A003229 -id:A003229 - cp's OEIS frontend

A077949 Expansion of 1/(1-x-2*x^3).

1, 1, 1, 3, 5, 7, 13, 23, 37, 63, 109, 183, 309, 527, 893, 1511, 2565, 4351, 7373, 12503, 21205, 35951, 60957, 103367, 175269, 297183, 503917, 854455, 1448821, 2456655, 4165565, 7063207, 11976517, 20307647, 34434061, 58387095, 99002389, 167870511, 284644701

Offset: 0

N. J. A. Sloane, Nov 17 2002

Row sums of the Riordan array (1, x*(1+2*x^2)). - Paul Barry, Jan 12 2006
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=3, 3*a(n-3) equals the number of 3-colored compositions of n with all parts >=3, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011
Number of compositions of n into parts 1 and two sorts of parts 2. - Joerg Arndt, Aug 29 2013
a(n+2) equals the number of words of length n on alphabet {0,1,2}, having at least two zeros between every two successive nonzero letters. - Milan Janjic, Feb 07 2015
Number of pairs of rabbits when there are 2 pairs per litter and offspring reach parenthood after 3 gestation periods; a(n) = a(n-1) + 2*a(n-3), with a(0) = a(1) = a(2) = 1. - Robert FERREOL, Oct 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,2).

Unsigned version of A077974. Cf. A003229.

a:=[1,1,1];; for n in [4..30] do a[n]:=a[n-1]+2*a[n-3]; od; a; # G. C. Greubel, Jun 22 2019

[n le 3 select 1 else Self(n-1)+2*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Mar 13 2014

a:= n-> (<<1|1|0>, <0|0|1>, <2|0|0>>^n)[1, 1]:
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 16 2008

CoefficientList[Series[1/(1-x-2*x^3), {x, 0, 50}], x] (* Jean-François Alcover, Mar 11 2014 *)
LinearRecurrence[{1, 0, 2}, {1, 1, 1}, 50] (* Robert G. Wilson v, Jul 12 2014 *)

Vec(1/(1-x-2*x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 23 2012

(1/(1-x-2*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 22 2019

a(n) = Sum_{k=0..floor(n/2)} C(n-2k, k)*2^k. - Paul Barry, Nov 18 2003
a(n) = Sum_{k=0..n} C(k, floor((n-k)/2))*2^((n-k)/2)*(1+(-1)^(n-k))/2. - Paul Barry, Jan 12 2006
a(n) = term (1,1) in the 3x3 matrix [1,1,0; 0,0,1; 2,0,0]^n. - Alois P. Heinz, Aug 16 2008
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(2*k+1 + 2*x^2)/( x*(2*k+2 + 2*x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 29 2013

A302680 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 4 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 5, 3, 4, 8, 8, 5, 8, 6, 16, 13, 7, 12, 7, 9, 32, 21, 13, 18, 20, 11, 14, 64, 34, 23, 40, 30, 33, 18, 22, 128, 55, 37, 94, 76, 63, 64, 29, 35, 256, 89, 63, 184, 217, 187, 125, 121, 47, 56, 512, 144, 109, 358, 509, 661, 453, 257, 231, 76, 90, 1024, 233, 183, 760

Offset: 1

R. H. Hardin, Apr 11 2018

Table starts
...1..2..3...5....8...13....21.....34......55.......89......144.......233
...2..3..3...5....7...13....23.....37......63......109......183.......309
...4..4..8..12...18...40....94....184.....358......760.....1594......3220
...8..6..7..20...30...76...217....509....1189.....3034.....7569.....18274
..16..9.11..33...63..187...661...1837....5075....15661....46975....135191
..32.14.18..64..125..453..2013...6725...21745....80985...295335...1015113
..64.22.29.121..257.1125..6311..25139...96728...439233..1942666...8017639
.128.35.47.231..528.2782.19497..92889..422915..2330640.12480973..61679118
.256.56.76.440.1085.6843.60253.343421.1847358.12346637.80210343.474618407

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

R. H. Hardin, Table of n, a(n) for n = 1..799

Column 1 is A000079(n-1).
Column 2 is A001611(n+1).
Row 1 is A000045(n+1).
Row 2 is A003229(n-1) for n>2.

