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 1-8 of 8 results.

A279977 T(n,k) is the number of n X k 0..1 arrays with no element equal to a strict majority of its horizontal and antidiagonal neighbors, with the exception of exactly two elements, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

0, 1, 0, 0, 3, 0, 3, 9, 9, 0, 3, 24, 50, 31, 0, 9, 62, 221, 296, 108, 0, 15, 134, 822, 1922, 1650, 366, 0, 31, 277, 2669, 10491, 15511, 8666, 1205, 0, 57, 542, 8068, 50690, 124030, 118857, 43543, 3873, 0, 108, 1035, 23169, 226771, 887491, 1393359, 876704, 211650
Offset: 1

Views

Author

R. H. Hardin, Dec 24 2016

Keywords

Examples

			Table starts:
.0.....1.......0.........3...........3............9.............15
.0.....3.......9........24..........62..........134............277
.0.....9......50.......221.........822.........2669...........8068
.0....31.....296......1922.......10491........50690.........226771
.0...108....1650.....15511......124030.......887491........5870751
.0...366....8666....118857.....1393359.....14787217......144819856
.0..1205...43543....876704....15071233....237386464.....3444870482
.0..3873..211650...6281773...158391708...3703836674....79672440007
.0.12207.1002602..43997218..1627160233..56499013470..1801951754910
.0.37859.4652327.302544617.16409869901.846166990079.40020022178950
...
Some solutions for n=4 and k=4:
..0..1..0..0. .0..1..0..0. .0..1..0..1. .0..0..1..0. .0..1..0..1
..0..0..1..0. .0..1..1..0. .0..1..0..0. .1..0..1..0. .1..0..1..1
..1..1..1..1. .0..0..1..1. .1..0..1..1. .0..1..0..1. .1..0..1..1
..1..0..0..1. .0..0..1..0. .0..0..0..1. .1..0..0..1. .0..1..0..1

Links

Crossrefs

Row 1 is A105423(n-2).

Formula

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 9*a(n-1) -30*a(n-2) +45*a(n-3) -30*a(n-4) +9*a(n-5) -a(n-6)
k=3: [order 9] for n>10
k=4: [order 24]
k=5: [order 38] for n>39
k=6: [order 96] for n>97
Empirical for row n:
n=1: a(n) = 3*a(n-1) -5*a(n-3) +3*a(n-5) +a(n-6)
n=2: [order 8] for n>10
n=3: [order 24] for n>30
n=4: [order 68] for n>78

A281802 T(n,k)=Number of nXk 0..1 arrays with no element unequal to a strict majority of its horizontal, diagonal and antidiagonal neighbors, with the exception of exactly two elements, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 3, 6, 9, 0, 3, 38, 75, 34, 0, 9, 157, 372, 324, 87, 0, 15, 524, 1725, 1916, 865, 194, 0, 31, 1631, 7293, 12318, 8354, 2272, 400, 0, 57, 4694, 29665, 71290, 71445, 32524, 5191, 790, 0, 108, 13006, 116539, 396185, 575062, 368408, 117401, 11141
Offset: 1

Views

Author

R. H. Hardin, Jan 30 2017

Keywords

Comments

Table starts
.0....1.....0.......3.........3..........9...........15.............31
.0....0.....6......38.......157........524.........1631...........4694
.0....9....75.....372......1725.......7293........29665.........116539
.0...34...324....1916.....12318......71290.......396185........2143364
.0...87...865....8354.....71445.....575062......4480480.......33526931
.0..194..2272...32524....368408....4173476.....45696162......473403162
.0..400..5191..117401...1770697...28237876....436154254.....6282549989
.0..790.11141..404594...8100180..181908580...3971817891....79584142302
.0.1511.22705.1345667..35778250.1130079060..34908793027...974107985517
.0.2830.44611.4351562.153961577.6826668428.298606378335.11610933931800

Examples

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

Links

Crossrefs

Column 2 is A279128.
Row 1 is A105423(n-2).

