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.

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A208050 T(n,k)=Number of nXk 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 8, 8, 5, 14, 32, 44, 32, 14, 41, 128, 244, 244, 128, 41, 122, 512, 1356, 1904, 1356, 512, 122, 365, 2048, 7540, 14976, 14976, 7540, 2048, 365, 1094, 8192, 41932, 118096, 168096, 118096, 41932, 8192, 1094, 3281, 32768, 233204, 931968
Offset: 1

Views

Author

R. H. Hardin, Feb 22 2012

Keywords

Comments

Equivalently, the number of colorings in the rhombic hexagonal square grid graph RH_(n,k) using 4 colors up to permutation of the colors. - Andrew Howroyd, Jun 25 2017

Examples

			Table starts
...1....1......2.......5........14.........41..........122...........365
...1....2......8......32.......128........512.........2048..........8192
...2....8.....44.....244......1356.......7540........41932........233204
...5...32....244....1904.....14976.....118096.......931968.......7356288
..14..128...1356...14976....168096....1897888.....21472544.....243113056
..41..512...7540..118096...1897888...30818432....502504448....8206614784
.122.2048..41932..931968..21472544..502504448..11838995200..279733684992
.365.8192.233204.7356288.243113056.8206614784.279733684992.9578237457408
...
Some solutions for n=4 k=3
..0..1..0....0..1..2....0..1..0....0..1..0....0..1..2....0..1..2....0..1..2
..2..3..1....2..3..0....2..3..1....2..3..1....2..0..3....2..3..0....2..0..3
..1..2..0....0..1..2....0..2..3....0..2..3....1..2..1....1..2..1....1..2..0
..3..1..2....2..3..1....1..0..1....3..0..2....3..0..3....3..0..2....3..1..2

Links

Crossrefs

Columns 1-7 are A007051(n-2), A004171(n-2), A208044, A208046, A208047-A208049.
Main diagonal is A208045.
Cf. A208054, A068271, A212162, A212163.

A081654 a(n) = 2*4^n - 0^n.

Original entry on oeis.org

1, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832
Offset: 0

Views

Author

Paul Barry, Mar 26 2003

Keywords

Comments

Binomial transform of A081632. Inverse binomial transform of A081655.

Examples

			a(0) = 2*4^0 - 0^0 = 2 - 1 = 1 (use 0^0 = 1).

Links

Crossrefs

Cf. A000244 (3^n), A187093.
Essentially the same as A004171.

Programs

Formula

a(0)=1, a(n) = 2*4^n, n>0
G.f.: (1+4*x)/(1-4*x).
E.g.f. 2*exp(4*x)-1.
With interpolated zeros, this is 2^n - 0^n + (-2)^n. - Paul Barry, Sep 06 2003
a(n) = A081294(n+1), n>0. - R. J. Mathar, Sep 17 2008
For n>0, a(n) = 2 * (1 + 3^(n-1) + Sum{x=1..n-2}Sum{k=0..x-1}(binomial(x-1,k)*(3^(k+1) + 3^(n-x+k)))). - J. Conrad, Dec 10 2015

A085350 Binomial transform of poly-Bernoulli numbers A027649.

Original entry on oeis.org

1, 5, 23, 101, 431, 1805, 7463, 30581, 124511, 504605, 2038103, 8211461, 33022991, 132623405, 532087943, 2133134741, 8546887871, 34230598205, 137051532983, 548593552421, 2195536471151, 8785632669005, 35152991029223
Offset: 0

Views

Author

Paul Barry, Jun 24 2003

Keywords

Comments

Binomial transform is A085351.
a(n) mod 10 = period 4:repeat 1,5,3,1 = A132400. - Paul Curtz, Nov 13 2009

Links

Crossrefs

a(n-1) = A080643(n)/2 = A081674(n+1) - A081674(n).
Cf. A000244 (3^n).

