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 11-16 of 16 results.

A197431 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 4,0,1,1,1 for x=0,1,2,3,4.

Original entry on oeis.org

1, 3, 3, 4, 9, 4, 5, 20, 20, 5, 9, 71, 68, 71, 9, 16, 227, 448, 448, 227, 16, 25, 664, 2152, 5197, 2152, 664, 25, 39, 2107, 10684, 43909, 43909, 10684, 2107, 39, 64, 6675, 55567, 410456, 720447, 410456, 55567, 6675, 64, 105, 20696, 280544, 4017895, 11769456
Offset: 1

Views

Author

R. H. Hardin Oct 14 2011

Keywords

Comments

Every 0 is next to 0 4's, every 1 is next to 1 0's, every 2 is next to 2 1's, every 3 is next to 3 1's, every 4 is next to 4 1's
Table starts
...1.....3.......4..........5............9..............16................25
...3.....9......20.........71..........227.............664..............2107
...4....20......68........448.........2152...........10684.............55567
...5....71.....448.......5197........43909..........410456...........4017895
...9...227....2152......43909.......720447........11769456.........195632425
..16...664...10684.....410456.....11769456.......337295264........9869104795
..25..2107...55567....4017895....195632425......9869104795......529968166606
..39..6675..280544...37566429...3187783369....292706228390....27927270266251
..64.20696.1425111..351740354..53026008675...8642552356445..1461399711177586
.105.65029.7280350.3301443327.881220874951.253114165333694.76357681634837889

Examples

			Some solutions containing all values 0 to 4 for n=6 k=4
..1..2..1..0....1..1..0..1....0..0..0..1....1..0..0..1....0..1..1..0
..0..2..1..0....0..2..0..1....0..2..1..3....3..1..1..1....2..1..4..1
..0..1..3..0....0..1..3..1....0..1..4..1....1..4..1..0....1..0..1..1
..1..4..1..1....1..4..1..0....0..3..1..0....0..1..4..1....2..1..3..0
..1..1..0..1....1..1..3..0....2..1..2..0....2..1..1..2....0..1..1..2
..0..2..1..2....0..0..1..1....1..0..0..0....1..0..0..0....1..2..0..1

Links

Crossrefs

Column 1 is A195971
Column 2 is A197403

A197679 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 2,0,1,0,1 for x=0,1,2,3,4.

Original entry on oeis.org

1, 3, 3, 4, 9, 4, 5, 16, 16, 5, 9, 33, 57, 33, 9, 16, 73, 202, 202, 73, 16, 25, 160, 721, 1111, 721, 160, 25, 39, 361, 2557, 5639, 5639, 2557, 361, 39, 64, 835, 8930, 29190, 45162, 29190, 8930, 835, 64, 105, 1966, 31326, 153715, 358209, 358209, 153715, 31326, 1966
Offset: 1

Views

Author

R. H. Hardin Oct 17 2011

Keywords

Comments

Every 0 is next to 0 2's, every 1 is next to 1 0's, every 2 is next to 2 1's, every 3 is next to 3 0's, every 4 is next to 4 1's
Table starts
...1....3......4........5..........9..........16............25..............39
...3....9.....16.......33.........73.........160...........361.............835
...4...16.....57......202........721........2557..........8930...........31326
...5...33....202.....1111.......5639.......29190........153715..........817865
...9...73....721.....5639......45162......358209.......2871646........22955155
..16..160...2557....29190.....358209.....4268977......51167876.......612121676
..25..361...8930...153715....2871646....51167876.....924904506.....16683567785
..39..835..31326...817865...22955155...612121676...16683567785....452341382317
..64.1966.110217..4363976..183536379..7361707007..302284521370..12312981845426
.105.4703.387627.23310791.1469158622.88656931195.5474405589906.335130309920001

Examples

			Some solutions containing all values 0 to 4 for n=6 k=4
..1..0..0..0....1..1..1..1....1..2..2..1....2..1..0..0....0..0..0..1
..1..0..3..0....0..0..0..0....0..1..1..0....1..4..1..1....3..0..3..2
..1..1..1..0....0..3..0..0....0..1..4..1....0..1..1..1....0..0..0..1
..0..1..1..0....0..1..1..1....0..1..1..1....0..3..0..0....1..1..1..1
..1..1..4..1....1..4..1..1....0..3..0..0....3..0..3..1....1..4..1..0
..1..0..1..2....2..1..0..0....0..0..0..0....0..0..0..1....0..1..1..0

Links

Crossrefs

Column 1 is A195971
Column 2 is A197531

A266067 T(n,k)=Number of nXk integer arrays with each element equal to the number of horizontal, vertical, diagonal and antidiagonal neighbors exactly one smaller than itself.

Original entry on oeis.org

1, 3, 3, 4, 17, 4, 5, 35, 35, 5, 9, 89, 34, 89, 9, 16, 323, 81, 81, 323, 16, 25, 1057, 809, 525, 809, 1057, 25, 39, 3027, 1750, 8262, 8262, 1750, 3027, 39, 64, 9257, 5071, 25663, 154939, 25663, 5071, 9257, 64, 105, 29835, 29256, 148323, 1066148, 1066148, 148323
Offset: 1

Views

Author

R. H. Hardin, Dec 20 2015

Keywords

Comments

Table starts
...1.....3......4.......5.........9........16........25.........39.........64
...3....17.....35......89.......323......1057......3027.......9257......29835
...4....35.....34......81.......809......1750......5071......29256......93170
...5....89.....81.....525......8262.....25663....148323....1480037....7361668
...9...323....809....8262....154939...1066148..11753249..164649556.1624261580
..16..1057...1750...25663...1066148..11211992.156829155.3678276863
..25..3027...5071..148323..11753249.156829155
..39..9257..29256.1480037.164649556
..64.29835..93170.7361668
.105.92561.300067

Examples

			Some solutions for n=4 k=4
..0..0..0..1....2..1..0..0....1..0..0..1....1..0..0..0....1..1..2..0
..0..2..3..2....1..3..2..0....2..2..2..2....1..2..3..0....0..2..2..1
..1..1..2..1....2..0..3..1....1..3..3..1....2..2..2..0....0..3..2..2
..2..3..2..0....1..2..1..2....0..1..1..0....0..1..2..1....0..0..0..1

Links

Crossrefs

Column 1 is A195971.

