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 1-10 of 23 results. Next

A240513 Number of n X 2 0..1 arrays with no element equal to exactly two horizontal and vertical neighbors, with new values 0..1 introduced in row major order.

Original entry on oeis.org

2, 3, 6, 10, 21, 42, 86, 179, 370, 770, 1601, 3330, 6930, 14419, 30006, 62442, 129941, 270410, 562726, 1171043, 2436962, 5071362, 10553601, 21962242, 45703842, 95110563, 197926886, 411889610, 857150101, 1783745642, 3712008566, 7724760339
Offset: 1

Views

Author

R. H. Hardin, Apr 06 2014

Keywords

Comments

Column 2 of A240519.

Examples

			All solutions for n=4:
..0..1....0..1....0..1....0..1....0..0....0..1....0..1....0..1....0..1....0..1
..0..0....0..1....1..0....0..1....1..1....1..0....1..1....1..0....1..0....1..0
..0..1....1..0....1..1....1..0....0..0....1..0....0..1....0..1....0..0....0..1
..1..0....1..0....1..0....0..1....1..1....0..1....1..0....1..0....1..0....0..1

Links

Crossrefs

Cf. A240519.

Formula

Empirical: a(n) = 2*a(n-1) + a(n-2) - a(n-3) - 2*a(n-4) + a(n-5).
Empirical g.f.: x*(2 - x)*(1 - x^2 - 2*x^3) / ((1 - x)*(1 - x - 2*x^2 - x^3 + x^4)). - Colin Barker, Feb 24 2018
Empirical: a(n) = 1+A105309(n). - R. J. Mathar, Nov 09 2018

A189435 T(n,k)=Number of nXk array permutations with each element not moving, or moving one space N, SW or SE.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 5, 9, 9, 1, 1, 8, 29, 31, 20, 1, 1, 13, 65, 140, 109, 41, 1, 1, 21, 181, 571, 841, 367, 85, 1, 1, 34, 441, 2413, 5680, 4653, 1245, 178, 1, 1, 55, 1165, 10069, 40065, 52241, 26589, 4247, 369, 1, 1, 89, 2929, 42205, 278105, 606201, 493941
Offset: 1

Views

Author

R. H. Hardin Apr 22 2011

Keywords

Comments

Table starts
.1...1.....1.......1.........1...........1.............1...............1
.1...2.....3.......5.........8..........13............21..............34
.1...5.....9......29........65.........181...........441............1165
.1...9....31.....140.......571........2413.........10069...........42205
.1..20...109.....841......5680.......40065........278105.........1940868
.1..41...367....4653.....52241......606201.......6944573........79826592
.1..85..1245...26589....493941.....9557077.....181540773......3467525301
.1.178..4247..151081...4681376...150278792....4742833745....150293731826
.1.369.14453..859264..44341381..2367212857..124239687001...6540976400913
.1.769.49167.4891841.420325171.37358187521.3261208487441.285499775348185

Examples

			Some solutions for 5X3
..0..4..5....0..4..5....3..1..2....0..1..5....0..4..5....0..4..5....0..4..5
..1..2..8....3..2..1....6..0..5....6..2..8....6..2..1....1..2..8....6..2..1
..9..3.11....6.10..8....4..7..8....4..3.11....9..3.11....6..3.11....9..3..8
..7..6.14....9.13..7....9.13.14....7.13.14....7..8.14....7.13.14....7.13.14
.12.13.10...12.11.14...12.11.10...12..9.10...12.13.10...12..9.10...12.11.10

Links

Crossrefs

Column 2 is A105309
Row 2 is A000045(n+1)
Row 3 is A006131

A297582 T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally, diagonally or antidiagonally adjacent to 1 or 4 neighboring 1s.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 11, 9, 1, 6, 17, 36, 20, 1, 9, 39, 72, 102, 41, 1, 13, 93, 188, 254, 370, 85, 1, 19, 183, 688, 1017, 1104, 1243, 178, 1, 28, 373, 2085, 5263, 5800, 4428, 3854, 369, 1, 41, 823, 5497, 20771, 47968, 31171, 17549, 13078, 769, 1, 60, 1741, 16037, 76340
Offset: 1