Programs

  • Magma
    [2*4^n-3^n: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011
  • Mathematica
    LinearRecurrence[{4,9,-36},{1,5,23},30] (* Harvey P. Dale, Nov 30 2011 *)
LinearRecurrence[{7, -12},{1, 5},23] (* Ray Chandler, Aug 03 2015 *)

Formula

G.f.: (1-2x)/((1-3x)(1-4x)).
E.g.f.: 2exp(4x) - exp(3x).
a(n) = 2*4^n-3^n.
From Paul Curtz, Nov 13 2009: (Start)
a(n) = 4*a(n-1) + 9*a(n-2) - 36*a(n-3);
a(n) = 4*a(n-1) + 3^(n-1), both like A005061 (note for A005061 dual formula a(n) = 3*a(n-1) + 4^(n-1) = 3*a(n-1) + A000302(n-1)).
a(n) = 3*a(n-1) + 2^(2n+1) = 3*a(n-1) + A004171(n).
a(n) = A005061(n) + A000302(n).
b(n) = mix(A005061, A085350) = 0,1,1,5,7,23,... = differences of (A167762 = 0,0,1,2,7,14,37,...); b(n) differences = A167784. (End)

A298970 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 5 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 219, 128, 16, 32, 512, 1575, 1575, 512, 32, 64, 2048, 11283, 21098, 11283, 2048, 64, 128, 8192, 80972, 280468, 280468, 80972, 8192, 128, 256, 32768, 581057, 3740381, 6892031, 3740381, 581057, 32768, 256, 512, 131072
Offset: 1

Views

Author

R. H. Hardin, Jan 30 2018

Keywords

Comments

Table starts
...1.....2.......4.........8...........16.............32...............64
...2.....8......32.......128..........512...........2048.............8192
...4....32.....219......1575........11283..........80972...........581057
...8...128....1575.....21098.......280468........3740381.........49885231
..16...512...11283....280468......6892031......170137416.......4200575252
..32..2048...80972...3740381....170137416.....7785598672.....356356552103
..64..8192..581057..49885231...4200575252...356356552103...30241030285680
.128.32768.4169867.665351771.103715870545.16312003284843.2566506415636896

Examples

			Some solutions for n=5 k=4
..0..0..0..1. .0..0..1..1. .0..0..0..1. .0..0..1..0. .0..0..0..0
..1..1..0..1. .0..0..0..0. .1..0..0..0. .1..1..1..0. .0..0..1..0
..0..1..0..1. .0..1..1..1. .1..1..0..1. .1..1..0..1. .0..1..1..0
..0..0..0..1. .1..1..0..1. .1..1..1..0. .1..0..0..0. .1..1..0..1
..0..1..1..0. .0..0..1..0. .0..0..0..1. .1..1..1..0. .1..0..0..1

Links

Crossrefs

Column 1 is A000079(n-1).
Column 2 is A004171(n-1).

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: [order 8]
k=4: [order 23]
k=5: [order 75]

A299654 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4, 5 or 6 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 247, 128, 16, 32, 512, 1924, 1924, 512, 32, 64, 2048, 14981, 29408, 14981, 2048, 64, 128, 8192, 116654, 448993, 448993, 116654, 8192, 128, 256, 32768, 908360, 6856789, 13431706, 6856789, 908360, 32768, 256, 512, 131072
Offset: 1

Views

Author

R. H. Hardin, Feb 15 2018

Keywords

Comments

Table starts
...1......2........4...........8.............16...............32
...2......8.......32.........128............512.............2048
...4.....32......247........1924..........14981...........116654
...8....128.....1924.......29408.........448993..........6856789
..16....512....14981......448993.......13431706........401989538
..32...2048...116654.....6856789......401989538......23582064542
..64...8192...908360...104711327....12030404350....1383316377321
.128..32768..7073213..1599074414...360039414559...81146123707386
.256.131072.55077652.24419877459.10775053220325.4760069868306954

Examples

			Some solutions for n=5 k=4
..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..1
..0..1..1..1. .0..0..0..1. .1..0..0..0. .1..0..0..1. .0..0..1..0
..1..0..0..1. .0..1..1..0. .0..1..1..1. .1..1..0..0. .0..1..1..1
..1..0..1..0. .1..1..1..0. .0..0..0..0. .1..0..1..0. .1..0..0..1
..0..1..1..1. .1..0..1..0. .0..0..1..0. .0..0..0..0. .1..1..0..0

Links

Crossrefs

Column 1 is A000079(n-1).
Column 2 is A004171(n-1).