Formula

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: [order 8] for n>9
k=3: [order 9] for n>16
k=4: [order 44] for n>51
k=5: [order 72] for n>86
Empirical for row n:
n=1: a(n) = 3*a(n-1) -5*a(n-3) +3*a(n-5) +a(n-6)
n=2: [order 14]
n=3: [order 60]

A105422 Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 1.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 2, 3, 0, 1, 3, 5, 3, 4, 0, 1, 5, 8, 9, 4, 5, 0, 1, 8, 15, 15, 14, 5, 6, 0, 1, 13, 26, 31, 24, 20, 6, 7, 0, 1, 21, 46, 57, 54, 35, 27, 7, 8, 0, 1, 34, 80, 108, 104, 85, 48, 35, 8, 9, 0, 1, 55, 139, 199, 209, 170, 125, 63, 44, 9, 10, 0, 1, 89, 240, 366, 404, 360
Offset: 0

Views

Author

Emeric Deutsch, Apr 07 2005

Keywords

Comments

T(n,k) is also the number of length n bit strings beginning with 0 having k singletons. Example: T(4,2)=3 because we have 0010, 0100 and 0110. - Emeric Deutsch, Sep 21 2008
The cyclic version of this array is A320341(n,k), which counts the (unmarked) cyclic compositions of n with exactly k parts equal to 1, with a minor exception for k=0. The sequence (A320341(n, k=0) - 1: n >= 1) counts the (unmarked) cyclic compositions of n with no parts equal to 1. - Petros Hadjicostas, Jan 08 2019
Also the convolution triangle of Fibonacci(n-2). # Peter Luschny, Oct 07 2022
T(n,k) is the number of length n+1 bit strings beginning and ending with 0 having k length 2 substrings 00. This is equivalent to the compositions interpretation because each m part corresponds to a length m+1 bit string beginning with 0 and ending with the next 0 bit. Thus a substring 00 corresponds to a 1 part. Example: T(4,2)=3 because we have 00010 for 112, 00100 for 121 and 01000 for 211. - Michael Somos, Sep 24 2024
In the Baccherini et al. 2008 link on page 81: "Bloom[3] studies the number of singles in all the 2^n n-length bit strings, where a single is any isolated 1 or 0, i.e., any run of length 1. Let R_{n,k} be the number of n-length bit strings beginning with 0 and having k singles." Here T(n,k) = R_{n,k}. This combinatorial interpretation is equivalent to my previous comment since a part of size k corresponds to run of k identical bits and also to a length k+1 bit string with 0s only at the beginning and end. - Michael Somos, Sep 25 2024

Examples

			T(6,2) = 9 because we have (1,1,4), (1,4,1), (4,1,1), (1,1,2,2), (1,2,1,2), (1,2,2,1), (2,1,1,2), (2,1,2,1) and (2,2,1,1).
Triangle begins:
00:    1;
01:    0,   1;
02:    1,   0,   1;
03:    1,   2,   0,   1;
04:    2,   2,   3,   0,   1;
05:    3,   5,   3,   4,   0,   1;
06:    5,   8,   9,   4,   5,   0,   1;
07:    8,  15,  15,  14,   5,   6,   0,   1;
08:   13,  26,  31,  24,  20,   6,   7,   0,  1;
09:   21,  46,  57,  54,  35,  27,   7,   8,  0,  1;
10:   34,  80, 108, 104,  85,  48,  35,   8,  9,  0,  1;
11:   55, 139, 199, 209, 170, 125,  63,  44,  9, 10,  0,  1;
12:   89, 240, 366, 404, 360, 258, 175,  80, 54, 10, 11,  0, 1;
13:  144, 413, 666, 780, 725, 573, 371, 236, 99, 65, 11, 12, 0, 1;
...

Links

Crossrefs

Column 0 yields A000045 (the Fibonacci numbers). Column 1 yields A006367. Column 2 yields A105423. Row sums yield A011782. Cyclic version is A320341.
T(2n,n) gives A222763.

Programs

  • Maple
    G:=(1-z)/(1-z-z^2-t*z+t*z^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 13 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
# second Maple program:
T:= proc(n) option remember; local j; if n=0 then 1
      else []; for j to n do zip((x, y)-> x+y, %,
      [`if`(j=1, 0, [][]), T(n-j)], 0) od; %[] fi
    end:
seq(T(n), n=0..20);  # Alois P. Heinz, Nov 05 2012
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> combinat:-fibonacci(n-2)); # Peter Luschny, Oct 07 2022
  • Mathematica
    nn = 10; a = x/(1 - x) - x + y x;
CoefficientList[CoefficientList[Series[1/(1 - a), {x, 0, nn}], x], y] // Flatten (* Geoffrey Critzer, Dec 23 2011 *)
T[ n_, k_] := Which[k<0 || k>n, 0, n<2, Boole[n==k], True, T[n, k] =  T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1] ]; (* Michael Somos, Sep 24 2024 *)
  • PARI
    {T(n, k) = if(k<0 || k>n, 0, n<2, n==k, T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-2, k-1) )}; /* Michael Somos, Sep 24 2024 */