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 2*a(n-1) -a(n-3)
k=3: a(n) = a(n-1) +a(n-2) for n>5
k=4: a(n) = a(n-1) +2*a(n-2) -a(n-4) for n>8
k=5: a(n) = a(n-1) +3*a(n-3) +2*a(n-4) +2*a(n-5) for n>10
k=6: a(n) = a(n-1) +2*a(n-2) +4*a(n-3) +a(n-4) -2*a(n-5) -a(n-6) for n>12
k=7: [order 12] for n>19
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-2)
n=2: a(n) = a(n-1) +2*a(n-3) for n>5
n=3: a(n) = a(n-1) +3*a(n-3) +3*a(n-4) for n>7
n=4: a(n) = a(n-1) +a(n-2) +5*a(n-3) +5*a(n-4) -3*a(n-5) -3*a(n-6) +2*a(n-7) for n>11
n=5: [order 11] for n>16
n=6: [order 17] for n>23
n=7: [order 31] for n>38

A303314 T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 4, 5 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 5, 3, 4, 8, 8, 5, 12, 6, 16, 13, 7, 17, 11, 9, 32, 21, 13, 24, 36, 19, 14, 64, 34, 23, 67, 50, 74, 34, 22, 128, 55, 37, 158, 128, 139, 165, 53, 35, 256, 89, 63, 298, 439, 410, 349, 361, 83, 56, 512, 144, 109, 595, 1085, 1799, 1221, 853, 783, 136, 90, 1024, 233

Offset: 1

R. H. Hardin, Apr 21 2018

Table starts
...1..2...3....5....8....13.....21......34.......55........89........144
...2..3...3....5....7....13.....23......37.......63.......109........183
...4..4..12...17...24....67....158.....298......595......1337.......2863
...8..6..11...36...50...128....439....1085.....2431......6452......17455
..16..9..19...74..139...410...1799....5907....16494.....53290.....184915
..32.14..34..165..349..1221...7096...30280...102683....403872....1783894
..64.22..53..361..853..3453..26184..148313...618149...2955145...16591424
.128.35..83..783.2180.10223.100128..746323..3851318..22515378..159560449
.256.56.136.1710.5525.30247.387892.3784002.23967605.171306353.1539204838

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

R. H. Hardin, Table of n, a(n) for n = 1..511

Column 1 is A000079(n-1).
Column 2 is A001611(n+1).
Row 1 is A000045(n+1).
Row 2 is A003229(n-1) for n>2.

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 2*a(n-1) -a(n-3)
k=3: a(n) = a(n-1) +a(n-3) +a(n-4) for n>7
k=4: a(n) = a(n-1) +a(n-2) +3*a(n-3) +2*a(n-4) -a(n-5) -2*a(n-6) -a(n-7) for n>10
k=5: a(n) = a(n-1) +9*a(n-3) +2*a(n-4) +4*a(n-5) -10*a(n-6) -6*a(n-7) +4*a(n-9) for n>12
k=6: [order 8] for n>11
k=7: [order 20] for n>23
Empirical for row n:
n=1: a(n) = a(n-1) +a(n-2)
n=2: a(n) = a(n-1) +2*a(n-3) for n>5
n=3: a(n) = a(n-1) +3*a(n-3) +4*a(n-4) for n>7
n=4: a(n) = a(n-1) +a(n-2) +5*a(n-3) +9*a(n-4) -3*a(n-5) -7*a(n-6) -2*a(n-7) for n>11
n=5: [order 12] for n>16
n=6: [order 23] for n>28
n=7: [order 46] for n>51

A183483 T(n,k)=Number of nXk 0..2 arrays with every element equal to either the sum mod 3 of its vertical neighbors or the sum mod 3 of its horizontal neighbors.

Original entry on oeis.org

1, 3, 3, 5, 15, 5, 7, 39, 39, 7, 13, 135, 117, 135, 13, 23, 495, 587, 587, 495, 23, 37, 1647, 2925, 4015, 2925, 1647, 37, 63, 5751, 12131, 40073, 40073, 12131, 5751, 63, 109, 20223, 58333, 270549, 706473, 270549, 58333, 20223, 109, 183, 70119, 270611

Offset: 1

R. H. Hardin Jan 05 2011

Table starts
...1......3.......5.........7..........13..........23..........37..........63
...3.....15......39.......135.........495........1647........5751.......20223
...5.....39.....117.......587........2925.......12131.......58333......270611
...7....135.....587......4015.......40073......270549.....1942323....15984991
..13....495....2925.....40073......706473.....7773025...107741253..1577932971
..23...1647...12131....270549.....7773025...116198719..2519745049.56697667073
..37...5751...58333...1942323...107741253..2519745049.84056287173
..63..20223..270611..15984991..1577932971.56697667073
.109..70119.1220877.119888365.20728674493
.183.244863.5724163.894632985

