A016116 -id:A016116 - cp's OEIS frontend

A216228 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=3, T(0,0) = T(0,1) = T(0,2) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 32

Offset: 0

Philippe Deléham, Mar 13 2013

An arithmetic hexagon of E. Lucas.

			Square array begins:
1, 1, 1, 0, 0,  0,  0,  0, ... row n=0
0, 1, 2, 2, 0,  0,  0,  0, ... row n=1
0, 0, 2, 4, 4,  0,  0,  0, ... row n=2
0, 0, 0, 4, 8,  8,  0,  0, ... row n=3
0, 0, 0, 0, 8, 16, 16,  0, ... row n=4
0, 0, 0, 0, 0, 16, 32, 32, ... row n=5
...

E. Lucas, Théorie des nombres, Albert Blanchard, Paris 1958, Tome 1, p.89

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, 1991, p.89

Cf. A000079, A011782, A016116, A068914, A084215.

T(n,n) = A011782(n).
T(n,n+1) = T(n,n+2) = 2^n = A000079(n).
Sum_{k, 0<=k<=n} T(n-k,k) = A016116(n).
Sum_{n, n>=0} T(n,k) = A084215(k).
Sum_{k, k>=0} T(n,k) = A084215(n+1), n>=1.

A240284 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or two plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

1, 2, 2, 2, 8, 3, 4, 19, 19, 4, 4, 76, 80, 38, 7, 8, 181, 570, 262, 114, 10, 8, 741, 2574, 3457, 1461, 251, 15, 16, 1779, 20764, 28654, 33183, 5443, 612, 24, 16, 7308, 97348, 443168, 484146, 218658, 24490, 1656, 35, 32, 17561, 802835, 3980245, 13490093, 5646644

Offset: 1

R. H. Hardin, Apr 03 2014

Table starts
..1....2.......2.........4............4.............8..............8
..2....8......19........76..........181...........741...........1779
..3...19......80.......570.........2574.........20764..........97348
..4...38.....262......3457........28654........443168........3980245
..7..114....1461.....33183.......484146......13490093......224906182
.10..251....5443....218658......5646644.....281488213.....8597299482
.15..612...24490...1851080.....88953626....8199368365...463717321235
.24.1656..117962..15760838...1357879302..225885684501.23104690637116
.35.3758..459193.110599613..17350066110.5291794810655
.54.9630.2147788.945852472.272318368893

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

R. H. Hardin, Table of n, a(n) for n = 1..111

Column 1 is A159288

Row 1 is A016116

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: [order 15]
Empirical for row n:
n=1: a(n) = 2*a(n-2)
n=2: a(n) = 12*a(n-2) -24*a(n-4) +31*a(n-6) -16*a(n-8)
n=3: [order 48] for n>51

A240295 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or two plus the sum of the elements above it or two plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

2, 4, 2, 8, 6, 4, 16, 26, 24, 4, 32, 90, 206, 56, 8, 64, 340, 1322, 974, 230, 8, 128, 1194, 9970, 12164, 8064, 552, 16, 256, 4424, 63892, 180886, 200864, 38252, 2270, 16, 512, 15766, 459420, 2228098, 5967512, 1867678, 316962, 5456, 32, 1024, 57754, 2983714

Offset: 1

R. H. Hardin, Apr 03 2014

Table starts
..2.....4........8..........16............32.............64............128
..2.....6.......26..........90...........340...........1194...........4424
..4....24......206........1322..........9970..........63892.........459420
..4....56......974.......12164........180886........2228098.......31650312
..8...230.....8064......200864.......5967512......144185218.....4068505132
..8...552....38252.....1867678.....109846410.....5231971212...293426742772
.16..2270...316962....31042576....3655313420...346662397488.38864331960018
.16..5456..1502948...289075166...67561324734.12734002906536
.32.22416.12468758..4812019390.2255342381120
.32.53864.59122266.44835430372

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

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

Column 1 is A016116(n+1)

Row 1 is A000079

Empirical for column k:
k=1: a(n) = 2*a(n-2)
k=2: a(n) = 12*a(n-2) -24*a(n-4) +31*a(n-6) -16*a(n-8) for n>9
k=3: [order 48] for n>50
Empirical for row n:
n=1: a(n) = 2*a(n-1)
n=2: [order 12]
n=3: [order 80] for n>81