Views

Author

R. H. Hardin, Jan 01 2018

Keywords

Comments

Table starts
.1...2.....3......4.......6.........9.........13..........19............28
.1...5....11.....17......39........93........183.........373...........823
.1...9....36.....72.....188.......688.......2085........5497.........16037
.1..20...102....254....1017......5263......20771.......76340........320326
.1..41...370...1104....5800.....47968.....284289.....1400065.......8274627
.1..85..1243...4428...31171....395011....3355439....21941552.....181405030
.1.178..3854..17549..171543...3230902...38609160...348140132....4059598106
.1.369.13078..71541..945046..27481626..476513137..5752782514...94310855136
.1.769.43861.288624.5175491.229676841.5731862594.92802629660.2136636243308

Examples

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

Links

Crossrefs

Column 2 is A105309(n+1).
Row 1 is A000930(n+1).

Formula

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3) -a(n-4)
k=3: a(n) = a(n-1) +2*a(n-2) +19*a(n-3) +4*a(n-4) -17*a(n-5) -8*a(n-6)
k=4: [order 16]
k=5: [order 30]
k=6: [order 57]
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-3)
n=2: a(n) = a(n-1) +4*a(n-3) +2*a(n-4)
n=3: a(n) = a(n-1) +a(n-2) +8*a(n-3) +18*a(n-4) +a(n-5) -11*a(n-6) -12*a(n-7) -a(n-8)
n=4: [order 17]
n=5: [order 41]
n=6: [order 94]

A297395 T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally, diagonally or antidiagonally adjacent to 1 neighboring 1.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 9, 9, 1, 6, 13, 19, 20, 1, 9, 33, 37, 57, 41, 1, 13, 69, 127, 126, 139, 85, 1, 19, 121, 323, 700, 385, 369, 178, 1, 28, 253, 763, 2569, 3175, 1243, 963, 369, 1, 41, 529, 2121, 7779, 14940, 15541, 3924, 2489, 769, 1, 60, 1013, 5557, 31081, 58901, 99682
Offset: 1

Views

Author

R. H. Hardin, Dec 29 2017

Keywords

Comments

Table starts
.1...2....3.....4.......6........9........13.........19...........28
.1...5....9....13......33.......69.......121........253..........529
.1...9...19....37.....127......323.......763.......2121.........5557
.1..20...57...126.....700.....2569......7779......31081.......117084
.1..41..139...385....3175....14940.....58901.....325922......1616869
.1..85..369..1243...15541....99682....514945....3977868.....27131403
.1.178..963..3924...74736...640562...4279111...46261441....428200086
.1.369.2489.12477..358341..4101278..35870939..540319235...6780786267
.1.769.6523.39625.1729617.26607999.302197213.6362528482.108762242579

Examples

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

Links

Crossrefs

Column 2 is A105309(n+1).
Row 1 is A000930(n+1).
Row 2 is A089977(n+1).

Formula

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3) -a(n-4)
k=3: a(n) = a(n-1) +4*a(n-2) +2*a(n-3) -4*a(n-4)
k=4: a(n) = a(n-1) +6*a(n-2) +4*a(n-3) -3*a(n-4) -a(n-5) -2*a(n-6) -a(n-7)
k=5: [order 20]
k=6: [order 25]
k=7: [order 55]
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-3)
n=2: a(n) = a(n-1) +4*a(n-3)
n=3: a(n) = a(n-1) +a(n-2) +8*a(n-3) +a(n-4) -2*a(n-6)
n=4: [order 8]
n=5: [order 21]
n=6: [order 31]
n=7: [order 69]

A297595 T(n,k) = Number of n X k 0..1 arrays with every 1 horizontally, diagonally or antidiagonally adjacent to 1 or 5 neighboring 1s.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 9, 9, 1, 6, 13, 25, 20, 1, 9, 33, 49, 69, 41, 1, 13, 69, 145, 154, 205, 85, 1, 19, 121, 443, 752, 577, 597, 178, 1, 28, 253, 1141, 3145, 3747, 1977, 1701, 369, 1, 41, 529, 3009, 10131, 23066, 18577, 6962, 4949, 769, 1, 60, 1013, 8455, 37929, 103673
Offset: 1