			Some solutions for 5X4
..2..2..0..0....0..0..2..2....1..1..0..0....0..0..0..0....1..2..1..2
..0..0..0..0....2..0..1..1....0..0..0..0....1..2..1..2....1..1..0..2
..2..0..1..0....2..0..2..2....2..0..0..0....1..1..1..2....0..0..0..0
..2..0..1..0....0..0..0..0....2..0..1..1....2..2..0..0....1..0..1..2
..0..0..0..0....0..0..2..2....0..0..0..0....2..2..2..0....1..0..1..2

R. H. Hardin, Table of n, a(n) for n = 1..97

Column 1 is A003229

A183526 T(n,k)=Number of nXk 0..2 arrays with each element equal to either the sum mod 3 of its horizontal and vertical neighbors or the sum mod 3 of its diagonal and antidiagonal neighbors.

Original entry on oeis.org

1, 3, 3, 5, 9, 5, 7, 31, 31, 7, 13, 95, 101, 95, 13, 23, 309, 543, 543, 309, 23, 37, 911, 2233, 3507, 2233, 911, 37, 63, 2803, 10003, 27609, 27609, 10003, 2803, 63, 109, 8673, 47685, 201833, 371691, 201833, 47685, 8673, 109, 183, 26619, 215451, 1521573, 4406933

Offset: 1

R. H. Hardin Jan 05 2011

Table starts
...1.....3.......5........7........13.........23.........37........63
...3.....9......31.......95.......309........911.......2803......8673
...5....31.....101......543......2233......10003......47685....215451
...7....95.....543.....3507.....27609.....201833....1521573..11678037
..13...309....2233....27609....371691....4406933...56562977.728459961
..23...911...10003...201833...4406933...89738569.1908431611
..37..2803...47685..1521573..56562977.1908431611
..63..8673..215451.11678037.728459961
.109.26619..994397.89238523
.183.81959.4603823

			Some solutions for 4X3
..1..2..1....0..2..0....2..2..0....2..1..2....1..0..0....2..0..1....0..2..2
..0..1..2....1..0..1....0..0..0....2..2..2....0..1..0....2..0..1....1..2..1
..2..1..0....2..2..2....2..1..1....1..0..1....0..0..0....0..2..0....2..1..0
..1..2..1....1..2..2....1..2..0....0..2..0....2..2..0....2..0..0....1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..83

Column 1 is A003229

A143453 Square array A(n,k) of numbers of length n ternary words with at least k 0-digits between any other digits (n,k >= 0), read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 3, 9, 1, 3, 5, 27, 1, 3, 5, 11, 81, 1, 3, 5, 7, 21, 243, 1, 3, 5, 7, 13, 43, 729, 1, 3, 5, 7, 9, 23, 85, 2187, 1, 3, 5, 7, 9, 15, 37, 171, 6561, 1, 3, 5, 7, 9, 11, 25, 63, 341, 19683, 1, 3, 5, 7, 9, 11, 17, 39, 109, 683, 59049, 1, 3, 5, 7, 9, 11, 13, 27, 57, 183, 1365, 177147

Offset: 0

Alois P. Heinz, Aug 16 2008

			A(3,1) = 11, because 11 ternary words of length 3 have at least 1 0-digit between any other digits: 000, 001, 002, 010, 020, 100, 101, 102, 200, 201, 202.
Square array A(n,k) begins:
     1,   1,  1,  1,  1,  1,  1,  1, ...
     3,   3,  3,  3,  3,  3,  3,  3, ...
     9,   5,  5,  5,  5,  5,  5,  5, ...
    27,  11,  7,  7,  7,  7,  7,  7, ...
    81,  21, 13,  9,  9,  9,  9,  9, ...
   243,  43, 23, 15, 11, 11, 11, 11, ...
   729,  85, 37, 25, 17, 13, 13, 13, ...
  2187, 171, 63, 39, 27, 19, 15, 15, ...

Alois P. Heinz, Rows n = 0..140, flattened

Column k=0: A000244, k=1: A001045(n+2), k=2: A003229(n+1) and A077949(n+2), k=3: A052942(n+3), k=4: A143447, k=5: A143448, k=6: A143449, k=7: A143450, k=8: A143451, k=9: A143452.

Diagonal: A005408.

A := proc (n::nonnegint, k::nonnegint) option remember; if k=0 then 3^n elif n<=k+1 then 2*n+1 else A(n-1, k) +2*A(n-k-1, k) fi end: seq(seq(A(n,d-n), n=0..d), d=0..14);

a[n_, 0] := 3^n; a[n_, k_] /; n <= k+1 := 2*n+1; a[n_, k_] := a[n, k] = a[n-1, k] + 2*a[n-k-1, k]; Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013 *)

G.f. of column k: 1/(x^k*(1-x-2*x^(k+1))).

