a leading column of 1's.

Examples

			Table starts:
1,  2,   3,     4,      5,      6,      7,      8,      9,     10
1,  3,   5,     7,      9,     11,     13,     15,     17,     19
1,  4,   8,    12,     16,     20,     24,     28,     32,     36
1,  5,  12,    20,     28,     36,     44,     52,     60,     68
1,  6,  17,    32,     48,     64,     80,     96,    112,    128
1,  7,  23,    49,     80,    112,    144,    176,    208,    240
1,  8,  30,    72,    129,    192,    256,    320,    384,    448
1,  9,  38,   102,    201,    321,    448,    576,    704,    832
1, 10,  47,   140,    303,    522,    769,   1024,   1280,   1536
1, 11,  57,   187,    443,    825,   1291,   1793,   2304,   2816
1, 12,  68,   244,    630,   1268,   2116,   3084,   4097,   5120
1, 13,  80,   312,    874,   1898,   3384,   5200,   7181,   9217
1, 14,  93,   392,   1186,   2772,   5282,   8584,  12381,  16398
1, 15, 107,   485,   1578,   3958,   8054,  13866,  20965,  28779
1, 16, 122,   592,   2063,   5536,  12012,  21920,  34831,  49744
1, 17, 138,   714,   2655,   7599,  17548,  33932,  56751,  84575
1, 18, 155,   852,   3369,  10254,  25147,  51480,  90683, 141326
1, 19, 173,  1007,   4221,  13623,  35401,  76627, 142163, 232009
1, 20, 192,  1180,   5228,  17844,  49024, 112028, 218790, 374172
1, 21, 212,  1372,   6408,  23072,  66868, 161052, 330818, 592962
1, 22, 233,  1584,   7780,  29480,  89940, 227920, 491870, 923780
1, 23, 255,  1817,   9364,  37260, 119420, 317860, 719790,1415650
1, 24, 278,  2072,  11181,  46624, 156680, 437280,1037650,2135440
1, 25, 302,  2350,  13253,  57805, 203304, 593960,1474930,3173090
1, 26, 327,  2652,  15603,  71058, 261109, 797264,2068890,4648020
1, 27, 353,  2979,  18255,  86661, 332167,1058373,2866154,6716910
1, 28, 380,  3332,  21234, 104916, 418828,1390540,3924527,9583064

Links

Crossrefs

Cf. A104734, A132379 (another transposed variant), A188553, A193605.

Programs

  • PARI
    T(n,m) = {sum(k=1, m, k*binomial(n,m-k))}
{ for(n=0, 10, for(m=1, 10, print1(T(n,m), ", ")); print) } \\ Andrew Howroyd, Apr 28 2020

Formula

T(n,k) = A193605(n,k).
T(n,m) = Sum_{k=1..m} k*binomial(n,m-k). - Vladimir Kruchinin, Apr 06 2018

Extensions

Offset corrected and terms a(55) and beyond from Andrew Howroyd, Apr 28 2020
Showing 1-7 of 7 results.

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