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: a(n) = 7*a(n-1) +5*a(n-2) +9*a(n-3) -a(n-4) -6*a(n-5)
k=4: [order 14]
k=5: [order 31]
k=6: [order 89]

A299661 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4, 5 or 7 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 227, 128, 16, 32, 512, 1642, 1642, 512, 32, 64, 2048, 11888, 22087, 11888, 2048, 64, 128, 8192, 86123, 297071, 297071, 86123, 8192, 128, 256, 32768, 624007, 4001253, 7411398, 4001253, 624007, 32768, 256, 512, 131072
Offset: 1

Views

Author

R. H. Hardin, Feb 15 2018

Keywords

Comments

Table starts
...1......2........4..........8............16..............32
...2......8.......32........128...........512............2048
...4.....32......227.......1642.........11888...........86123
...8....128.....1642......22087........297071.........4001253
..16....512....11888.....297071.......7411398.......185302633
..32...2048....86123....4001253.....185302633......8607101770
..64...8192...624007...53909088....4634931975....400012077773
.128..32768..4521433..726363190..115940148451..18591844096646
.256.131072.32761769.9787119222.2900256951630.864147943636240

Examples

			Some solutions for n=5 k=5
..0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0
..0..0..0..0..0. .0..0..1..1..0. .0..0..1..1..0. .0..1..0..0..0
..1..0..0..1..0. .1..1..1..1..0. .0..0..0..1..0. .0..0..1..1..1
..1..1..0..1..0. .1..1..0..0..0. .0..1..1..1..1. .0..1..1..0..0
..1..1..0..1..1. .0..0..0..1..0. .0..0..0..1..0. .0..0..1..1..1

Links

Crossrefs

Column 1 is A000079(n-1).
Column 2 is A004171(n-1).

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: [order 8]
k=4: [order 24]
k=5: [order 89]

A299740 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4, 5 or 8 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 220, 128, 16, 32, 512, 1578, 1578, 512, 32, 64, 2048, 11303, 21111, 11303, 2048, 64, 128, 8192, 81105, 280642, 280642, 81105, 8192, 128, 256, 32768, 582032, 3742524, 6896530, 3742524, 582032, 32768, 256, 512, 131072
Offset: 1

Views

Author

R. H. Hardin, Feb 18 2018

Keywords

Comments

Table starts
...1.....2.......4.........8...........16.............32...............64
...2.....8......32.......128..........512...........2048.............8192
...4....32.....220......1578........11303..........81105...........582032
...8...128....1578.....21111.......280642........3742524.........49914496
..16...512...11303....280642......6896530......170243005.......4203272237
..32..2048...81105...3742524....170243005.....7790212998.....356575568843
..64..8192..582032..49914496...4203272237...356575568843...30260326859957
.128.32768.4177161.665759775.103785926879.16322570202905.2568233775595684

Examples

			Some solutions for n=5 k=4
..0..0..0..1. .0..0..1..1. .0..0..0..0. .0..0..1..1. .0..0..0..0
..0..0..0..1. .0..0..0..1. .1..1..1..0. .0..0..0..1. .0..0..0..0
..0..0..0..1. .1..1..1..0. .0..0..0..1. .1..1..0..0. .0..0..0..0
..1..1..1..0. .0..0..0..0. .0..0..0..1. .1..0..1..0. .1..1..1..1
..1..1..1..0. .1..0..0..1. .0..0..0..1. .1..1..0..1. .0..1..1..1

Links

Crossrefs

Column 1 is A000079(n-1).
Column 2 is A004171(n-1).

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: [order 8]
k=4: [order 24]
k=5: [order 79]

A300182 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4, 5, 6 or 7 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 255, 128, 16, 32, 512, 2033, 2033, 512, 32, 64, 2048, 16208, 32321, 16208, 2048, 64, 128, 8192, 129217, 513832, 513832, 129217, 8192, 128, 256, 32768, 1030173, 8168705, 16288960, 8168705, 1030173, 32768, 256, 512, 131072
Offset: 1

Views

Author

R. H. Hardin, Feb 27 2018

Keywords

Comments

Table starts
...1......2........4...........8.............16...............32
...2......8.......32.........128............512.............2048
...4.....32......255........2033..........16208...........129217
...8....128.....2033.......32321.........513832..........8168705
..16....512....16208......513832.......16288960........516368256
..32...2048...129217.....8168705......516368256......32640586945
..64...8192..1030173...129863167....16369174784....2063278351093
.128..32768..8212978..2064518282...518912313824..130424025161538
.256.131072.65477359.32820974441.16449820393120.8244367994118153

Examples

			Some solutions for n=5 k=4
..0..0..0..1. .0..0..0..1. .0..0..0..0. .0..0..0..1. .0..0..0..1
..1..0..0..0. .0..0..1..0. .1..0..1..1. .0..1..1..1. .0..1..1..0
..1..1..1..0. .1..0..0..1. .0..1..1..1. .0..1..0..0. .1..0..0..1
..1..0..0..0. .1..0..0..1. .1..1..1..0. .1..0..0..0. .1..0..1..1
..1..1..0..0. .0..0..0..0. .1..1..1..0. .1..0..0..0. .1..1..1..0

Links

Crossrefs

Column 1 is A000079(n-1).
Column 2 is A004171(n-1).