A(n,k) = 3^n if k=0, else A(n,k) = 2*n+1 if n<=k+1, else A(n,k) = A(n-1,k) + 2*A(n-k-1,k).

A183342 T(n,k)=Number of nXk binary arrays with each 1 adjacent to exactly one 1 vertically and one 1 horizontally.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 6, 7, 7, 6, 1, 1, 9, 13, 12, 13, 9, 1, 1, 13, 23, 26, 26, 23, 13, 1, 1, 19, 37, 51, 72, 51, 37, 19, 1, 1, 28, 63, 97, 175, 175, 97, 63, 28, 1, 1, 41, 109, 193, 407, 513, 407, 193, 109, 41, 1, 1, 60, 183, 380, 1005, 1397, 1397, 1005, 380

Offset: 1

R. H. Hardin Jan 04 2011

Equivalent to all 1s connected only in 2X2 blocks
Table starts
.1..1...1...1....1.....1......1.......1.......1........1.........1..........1
.1..2...3...4....6.....9.....13......19......28.......41........60.........88
.1..3...5...7...13....23.....37......63.....109......183.......309........527
.1..4...7..12...26....51.....97.....193.....380......741......1456.......2860
.1..6..13..26...72...175....407....1005....2450.....5893.....14318......34780
.1..9..23..51..175...513...1397....4133...12075....34521....100047.....290287
.1.13..37..97..407..1397...4531...16029...55471...188735....651517....2246015
.1.19..63.193.1005..4133..16029...68662..286079..1170324...4869491...20218182
.1.28.109.380.2450.12075..55471..286079.1429824..6989477..34856362..173451524
.1.41.183.741.5893.34521.188735.1170324.6989477.40840428.243887497.1451641573

			Some solutions for 6X5
..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....1..1..0..0..0....0..0..0..0..0
..0..0..1..1..0....0..0..1..1..0....1..1..0..0..0....0..0..0..0..0
..0..0..1..1..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....0..0..0..0..0....0..1..1..0..0
..0..0..0..0..0....1..1..0..0..0....0..0..0..0..0....0..1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..839

Diagonal is A145857
Column 2 is A000930(n+1)
Column 3 is A003229

A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.

Original entry on oeis.org

1, 6, 9, 5, 6, 2, 0, 7, 6, 9, 5, 5, 9, 8, 6, 2, 0, 5, 7, 4, 1, 6, 3, 6, 7, 1, 0, 0, 1, 1, 7, 5, 3, 5, 3, 4, 2, 6, 1, 8, 1, 7, 9, 3, 8, 8, 2, 0, 8, 5, 0, 7, 7, 3, 0, 2, 2, 1, 8, 7, 0, 7, 2, 8, 4, 4, 5, 2, 4, 4, 5, 3, 4, 5, 4, 0, 8, 0, 0, 7, 2, 2, 1, 3, 9, 9

Offset: 1

Clark Kimberling, Jul 14 2017

			1.6956207695598620574163671001175353426181793882085077...

D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves, unpublished, 1976, end of section 2. See links in A003229.

Angel Chang and Tianrong Zhang, The Fractal Geometry of the Boundary of Dragon Curves, Journal of Recreational Mathematics, volume 30, number 1, 1999-2000, pages 9-22.
Index entries for algebraic numbers, degree 3.

Sequences growing as this power: A003229, A003476, A003479, A052537, A077949, A144181, A164395, A164399, A164410, A164414, A164471, A203175, A227036, A289260, A292764.

Cf. A078140 (includes guide to constants similar to A289260).

z = 2000; r = 8/5;
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A289260 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]] (* A289265 *)

solve(x=1, 2, x^3 - x^2 - 2) \\ Michel Marcus, Oct 26 2019

r = D^(1/3) + (1/9)*D^(-1/3) + 1/3 where D = 28/27 + (1/9)*sqrt(29*3) [Chang and Zhang] from the usual cubic solution formula. Or similarly r = (1/3)*(1 + C + 1/C) where C = (28 + sqrt(29*27))^(1/3). - Kevin Ryde, Oct 25 2019