Formula

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-3) +a(n-4)
k=2: [order 18]
k=3: [order 78]

A303111 Array read by antidiagonals: T(m,n) = number of total dominating sets in the grid graph P_m X P_n.

Original entry on oeis.org

0, 1, 1, 3, 9, 3, 4, 25, 25, 4, 5, 81, 161, 81, 5, 9, 289, 961, 961, 289, 9, 16, 961, 6235, 11236, 6235, 961, 16, 25, 3249, 39601, 137641, 137641, 39601, 3249, 25, 39, 11025, 251433, 1677025, 3270375, 1677025, 251433, 11025, 39
Offset: 1

Views

Author

Andrew Howroyd, Apr 18 2018

Keywords

Comments

Equivalently, the number of n X m binary matrices with every element adjacent to some 0 horizontally or vertically.

Examples

			Table begins:
=======================================================================
m\n|  1    2      3        4          5            6              7
---|-------------------------------------------------------------------
1  |  0    1      3        4          5            9             16 ...
2  |  1    9     25       81        289          961           3249 ...
3  |  3   25    161      961       6235        39601         251433 ...
4  |  4   81    961    11236     137641      1677025       20430400 ...
5  |  5  289   6235   137641    3270375     76405081     1783064069 ...
6  |  9  961  39601  1677025   76405081   3416753209   152598828321 ...
7  | 16 3249 251433 20430400 1783064069 152598828321 13057656650476 ...
...

Links

Crossrefs

Rows 1..2 are A195971(n-1), A141583(n+1).
Main diagonal is A133793.
Cf. A218354 (dominating sets), A291872 (connected dominating sets).
Cf. A303114 (king graph), A303118 (minimal total dominating sets).

A303114 Array read by antidiagonals: T(m,n) = number of total dominating sets in the n X m king graph.

Original entry on oeis.org

0, 1, 1, 3, 11, 3, 4, 47, 47, 4, 5, 165, 353, 165, 5, 9, 625, 2545, 2545, 625, 9, 16, 2435, 19651, 35458, 19651, 2435, 16, 25, 9367, 150719, 538977, 538977, 150719, 9367, 25, 39, 35901, 1149593, 8213971, 16322279, 8213971, 1149593, 35901, 39
Offset: 1

Views

Author

Andrew Howroyd, Apr 18 2018

Keywords

Examples

			Table begins:
============================================================================
m\n|  1    2       3         4           5             6               7
---|------------------------------------------------------------------------
1  |  0    1       3         4           5             9              16 ...
2  |  1   11      47       165         625          2435            9367 ...
3  |  3   47     353      2545       19651        150719         1149593 ...
4  |  4  165    2545     35458      538977       8213971       124153394 ...
5  |  5  625   19651    538977    16322279     496873689     14980146565 ...
6  |  9 2435  150719   8213971   496873689   30158547693   1812834702647 ...
7  | 16 9367 1149593 124153394 14980146565 1812834702647 217221533288240 ...
...

Links

Crossrefs

Rows 1..2 are A195971(n-1), A219079.
Main diagonal is A303116.
Cf. A218663 (dominating sets), A291873 (connected dominating sets).
Cf. A303111 (grid graph).

A362388 a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 7.

Original entry on oeis.org

1, 2, 5, 7, 10, 17, 29, 46, 73, 119, 194, 313, 505, 818, 1325, 2143, 3466, 5609, 9077, 14686, 23761, 38447, 62210, 100657, 162865, 263522, 426389, 689911, 1116298, 1806209, 2922509, 4728718, 7651225, 12379943, 20031170, 32411113, 52442281, 84853394
Offset: 0

Views

Author

Greg Dresden and Jiaqi Wang, Jun 18 2023

Keywords

Comments

For n >= 3, a(n) is also the number of ways to tile this "central staircase" figure of length n with squares and dominoes; this is the picture for length n=10:
_
_______|_|_____
|||_|||_|||_|_|
|_|

Examples

			Here is one of the a(10)=194 tilings for length n=10:
             _
   _________|_|_______
  |___|___| |___|_|___|
          |_|

Links

Crossrefs

Cf. A000032, A000045, A195971, A295681, A347902.

Programs

  • Mathematica
    LinearRecurrence[{1, 0, 1, 1}, { 1, 2, 5, 7}, 50]

Formula

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 7.
G.f.: (1 + x + 3*x^2 + x^3)/((1 +x^2)*(1-x-x^2)).
a(2*n) = F(n+2)^2 + F(n)^2, a(2*n+1) = F(n+2)^2 + F(n+1)*L(n+1) for F(n) and L(n) the Fibonacci and Lucas numbers.
Sum_{k=0..n} a(k) = A295681(n+5) - 3.
5*a(n) = 3*A000032(n+2) -2*A000034(n+1)*(-1)^floor(n/2). - R. J. Mathar, Jun 22 2023
a(n)+a(n+2) = 3*A000045(n+3). - R. J. Mathar, Jun 22 2023
Previous Showing 11-16 of 16 results.

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

Content is available under The OEIS End-User License Agreement.