A240394 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it or two plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 1, 7, 12, 4, 1, 9, 32, 50, 4, 1, 11, 62, 258, 120, 8, 1, 13, 118, 954, 1232, 493, 8, 1, 15, 206, 3064, 8656, 10291, 1184, 16, 1, 17, 351, 9075, 50756, 142016, 48826, 4863, 16, 1, 19, 568, 27120, 263816, 1568581, 1314136, 405404, 11684, 32, 1, 21, 882

Offset: 1

R. H. Hardin, Apr 04 2014

Table starts
..1.....1........1..........1............1.............1.............1
..2.....5........7..........9...........11............13............15
..2....12.......32.........62..........118...........206...........351
..4....50......258........954.........3064..........9075.........27120
..4...120.....1232.......8656........50756........263816.......1378418
..8...493....10291.....142016......1568581......14958462.....138422394
..8..1184....48826....1314136.....27938412.....502139307....8505203924
.16..4863...405404...21792634....910553970...31597696508.1003858789731
.16.11684..1922824..202647943..16647316316.1127672103190
.32.47994.15957927.3370994234.551914186148

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

R. H. Hardin, Table of n, a(n) for n = 1..112

Column 1 is A016116

Empirical for column k:
k=1: a(n) = 2*a(n-2)
k=2: a(n) = 12*a(n-2) -24*a(n-4) +31*a(n-6) -16*a(n-8)
k=3: [order 48] for n>49
Empirical for row n:
n=1: a(n) = 1
n=2: a(n) = 2*n + 1 for n>1
n=3: a(n) = (1/6)*n^4 - (5/6)*n^3 + (13/3)*n^2 + (31/3)*n - 48 for n>5
n=4: [polynomial of degree 14] for n>13
n=5: [polynomial of degree 44] for n>38

A056503 Number of periodic palindromic structures of length n using a maximum of two different symbols.

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 14, 16, 26, 32, 51, 64, 100, 128, 198, 256, 392, 512, 778, 1024, 1552, 2048, 3091, 4096, 6176, 8192, 12324, 16384, 24640, 32768, 49222, 65536, 98432, 131072, 196744, 262144, 393472, 524288, 786698, 1048576, 1573376, 2097152, 3146256, 4194304

Offset: 1

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.
A periodic palindrome is just a necklace that is equivalent to its reverse. The number of binary periodic palindromes of length n is given by A164090(n). A binary periodic palindrome can only be equivalent to its complement when there are an equal number of 0's and 1's. - Andrew Howroyd, Sep 29 2017
Number of cyclic compositions (necklaces of positive integers) summing to n that can be rotated to form a palindrome. - Gus Wiseman, Sep 16 2018

			From _Gus Wiseman_, Sep 16 2018: (Start)
The sequence of palindromic cyclic compositions begins:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (113)    (33)      (115)
                    (112)   (122)    (114)     (133)
                    (1111)  (11111)  (222)     (223)
                                     (1122)    (11113)
                                     (11112)   (11212)
                                     (111111)  (11122)
                                               (1111111)
(End)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of A179181.
Cf. A016116, A045674, A056508, A164090, A285012.
Cf. A000740, A000837, A008965, A025065, A059966, A242414, A296302, A317085, A317086, A317087, A318731.

(* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1; c[n_] := c[n] = If[EvenQ[n], 2^(n/2-1) + c[n/2], 2^((n-1)/2)];
a[n_?OddQ] := b[n]/2; a[n_?EvenQ] := (1/2)*(b[n] + c[n/2]);
Array[a, 45] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Function[q,And[Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And],Array[SameQ[RotateRight[q,#],Reverse[RotateRight[q,#]]]&,Length[q],1,Or]]]]],{n,15}] (* Gus Wiseman, Sep 16 2018 *)

a(2n+1) = A164090(2n+1)/2 = 2^n, a(2n) = (A164090(2n) + A045674(n))/2. - Andrew Howroyd, Sep 29 2017

a(17)-a(45) from Andrew Howroyd, Apr 07 2017

A060656 a(n) = 2a(n-1)a(n-2)/a(n-3), with a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 2, 4, 16, 64, 512, 4096, 65536, 1048576, 33554432, 1073741824, 68719476736, 4398046511104, 562949953421312, 72057594037927936, 18446744073709551616, 4722366482869645213696, 2417851639229258349412352