A303719 T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1 or 5 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 5, 3, 3, 5, 8, 5, 5, 5, 8, 13, 7, 8, 8, 7, 13, 21, 13, 14, 17, 14, 13, 21, 34, 23, 24, 36, 36, 24, 23, 34, 55, 37, 40, 76, 81, 76, 40, 37, 55, 89, 63, 68, 161, 169, 169, 161, 68, 63, 89, 144, 109, 116, 349, 361, 343, 361, 349, 116, 109, 144, 233, 183, 196, 749

Offset: 1

R. H. Hardin, Apr 29 2018

Table starts
..1..2...3...5....8...13...21....34....55....89....144....233....377.....610
..2..1...3...5....7...13...23....37....63...109....183....309....527.....893
..3..3...5...8...14...24...40....68...116...196....332....564....956....1620
..5..5...8..17...36...76..161...349...749..1604...3449...7412..15912...34177
..8..7..14..36...81..169..361...784..1681..3600...7744..16641..35721...76729
.13.13..24..76..169..343..741..1618..3451..7390..15924..34201..73387..157681
.21.23..40.161..361..741.1592..3469..7416.15880..34193..73457.157645..338676
.34.37..68.349..784.1618.3469..7551.16159.34602..74481.160024.343450..737809
.55.63.116.749.1681.3451.7416.16159.34546.73974.159281.342185.734359.1577656

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

R. H. Hardin, Table of n, a(n) for n = 1..6123

Column 1 is A000045(n+1).

Column 2 is A003229(n-1).

Empirical for diagonal:
Diagonal: [linear recurrence of order 15] for n>18
Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = a(n-1) +2*a(n-3) for n>4
k=3: a(n) = a(n-1) +2*a(n-3) for n>4
k=4: a(n) = a(n-1) +a(n-2) +3*a(n-3) +a(n-4) -a(n-5) -a(n-6)
k=5: a(n) = a(n-1) +a(n-2) +3*a(n-3) +a(n-4) -a(n-5) -a(n-6) for n>9
k=6: a(n) = a(n-1) +a(n-2) +3*a(n-3) +a(n-4) -a(n-5) -a(n-6) for n>9
k=7: a(n) = a(n-1) +a(n-2) +3*a(n-3) +a(n-4) -a(n-5) -a(n-6) for n>9

A304270 T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1, 5 or 8 king-move adjacent elements, with upper left element zero.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 5, 3, 3, 5, 8, 5, 6, 5, 8, 13, 7, 10, 10, 7, 13, 21, 13, 19, 21, 19, 13, 21, 34, 23, 37, 50, 50, 37, 23, 34, 55, 37, 67, 116, 146, 116, 67, 37, 55, 89, 63, 124, 259, 404, 404, 259, 124, 63, 89, 144, 109, 235, 601, 1074, 1246, 1074, 601, 235, 109, 144, 233

Offset: 1

R. H. Hardin, May 09 2018

Table starts
..1..2...3....5....8....13.....21......34......55.......89.......144.......233
..2..1...3....5....7....13.....23......37......63......109.......183.......309
..3..3...6...10...19....37.....67.....124.....235......436.......808......1513
..5..5..10...21...50...116....259.....601....1397.....3196......7359.....17016
..8..7..19...50..146...404...1074....2990....8316....22660.....62314....172244
.13.13..37..116..404..1246...3788...12342...39252...122156....388150...1234248
.21.23..67..259.1074..3788..13767...53839..201274...741275...2806825..10575217
.34.37.124..601.2990.12342..53839..252509.1118021..4901568..22174589..99427544
.55.63.235.1397.8316.39252.201274.1118021.5755625.29475545.156881417.822798993

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

R. H. Hardin, Table of n, a(n) for n = 1..1012

Column 1 is A000045(n+1).

Column 2 is A003229(n-1).

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = a(n-1) +2*a(n-3) for n>4
k=3: a(n) = a(n-1) +3*a(n-3) for n>4
k=4: a(n) = a(n-1) +a(n-2) +5*a(n-3) +a(n-4) -3*a(n-5) -3*a(n-6) for n>7
k=5: [order 9] for n>10
k=6: [order 12] for n>13
k=7: [order 24] for n>25

cp's OEIS Frontend

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A183526 T(n,k)=Number of nXk 0..2 arrays with each element equal to either the sum mod 3 of its horizontal and vertical neighbors or the sum mod 3 of its diagonal and antidiagonal neighbors.

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A143453 Square array A(n,k) of numbers of length n ternary words with at least k 0-digits between any other digits (n,k >= 0), read by antidiagonals.

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A183342 T(n,k)=Number of nXk binary arrays with each 1 adjacent to exactly one 1 vertically and one 1 horizontally.

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A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.

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