Views

Author

R. H. Hardin, Jan 01 2018

Keywords

Comments

Table starts
.1...2.....3.....4.......6........9........13..........19...........28
.1...5.....9....13......33.......69.......121.........253..........529
.1...9....25....49.....145......443......1141........3009.........8455
.1..20....69...154.....752.....3145.....10131.......37929.......150388
.1..41...205...577....3747....23066....103673......514290......2834897
.1..85...597..1977...18577...163704....975485.....6551844.....50398161
.1.178..1701..6962...93150..1172288...9403199....85828150....919035936
.1.369..4949.24441..464697..8419996..90862063..1120526916..16723808887
.1.769.14389.85803.2320289.60354437.875241087.14592832760.303459238317

Examples

			Some solutions for n=6 k=4
..0..0..0..0. .0..1..0..0. .1..0..0..0. .1..0..0..0. .0..0..1..0
..0..1..1..0. .0..0..1..0. .0..1..0..0. .0..1..0..0. .0..1..0..0
..0..0..0..0. .0..0..0..0. .0..1..0..0. .0..0..0..0. .0..0..0..0
..0..1..0..1. .0..0..1..1. .1..0..0..0. .0..0..0..0. .0..0..0..0
..0..1..1..1. .0..0..0..0. .1..0..0..0. .0..0..0..0. .1..0..0..0
..0..0..0..1. .1..1..0..0. .0..1..0..0. .0..0..0..0. .0..1..0..0

Links

Crossrefs

Column 2 is A105309(n+1).
Row 1 is A000930(n+1).
Row 2 is A089977(n+1).

Formula

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3) -a(n-4)
k=3: a(n) = a(n-1) +2*a(n-2) +10*a(n-3) +4*a(n-4) -8*a(n-5) -8*a(n-6)
k=4: [order 9]
k=5: [order 22]
k=6: [order 40]
k=7: [order 83]
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-3)
n=2: a(n) = a(n-1) +4*a(n-3)
n=3: a(n) = a(n-1) +a(n-2) +8*a(n-3) +7*a(n-4) -8*a(n-6) -6*a(n-7)
n=4: [order 12]
n=5: [order 26]
n=6: [order 49]

A196957 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 2,1,3,4,0 for x=0,1,2,3,4.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 9, 4, 6, 20, 26, 20, 6, 9, 41, 87, 87, 41, 9, 13, 85, 282, 492, 282, 85, 13, 19, 178, 919, 2687, 2687, 919, 178, 19, 28, 369, 2987, 14509, 23956, 14509, 2987, 369, 28, 41, 769, 9722, 78717, 214124, 214124, 78717, 9722, 769, 41, 60, 1600, 31643
Offset: 1

Views

Author

R. H. Hardin Oct 08 2011

Keywords

Comments

Every 0 is next to 0 2's, every 1 is next to 1 1's, every 2 is next to 2 3's, every 3 is next to 3 4's, every 4 is next to 4 0's
Table starts
..1....2......3........4..........6............9.............13
..2....5......9.......20.........41...........85............178
..3....9.....26.......87........282..........919...........2987
..4...20.....87......492.......2687........14509..........78717
..6...41....282.....2687......23956.......214124........1918608
..9...85....919....14509.....214124......3166711.......46887039
.13..178...2987....78717....1918608.....46887039.....1147966466
.19..369...9722...427700...17197531....695103098....28133304588
.28..769..31643..2320738..154075730..10297360567...688970472958
.41.1600.102962.12593583.1380294235.152533682507.16871844816740

Examples

			Some solutions for n=6 k=4
..0..0..0..0....0..1..0..0....1..1..0..0....0..1..0..0....0..0..0..0
..1..0..1..1....0..1..0..0....0..0..1..1....0..1..0..0....0..0..1..0
..1..0..0..0....0..0..0..0....1..1..0..0....1..0..0..0....0..0..1..0
..0..0..1..1....0..0..0..0....0..0..4..0....1..0..4..0....0..0..0..1
..1..0..0..0....0..1..0..0....0..0..0..0....0..4..0..0....1..1..0..1
..1..0..0..0....0..1..0..0....0..1..1..0....0..0..1..1....0..0..0..0