Formula

G.f.: (1-z)/(1 - z - z^2 - tz + tz^2).
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0)=1, T(1,0)=0. - Philippe Deléham, Nov 18 2009
If the triangle's columns are transposed into rows, then T(n,k) can be interpreted as the number of compositions of n+k having exactly k 1's. Then g.f.: ((1-x)/(1-x-x^2))^(k-1) and T(n,k) = T(n-2,k) + T(n-1,k) - T(n-1, k-1) + T(n, k-1). Also, Sum_{j=1..n} T(n-j, j) = 2^(n-1), the number of compositions of n. - Gregory L. Simay, Jun 28 2019

A279134 T(n,k)=Number of nXk 0..1 arrays with no element equal to a strict majority of its horizontal and vertical neighbors, with the exception of exactly two elements, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 3, 9, 9, 3, 3, 34, 66, 34, 3, 9, 87, 256, 256, 87, 9, 15, 194, 820, 1324, 820, 194, 15, 31, 400, 2551, 6396, 6396, 2551, 400, 31, 57, 790, 7491, 30074, 47452, 30074, 7491, 790, 57, 108, 1511, 21131, 129264, 316516, 316516, 129264, 21131, 1511, 108
Offset: 1

Views

Author

R. H. Hardin, Dec 06 2016

Keywords

Comments

Table starts
...0....1......0.......3.........3...........9...........15.............31
...1....0......9......34........87.........194..........400............790
...0....9.....66.....256.......820........2551.........7491..........21131
...3...34....256....1324......6396.......30074.......129264.........535814
...3...87....820....6396.....47452......316516......2017028.......12376570
...9..194...2551...30074....316516.....3125600.....29410145......266502710
..15..400...7491..129264...2017028....29410145....409061044.....5488392521
..31..790..21131..535814..12376570...266502710...5488392521...109117856920
..57.1511..57971.2150797..73672888..2346800921..71618045798..2111166039927
.108.2830.155551.8418336.428568648.20197483932.913912909445.39962431131266

Examples

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

Links

Crossrefs

Column 1 is A105423(n-2).

Formula

Empirical for column k:
k=1: a(n) = 3*a(n-1) -5*a(n-3) +3*a(n-5) +a(n-6)
k=2: [order 8] for n>9
k=3: [order 11] for n>17
k=4: [order 43] for n>50
k=5: [order 88] for n>108

A279168 T(n,k)=Number of nXk 0..1 arrays with no element equal to a strict majority of its king-move neighbors, with the exception of exactly two elements, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 3, 0, 0, 3, 3, 8, 16, 8, 3, 9, 24, 117, 117, 24, 9, 15, 88, 483, 864, 483, 88, 15, 31, 284, 2001, 5628, 5628, 2001, 284, 31, 57, 772, 7709, 34764, 57248, 34764, 7709, 772, 57, 108, 2000, 28139, 203226, 557163, 557163, 203226, 28139, 2000, 108, 199
Offset: 1

Views

Author

R. H. Hardin, Dec 07 2016

Keywords

Comments

Table starts
...0....1......0........3..........3............9.............15
...1....0......0........8.........24...........88............284
...0....0.....16......117........483.........2001...........7709
...3....8....117......864.......5628........34764.........203226
...3...24....483.....5628......57248.......557163........5159514
...9...88...2001....34764.....557163......8426362......121098448
..15..284...7709...203226....5159514....121098448.....2708250146
..31..772..28139..1143396...45915548...1686053298....58954576326
..57.2000..99519..6219491..396958758..22825771952..1248383818884
.108.5008.343156.33013384.3354431037.302051174586.25842455526113

Examples

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

Links

Crossrefs

Column 1 is A105423(n-2).