Offset: 0

Henry Bottomley, Apr 18 2001

a(n+1) is the Hankel transform of A135052. - Paul Barry, Nov 15 2007
a(n+1) is the Hankel transform of the aerated large Schroeder numbers. a(n) and a(n+1) both satisfy the trivial Somos-4 recurrence u(n)=4*u(n-2)^2/u(n-4). Associated with the elliptic curve y^2=1-6x^2+x^4 via Schroeder numbers. - Paul Barry, Dec 08 2009
Hankel transform of A089324. - Paul Barry, Mar 01 2010
a(n+1) is the number of n X n binary matrices that are symmetric about both diagonals (bisymmetric). For the derivation of this result, see the link below. - Dennis P. Walsh, Apr 03 2014
1 followed by {a(n-1)}A078495).%20-%20_Vladimir%20Shevelev">(n>=1) is the Somos-3 sequence: b(0)=b(1)=b(2)=1;for n>=3, b(n)=2*b(n-1)*b(n-2)/b(n-3) (cf. comment in A078495). - _Vladimir Shevelev, Apr 20 2016
If the Hankel transform is defined as in the link 'Sequence transformations' then a(n) is the Hankel transform of A151374. - Peter Luschny, Nov 30 2016

			a(6) = 2*64*16/4 = 512.
G.f. = 1 + x + 2*x^2 + 4*x^3 + 16*x^4 + 64*x^5 + 512*x^6 + 4096*x^7 + ...

Harry J. Smith, Table of n, a(n) for n = 0..100
Peter Luschny, Sequence transformations
Dennis P. Walsh, Notes on binary bisymmetric matrices

Cf. A002416, A002620, A016116, A038754, A089324, A135052, A262666, A078495, A151374.

A060656:=n->2^floor(n^2/4); seq(A060656(n), n=0..20); # Wesley Ivan Hurt, Apr 30 2014

a[ n_] := 2^Quotient[n^2, 4]; (* Michael Somos, Jan 24 2014 *)
nxt[{a_,b_,c_}]:={b,c,(2c*b)/a}; NestList[nxt,{1,1,2},20][[All,1]] (* Harvey P. Dale, Nov 26 2017 *)

{ for (n=0, 100, write("b060656.txt", n, " ", 2^(n^2\4)); ) } \\ Harry J. Smith, Jul 09 2009

{a(n) = 2^(n^2\4)}; /* Michael Somos, Jan 24 2014 */

a(n) = 2^floor( n^2/4 ) = a(n - 1) * 2^floor( n/2 ) = a(n - 2) * 2^(n - 1) = a(n - 1) * A016116(n) = 2^A002620(n).
0 = a(n) * a(n+3) + a(n+1) * ( -2*a(n+2) ) for all n in Z. - Michael Somos, Jan 24 2014
0 = a(n) * a(n+4) + a(n+2) * ( -4*a(n+2) ) for all n in Z. - Michael Somos, Jan 24 2014

A068913 Square array read by antidiagonals of number of k step walks (each step +-1 starting from 0) which are never more than n or less than -n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 2, 1, 0, 4, 4, 2, 1, 0, 4, 6, 4, 2, 1, 0, 8, 12, 8, 4, 2, 1, 0, 8, 18, 14, 8, 4, 2, 1, 0, 16, 36, 28, 16, 8, 4, 2, 1, 0, 16, 54, 48, 30, 16, 8, 4, 2, 1, 0, 32, 108, 96, 60, 32, 16, 8, 4, 2, 1, 0, 32, 162, 164, 110, 62, 32, 16, 8, 4, 2, 1, 0, 64, 324, 328, 220, 124, 64, 32, 16, 8, 4, 2, 1

Offset: 0

Henry Bottomley, Mar 06 2002

			Rows start:
  1,  0,  0,  0,  0, ...
  1,  2,  2,  4,  4, ...
  1,  2,  4,  6, 12, ...
  1,  2,  4,  8, 14, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Cf. early rows: A000007, A016116 (without initial term), A068911, A068912, A216212, A216241, A235701.