Links

Crossrefs

Column 1 is A000930(n+1)
Column 2 is A105309(n+1)

A295120 T(n,k)=Number of nXk 0..1 arrays with each 1 horizontally or vertically adjacent to 1 or 4 1s.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 9, 4, 6, 20, 26, 20, 6, 9, 41, 77, 77, 41, 9, 13, 85, 226, 326, 226, 85, 13, 19, 178, 665, 1373, 1373, 665, 178, 19, 28, 369, 1960, 5793, 8257, 5793, 1960, 369, 28, 41, 769, 5769, 24347, 49302, 49302, 24347, 5769, 769, 41, 60, 1600, 16983, 102398
Offset: 1

Views

Author

R. H. Hardin, Nov 15 2017

Keywords

Comments

Table starts
..1...2.....3......4........6.........9.........13...........19............28
..2...5.....9.....20.......41........85........178..........369...........769
..3...9....26.....77......226.......665.......1960.........5769.........16983
..4..20....77....326.....1373......5793......24347.......102398........431050
..6..41...226...1373.....8257.....49302.....295083......1768323......10586331
..9..85...665...5793....49302....420519....3590821.....30650456.....261518933
.13.178..1960..24347...295083...3590821...43655680....530696748....6452840307
.19.369..5769.102398..1768323..30650456..530696748...9196006628..159316011413
.28.769.16983.431050.10586331.261518933.6452840307.159316011413.3931962260999

Examples

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

Links

Crossrefs

Column 1 is A000930(n+1).
Column 2 is A105309(n+1).

Formula

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-3)
k=2: a(n) = a(n-1) +2*a(n-2) +a(n-3) -a(n-4)
k=3: [order 10]
k=4: [order 14]
k=5: [order 40]
k=6: [order 78]

A196436 T(n,k) = Number of n X k 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 2,1,0,3,4 for x=0,1,2,3,4.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 9, 4, 6, 20, 25, 20, 6, 9, 41, 75, 75, 41, 9, 13, 85, 213, 314, 213, 85, 13, 19, 178, 621, 1283, 1283, 621, 178, 19, 28, 369, 1801, 5311, 7358, 5311, 1801, 369, 28, 41, 769, 5219, 21803, 42604, 42604, 21803, 5219, 769, 41, 60, 1600, 15133, 89640
Offset: 1

Views

Author

R. H. Hardin, Oct 02 2011

Keywords

Comments

Every 0 is next to zero 2's, every 1 is next to one 1's, every 2 is next to two 2's, every 3 is next to three 3's, every 4 is next to four 4's.
Table starts:
..1....2.....3.......4........6..........9..........13............19
..2....5.....9......20.......41.........85.........178...........369
..3....9....25......75......213........621........1801..........5219
..4...20....75.....314.....1283.......5311.......21803.........89640
..6...41...213....1283.....7358......42604......246463.......1427278
..9...85...621....5311....42604.....349511.....2856711......23313460
.13..178..1801...21803...246463....2856711....32855368.....378154308
.19..369..5219...89640..1427278...23313460...378154308....6141909014
.28..769.15133..369032..8257394..190423461..4356726770...99754551113
.41.1600.43867.1517961.47782109.1555501171.50163397373.1619739781963

Examples

			Some solutions for n=6, k=4:
..0..0..1..1....1..0..1..1....0..1..0..0....0..0..0..0....1..1..0..0
..0..1..0..0....1..0..0..0....0..1..0..0....0..0..1..1....0..0..1..1
..0..1..0..1....0..1..0..1....0..0..0..0....0..0..0..0....0..0..0..0
..0..0..0..1....0..1..0..1....0..0..0..0....0..0..0..0....0..0..0..0
..0..0..0..0....1..0..0..0....1..1..0..0....0..0..1..1....1..1..0..0
..1..1..0..0....1..0..1..1....0..0..0..0....0..0..0..0....0..0..1..1

Links

Crossrefs

Column 1 is A000930(n+1).
Column 2 is A105309(n+1).
Diagonal is A145773.

A101400 a(n) = a(n-1) + 2*a(n-2) + a(n-3) - a(n-4).