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: a(n) = 7*a(n-1) +7*a(n-2) +6*a(n-3)
k=4: a(n) = 14*a(n-1) +27*a(n-2) +51*a(n-3) -10*a(n-4) -a(n-5) -10*a(n-6)
k=5: [order 9]
k=6: [order 15]
k=7: [order 36]

A300208 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4, 5, 6 or 8 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 248, 128, 16, 32, 512, 1933, 1933, 512, 32, 64, 2048, 15070, 29561, 15070, 2048, 64, 128, 8192, 117494, 451996, 451996, 117494, 8192, 128, 256, 32768, 916061, 6912249, 13548425, 6912249, 916061, 32768, 256, 512, 131072
Offset: 1

Views

Author

R. H. Hardin, Feb 28 2018

Keywords

Comments

Table starts
...1......2........4...........8.............16...............32
...2......8.......32.........128............512.............2048
...4.....32......248........1933..........15070...........117494
...8....128.....1933.......29561.........451996..........6912249
..16....512....15070......451996.......13548425........406228452
..32...2048...117494.....6912249......406228452......23883950854
..64...8192...916061...105708560....12180207076....1404241937017
.128..32768..7142233..1616600364...365209429387...82562348427139
.256.131072.55685704.24722667407.10950385677546.4854250928660105

Examples

			Some solutions for n=5 k=4
..0..0..0..1. .0..0..0..1. .0..0..0..0. .0..0..0..0. .0..0..0..1
..1..0..1..0. .1..0..0..0. .0..1..0..1. .0..1..0..1. .0..0..1..0
..0..1..1..0. .0..0..1..1. .1..0..1..1. .0..0..0..0. .0..0..1..0
..1..1..1..0. .0..0..0..0. .1..0..0..0. .1..0..0..0. .0..0..0..1
..0..1..0..0. .0..1..1..1. .0..1..1..1. .1..1..0..1. .1..0..1..0

Links

Crossrefs

Column 1 is A000079(n-1).
Column 2 is A004171(n-1).

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: a(n) = 8*a(n-1) -2*a(n-2) +5*a(n-3) -13*a(n-4) -6*a(n-5)
k=4: [order 15]
k=5: [order 38]

A300215 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4, 5, 7 or 8 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 32, 32, 8, 16, 128, 228, 128, 16, 32, 512, 1651, 1651, 512, 32, 64, 2048, 11965, 22194, 11965, 2048, 64, 128, 8192, 86775, 298600, 298600, 86775, 8192, 128, 256, 32768, 629440, 4023881, 7452385, 4023881, 629440, 32768, 256, 512, 131072
Offset: 1

Views

Author

R. H. Hardin, Feb 28 2018

Keywords

Comments

Table starts
...1......2........4..........8............16..............32
...2......8.......32........128...........512............2048
...4.....32......228.......1651.........11965...........86775
...8....128.....1651......22194........298600.........4023881
..16....512....11965.....298600.......7452385.......186427449
..32...2048....86775....4023881.....186427449......8664017314
..64...8192...629440...54246856....4666136435....402932952786
.128..32768..4566023..731384148..116802113451..18741286700978
.256.131072.33122989.9861234001.2923892905217.871742323938453

Examples

			Some solutions for n=5 k=4
..0..0..0..0. .0..0..0..0. .0..0..0..0. .0..0..0..1. .0..0..0..0
..0..1..0..1. .0..0..1..1. .1..1..1..1. .1..1..0..1. .0..1..0..0
..0..0..1..0. .1..0..1..0. .1..0..1..1. .1..1..0..1. .1..1..1..0
..0..1..1..0. .0..0..1..0. .0..0..0..0. .0..0..0..1. .0..1..1..1
..0..0..1..1. .0..0..0..0. .0..0..1..0. .1..0..0..0. .0..0..1..0

Links

Crossrefs

Column 1 is A000079(n-1).
Column 2 is A004171(n-1).

Formula

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1)
k=3: [order 8]
k=4: [order 24]
k=5: [order 89]
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