Central and lower diagonals are A000079, higher diagonals include A000918, A028399.

T[n_,0]=1; T[n_,k_]:=2^k/(n+1) Sum[(-1)^r Cos[(Pi (2r-1))/(2 (n+1))]^k Cot[(Pi (1-2r))/(4 (n+1))],{r,1,n+1}]; Table[T[r,n-r],{n,0,20},{r,0,n}]//Round//Flatten (* Herbert Kociemba, Sep 23 2020 *)

Starting with T(n, 0) = 1, if (k-n) is negative or even then T(n, k) = 2*T(n, k-1), otherwise T(n, k) = 2*T(n, k-1) - A061897(n+1, (k-n-1)/2). So for n>=k, T(n, k) = 2^k. [Corrected by Sean A. Irvine, Mar 23 2024]

T(n,0) = 1, T(n,k) = (2^k/(n+1))*Sum_{r=1..n+1} (-1)^r*cos((Pi*(2*r-1))/(2*(n+1)))^k*cot((Pi*(1-2*r))/(4*(n+1))). - Herbert Kociemba, Sep 23 2020

A164298 a(n) = ((1+4sqrt(2))(2+sqrt(2))^n + (1-4sqrt(2))(2-sqrt(2))^n)/2.

Original entry on oeis.org

1, 10, 38, 132, 452, 1544, 5272, 18000, 61456, 209824, 716384, 2445888, 8350784, 28511360, 97343872, 332352768, 1134723328, 3874187776, 13227304448, 45160842240, 154188760064, 526433355776, 1797355902976, 6136556900352, 20951515795456, 71532949381120

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

Binomial transform of A048696. Second binomial transform of A164587. Inverse binomial transform of A164299.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Sequences in the class a(n, m): this sequence (m=1), A164299 (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).
Cf. A007070, A016921, A048696, A056236, A112032, A164587, A241204.
Cf. A016116(n+1).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(2+r)^n+(1-4*r)*(2-r)^n)/2: n in [0..27] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+6*x)/(1-4*x+2*x^2) )); // G. C. Greubel, Dec 14 2018

a:=n->((1+4*sqrt(2))*(2+sqrt(2))^n+(1-4*sqrt(2))*(2-sqrt(2))^n)/2: seq(floor(a(n)),n=0..25); # Muniru A Asiru, Dec 15 2018

LinearRecurrence[{4,-2}, {1,10}, 50] (* or *) CoefficientList[Series[(1 + 6*x)/(1 - 4*x + 2*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 12 2017 *)

my(x='x+O('x^50)); Vec((1+6*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Sep 12 2017

[( (1+6*x)/(1-4*x+2*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Dec 14 2018; Mar 12 2021

a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0)=1, a(1)=10.
G.f.: (1+6*x)/(1-4*x+2*x^2).
E.g.f.: (cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(2*x). - G. C. Greubel, Sep 12 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = A056236(n) + 8*A007070(n-1).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009

A240381 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or two plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

2, 2, 4, 4, 10, 10, 4, 38, 44, 22, 8, 90, 330, 148, 50, 8, 366, 1494, 2066, 636, 114, 16, 878, 12234, 17550, 16994, 2430, 258, 16, 3606, 57722, 279886, 281186, 116030, 9648, 586, 32, 8666, 477574, 2545618, 8802558, 3502886, 884792, 37946, 1330, 32, 35602

Offset: 1

R. H. Hardin, Apr 04 2014

Table starts
....2......2.........4............4.............8..............8.............16
....4.....10........38...........90...........366............878...........3606
...10.....44.......330.........1494.........12234..........57722.........477574
...22....148......2066........17550........279886........2545618.......41758418
...50....636.....16994.......281186.......8802558......157432290.....5145703760
..114...2430....116030......3502886.....207932244.....7149227810...457988195982
..258...9648....884792.....52375114....6169009514...422227556156.54164128056204
..586..37946...6273952....672652728..148090588518.19320096061230
.1330.149336..46648918...9771038498.4275011910288
.3018.588102.335571098.127878632630

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

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

Column 1 is A078040

Row 1 is A016116(n+1)

Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2) +2*a(n-3)
k=2: [order 14] for n>15
Empirical for row n:
n=1: a(n) = 2*a(n-2)
n=2: a(n) = 12*a(n-2) -24*a(n-4) +31*a(n-6) -16*a(n-8)
n=3: [order 48] for n>50