Formula

Empirical for column k:
k=1: a(n) = 3*a(n-1) -5*a(n-3) +3*a(n-5) +a(n-6)
k=2: [order 11]
k=3: [order 33] for n>36
k=4: [order 84] for n>88

A279494 T(n,k)=Number of nXk 0..1 arrays with no element equal to a strict majority of its horizontal, vertical and antidiagonal neighbors, with the exception of exactly two elements, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 3, 5, 5, 3, 3, 18, 38, 18, 3, 9, 68, 254, 254, 68, 9, 15, 235, 1433, 2684, 1433, 235, 15, 31, 801, 8330, 26157, 26157, 8330, 801, 31, 57, 2678, 46095, 246237, 425588, 246237, 46095, 2678, 57, 108, 8777, 250440, 2241332, 6559816, 6559816
Offset: 1

Views

Author

R. H. Hardin, Dec 13 2016

Keywords

Comments

Table starts
...0.....1.......0..........3............3..............9...............15
...1.....0.......5.........18...........68............235..............801
...0.....5......38........254.........1433...........8330............46095
...3....18.....254.......2684........26157.........246237..........2241332
...3....68....1433......26157.......425588........6559816.........98128162
...9...235....8330.....246237......6559816......166229033.......4065077852
..15...801...46095....2241332.....98128162.....4065077852.....162346342126
..31..2678..250440...19885341...1427793365....96674459373....6294925767121
..57..8777.1332366..172968364..20331614084..2247110811000..238434528280477
.108.28343.6989712.1481176019.284582186755.51300573165630.8863487201716422

Examples

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

Links

Crossrefs

Column 1 is A105423(n-2).

Formula

Empirical for column k:
k=1: a(n) = 3*a(n-1) -5*a(n-3) +3*a(n-5) +a(n-6)
k=2: [order 16]
k=3: [order 54] for n>55

A280180 T(n,k)=Number of nXk 0..1 arrays with no element unequal to a strict majority of its horizontal, vertical and antidiagonal neighbors, with the exception of exactly two elements, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 3, 12, 12, 3, 3, 41, 60, 41, 3, 9, 103, 279, 279, 103, 9, 15, 263, 1082, 1633, 1082, 263, 15, 31, 656, 3931, 8759, 8759, 3931, 656, 31, 57, 1618, 13720, 43094, 63336, 43094, 13720, 1618, 57, 108, 3931, 46467, 202693, 421214, 421214, 202693, 46467
Offset: 1

Views

Author

R. H. Hardin, Dec 28 2016

Keywords

Comments

Table starts
...0....1......0........3.........3...........9...........15.............31
...1....2.....12.......41.......103.........263..........656...........1618
...0...12.....60......279......1082........3931........13720..........46467
...3...41....279.....1633......8759.......43094.......202693.........919058
...3..103...1082.....8759.....63336......421214......2665301.......16203600
...9..263...3931....43094....421214.....3755997.....31821879......258976696
..15..656..13720...202693...2665301....31821879....359880117.....3910652938
..31.1618..46467...919058..16203600...258976696...3910652938....56789421603
..57.3931.153650..4057457..95738359..2046791216..41205820599...798794739075
.108.9459.499289.17554353.553426602.15814457993.424186764568.10974204206787

Examples

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

Links

Crossrefs

Column 1 is A105423(n-2).

Formula

Empirical for column k:
k=1: a(n) = 3*a(n-1) -5*a(n-3) +3*a(n-5) +a(n-6)
k=2: [order 10] for n>14
k=3: [order 21] for n>28
k=4: [order 45] for n>54

A210553 Triangle of coefficients of polynomials v(n,x) jointly generated with A210552; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 3, 5, 3, 5, 4, 9, 8, 5, 6, 5, 14, 15, 15, 8, 7, 6, 20, 24, 31, 26, 13, 8, 7, 27, 35, 54, 57, 46, 21, 9, 8, 35, 48, 85, 104, 108, 80, 34, 10, 9, 44, 63, 125, 170, 209, 199, 139, 55, 11, 10, 54, 80, 175, 258, 360, 404, 366, 240, 89, 12, 11, 65, 99
Offset: 1

Views

Author

Clark Kimberling, Mar 22 2012

Keywords

Comments

Let T(n,k) denote the term in row n, column k.
T(n,n): A000045 (Fibonacci numbers)
T(n,n-1): A006367
T(n,n-2): A105423
T(n,1): 1,2,3,4,5,6,7,8,9,...
T(n,2): 1,2,3,4,5,6,7,8,9,...
T(n,3): A000096
T(n,4): A005563
T(n,5): A055831
T(n,6): A111694
Row sums: A000225
Alternating row sums: A052551
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
2...1
3...2...2
4...3...5...3
5...4...9...8...5
First three polynomials v(n,x): 1, 2 + x , 3 + 2x + 2x^2.

Crossrefs

Cf. A210552, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210552 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210553 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)

Formula

u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Showing 1-8 of 8 results.

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