A003945 -id:A003945 - cp's OEIS frontend

A252753 Tree of Eratosthenes: a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 21, 16, 11, 14, 27, 20, 35, 30, 33, 24, 49, 50, 51, 36, 55, 42, 45, 32, 13, 22, 39, 28, 65, 54, 57, 40, 77, 70, 87, 60, 85, 66, 69, 48, 121, 98, 147, 100, 125, 102, 105, 72, 91, 110, 123, 84, 115, 90, 93, 64, 17, 26, 63, 44, 95, 78, 81, 56, 119, 130, 159, 108, 145, 114, 117, 80

Offset: 0

Antti Karttunen, Jan 02 2015

This sequence can be represented as a binary tree. Each child to the left is obtained by applying A250469 to the parent, and each child to the right is obtained by doubling the parent:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........6                   9......../ \........8
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  7       10         15       12         25       18         21       16
11 14   27  20     35  30   33  24     49  50   51  36     55  42   45  32
etc.
Sequence A252755 is the mirror image of the same tree. A253555(n) gives the distance of n from 1 in both trees.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A252754.
Row sums: A253787, products: A253788.
Fixed points of a(n-1): A253789.
Similar permutations: A005940, A252755, A054429, A250245.
Cf. also A001248, A001511, A055396, A083221, A181565, A250469, A253555.

(* b = A250469 *)
b[1] = 1; b[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[ 1, 1]]; For[k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1+2 == k2, Return[m2]]]];
a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ[n], b[a[n/2]], 2 a[(n-1)/2]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2016 *)

a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).
As a composition of related permutations:
a(n) = A252755(A054429(n)).
a(n) = A250245(A005940(1+n)).
Other identities. For all n >= 1:
A055396(a(n)) = A001511(n). [A005940 has the same property.]
a(A003945(n)) = A001248(n) for n>=1. - Peter Luschny, Jan 13 2015

A072405 Triangle T(n, k) = C(n,k) - C(n-2,k-1) for n >= 3 and T(n, k) = 1 otherwise, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 1, 1, 4, 7, 7, 4, 1, 1, 5, 11, 14, 11, 5, 1, 1, 6, 16, 25, 25, 16, 6, 1, 1, 7, 22, 41, 50, 41, 22, 7, 1, 1, 8, 29, 63, 91, 91, 63, 29, 8, 1, 1, 9, 37, 92, 154, 182, 154, 92, 37, 9, 1, 1, 10, 46, 129, 246, 336, 336, 246, 129, 46, 10, 1, 1, 11, 56, 175, 375, 582, 672, 582, 375, 175, 56, 11, 1

Offset: 0

Henry Bottomley, Jun 16 2002

Starting 1,0,1,1,1,... this is the Riordan array ((1-x+x^2)/(1-x), x/(1-x)). Its diagonal sums are A006355. Its inverse is A106509. - Paul Barry, May 04 2005

			Rows start as:
  1;
  1, 1;
  1, 1,  1; (key row for starting the recurrence)
  1, 2,  2,  1;
  1, 3,  4,  3,  1;
  1, 4,  7,  7,  4, 1;
  1, 5, 11, 14, 11, 5, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Row sums give essentially A003945, A007283, or A042950.
Cf. A072406 for number of odd terms in each row.
Cf. A051597, A096646, A122218 (identical for n > 1).
Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), A173118 (q=2), A173119 (q=3), A173120 (q=-4).

T:= func< n,k | n lt 3 select 1 else Binomial(n,k) - Binomial(n-2,k-1) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 28 2021

t[2, 1] = 1; t[n_, n_] = t[, 0] = 1; t[n, k_] := t[n, k] = t[n-1, k-1] + t[n-1, k]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2013, after Ralf Stephan *)

A072405(n, k) = if(n>2, binomial(n, k)-binomial(n-2, k-1), 1) \\ M. F. Hasler, Jan 06 2024

def T(n,k): return 1 if n<3 else binomial(n,k) - binomial(n-2,k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 28 2021

T(n, k) = C(n,k) - C(n-2,k-1) for n >= 3 and T(n, k) = 1 otherwise.
T(n, k) = T(n-1, k-1) + T(n-1, k) starting with T(2, 0) = T(2, 1) = T(2, 2) = 1 and T(n, 0) = T(n, n) = 1.
G.f.: (1-x^2*y) / (1 - x*(1+y)). - Ralf Stephan, Jan 31 2005
From G. C. Greubel, Apr 28 2021: (Start)
Sum_{k=0..n} T(n, k) = (n+1)*[n<3] + 3*2^(n-2)*[n>=3].
T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = -1. (End)

A287825 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 1.