A305749 T(n,k) is the number of achiral color patterns (set partitions) in a row or loop of length n with k or fewer colors (sets).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 6, 4, 1, 1, 2, 3, 7, 9, 8, 1, 1, 2, 3, 7, 11, 18, 8, 1, 1, 2, 3, 7, 12, 27, 27, 16, 1, 1, 2, 3, 7, 12, 30, 43, 54, 16, 1, 1, 2, 3, 7, 12, 31, 55, 107, 81, 32, 1, 1, 2, 3, 7, 12, 31, 58, 141, 171, 162, 32, 1, 1, 2, 3, 7, 12, 31, 59, 159, 266, 427, 243, 64, 1, 1, 2, 3, 7, 12, 31, 59, 163, 312, 688, 683, 486, 64, 1

Offset: 1

Robert A. Russell, Jun 09 2018

An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABC are equivalent, as are AAABB and BBBAA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a loop are equivalent, so for loops AAABCB = BAAABC = CBAAAB.

			The array begins at T(1,1):
1  1   1    1    1     1     1     1     1     1     1     1     1 ...
1  2   2    2    2     2     2     2     2     2     2     2     2 ...
1  2   3    3    3     3     3     3     3     3     3     3     3 ...
1  4   6    7    7     7     7     7     7     7     7     7     7 ...
1  4   9   11   12    12    12    12    12    12    12    12    12 ...
1  8  18   27   30    31    31    31    31    31    31    31    31 ...
1  8  27   43   55    58    59    59    59    59    59    59    59 ...
1 16  54  107  141   159   163   164   164   164   164   164   164 ...
1 16  81  171  266   312   334   338   339   339   339   339   339 ...
1 32 162  427  688   883   963   993   998   999   999   999   999 ...
1 32 243  683 1313  1774  2069  2169  2204  2209  2210  2210  2210 ...
1 64 486 1707 3407  5103  6119  6634  6789  6834  6840  6841  6841 ...
1 64 729 2731 6532 10368 13524 15080 15790 15975 16026 16032 16033 ...
a(n) are the terms of this array read by antidiagonals.
For T(4,3)=6, the achiral pattern rows are AAAA, AABB, ABAB, ABBA, ABBC, and ABCA. The achiral pattern loops are AAAA, AAAB, AABB, ABAB, AABC, and ABAC.

Columns 1-6 are A057427, A016116, A182522, A305750, A305751, and A305752.

Columns converge to the right to A080107.

Ach[n_, k_] := Ach[n,k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] +
  Ach[n-2,k-1] + Ach[n-2,k-2]]; (* A304972 *)
Table[Sum[Ach[n, j], {j, 1, k - n + 1}], {k, 1, 15}, {n, 1, k}] // Flatten

T(n,k) = Sum_{j=0..k} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0 <= n <= 1 & n==k].

T(n,k) = Sum_{j=1..k} A304972(n,j).

cp's OEIS Frontend

A216228 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=1 or if k-n>=3, T(0,0) = T(0,1) = T(0,2) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A240284 T(n,k)=Number of nXk 0..3 arrays with no element equal to one plus the sum of elements to its left or two plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest, modulo 4.

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A240295 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or two plus the sum of the elements above it or two plus the sum of the elements diagonally to its northwest, modulo 4.

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A240394 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it or two plus the sum of the elements diagonally to its northwest, modulo 4.

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A056503 Number of periodic palindromic structures of length n using a maximum of two different symbols.

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A060656 a(n) = 2*a(n-1)*a(n-2)/a(n-3), with a(0)=a(1)=1.

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PARI

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A068913 Square array read by antidiagonals of number of k step walks (each step +-1 starting from 0) which are never more than n or less than -n.

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A164298 a(n) = ((1+4*sqrt(2))*(2+sqrt(2))^n + (1-4*sqrt(2))*(2-sqrt(2))^n)/2.

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A060656 a(n) = 2a(n-1)a(n-2)/a(n-3), with a(0)=a(1)=1.

A164298 a(n) = ((1+4sqrt(2))(2+sqrt(2))^n + (1-4sqrt(2))(2-sqrt(2))^n)/2.