Original entry on oeis.org

1, 2, 5, 10, 21, 44, 91, 190, 395, 822, 1711, 3560, 7409, 15418, 32085, 66770, 138949, 289156, 601739, 1252230, 2605915, 5422958, 11285279, 23484880, 48872481, 101704562, 211649125, 440445850, 916576181, 1907412444, 3969361531
Offset: 0

Views

Author

Jeroen F.J. Laros, Jan 15 2005

Keywords

Comments

Lengths of successive words (starting with a) under the substitution: {a -> ab, b -> aac, c -> d, d -> b}.

Examples

			a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 10, a(4) = 21, a(5) = 44

Links

Crossrefs

Cf. A092886, A105309.

Programs

  • GAP
    a:=[1,2,5,10];; for n in [5..35] do a[n]:=a[n-1]+2*a[n-2]+a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Apr 03 2018
  • Magma
    I:=[1,2,5,10]; [n le 4 select I[n] else Self(n-1) + 2*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // G. C. Greubel, Apr 03 2018
  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x+x^2)/(1-x-2*x^2-x^3+x^4))); // G. C. Greubel, Apr 03 2018
  • Mathematica
    a[0] = 1; a[1] = 2; a[2] = 5; a[3] = 10; a[n_] := a[n] = a[n - 1] + 2a[n - 2] + a[n - 3] - a[n - 4]; Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 15 2005 *)
LinearRecurrence[{1,2,1,-1},{1,2,5,10},40] (* Harvey P. Dale, Oct 24 2017 *)
  • PARI
    x='x+O('x^30); Vec((1+x+x^2)/(1-x-2*x^2-x^3+x^4)) \\ G. C. Greubel, Apr 03 2018

Formula

G.f.: (1+x+x^2)/(1-x-2*x^2-x^3+x^4). - G. C. Greubel, Apr 03 2018

Extensions

More terms from Robert G. Wilson v, Jan 15 2005

A197199 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 4,1,3,2,0 for x=0,1,2,3,4.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 9, 4, 6, 20, 25, 20, 6, 9, 41, 75, 75, 41, 9, 13, 85, 213, 314, 213, 85, 13, 19, 178, 621, 1283, 1283, 621, 178, 19, 28, 369, 1801, 5311, 7363, 5311, 1801, 369, 28, 41, 769, 5219, 21803, 42664, 42664, 21803, 5219, 769, 41, 60, 1600, 15133, 89640
Offset: 1

Views

Author

R. H. Hardin Oct 11 2011

Keywords

Comments

Every 0 is next to 0 4's, every 1 is next to 1 1's, every 2 is next to 2 3's, every 3 is next to 3 2's, every 4 is next to 4 0's
Table starts
..1....2.....3.......4........6..........9..........13............19
..2....5.....9......20.......41.........85.........178...........369
..3....9....25......75......213........621........1801..........5219
..4...20....75.....314.....1283.......5311.......21803.........89640
..6...41...213....1283.....7363......42664......247053.......1431868
..9...85...621....5311....42664.....350711.....2873141......23497244
.13..178..1801...21803...247053....2873141....33176391.....383262156
.19..369..5219...89640..1431868...23497244...383262156....6258308430
.28..769.15133..369032..8291425..192348965..4432780869..102222467201
.41.1600.43867.1517961.48020216.1574634137.51234398070.1669087121069

Examples

			Some solutions for n=6 k=4
..0..1..1..0....1..1..0..1....0..0..0..0....0..1..0..0....1..0..0..1
..1..0..0..1....0..0..0..1....0..1..1..0....0..1..0..0....1..0..0..1
..1..0..0..1....0..0..0..0....1..0..0..0....0..0..0..0....0..0..0..0
..0..0..0..0....1..0..0..0....1..0..0..1....1..1..0..0....1..1..0..0
..0..0..0..0....1..0..0..0....0..1..0..1....0..0..0..1....0..0..0..0
..0..0..0..0....0..0..1..1....0..1..0..0....0..0..0..1....0..0..1..1

Links

Crossrefs

Column 1 is A000930(n+1)
Column 2 is A105309(n+1)
Column 3 is A196431
Column 4 is A196432
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