Original entry on oeis.org

1, 10, 82, 674, 5540, 45538, 374316, 3076828, 25291120, 207889674, 1708825732, 14046322404, 115458919774, 949057110644, 7801124426174, 64124215108032, 527092600834054, 4332631742719370, 35613662169258228, 292739611493034596, 2406281042646218328

Offset: 0

David Nacin, Jun 02 2017

Index entries for linear recurrences with constant coefficients, signature (9, -4, -21, 9, 5).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287831.

LinearRecurrence[{9, -4, -21, 9, 5}, {1, 10, 82, 674, 5540, 45538}, 40]

def a(n):
    if n in [0, 1, 2, 3, 4, 5]:
        return [1, 10, 82, 674, 5540, 45538][n]
    return 9*a(n-1) - 4*a(n-2) - 21*a(n-3) + 9*a(n-4) + 5*a(n-5)

For n>5, a(n) = 9*a(n-1) - 4*a(n-2) - 21*a(n-3) + 9*a(n-4) + 5*a(n-5), a(0)=1, a(1)=10, a(2)=82, a(3)=674, a(4)=5540, a(5)=45538.

G.f.: (-1 - x + 4*x^2 + 3*x^3 - 3*x^4 - x^5)/(-1 + 9*x - 4*x^2 - 21*x^3 + 9*x^4 + 5*x^5).

A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

Original entry on oeis.org

0, 1, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 0

Labos Elemer, Apr 28 2003

a(n) is the least number x such that gcd(2^x, x-phi(x)) = 2^n. If cototient is replaced by totient, analogous values are different: A053576.

			G.f. = x + 6*x^2 + 12*x^3 + 24*x^4 + 48*x^5 + 96*x^6 + 192*x^7 + 384*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000010, A007283, A009195, A050339, A134059.

Essentially the same as A003945 (and perhaps also A058764).

[0, 1] cat [ &+[ 3*Binomial(n,k): k in [0..n] ]: n in [1..30] ]; // Klaus Brockhaus, Dec 02 2009

0,1,seq(3*2^(n-1), n=2..40); # G. C. Greubel, Apr 27 2021

{0}~Join~Map[Total, {{1}}~Join~Table[3 Binomial[n, k], {n, 30}, {k, 0, n}]] (* Michael De Vlieger, Jul 03 2016, after Harvey P. Dale at A134059 *)
Table[3*2^(n-1) -(3/2)*Boole[n==0] -2*Boole[n==1], {n,0,40}] (* G. C. Greubel, Apr 27 2021 *)
Join[{0,1},NestList[2#&,6,30]] (* Harvey P. Dale, Jan 22 2024 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-6*k + 16) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

a(n)=if(n<2,n,3<<(n-1)) \\ Charles R Greathouse IV, Jun 16 2012

[0,1]+[3*2^(n-1) for n in (2..40)] # G. C. Greubel, Apr 27 2021

a(n) = A007283(n-1) for n>1, with a(0) = 0 and a(1) = 1.
G.f.: x * (1 + 4*x) / (1 - 2*x) = x / (1 - 6*x / (1 + 4*x)). - Michael Somos, Jun 15 2012
Starting (1, 6, 12, 24, 48, ...) = binomial transform of [1, 5, 1, 5, 1, 5, ...]. - Gary W. Adamson, Nov 18 2007
a(n+1) = Sum_{k=0..n} A109466(n,k)*A144706(k). - Philippe Deléham, Oct 30 2008
a(n) = (-6*n + 16) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
E.g.f.: (-3 - 4*x + 3*exp(2*x))/2. - Ilya Gutkovskiy, Jul 04 2016
a(n) = 3*2^(n-1) - (3/2)*[n=0] - 2*[n=1]. - G. C. Greubel, Apr 27 2021

More terms from Klaus Brockhaus, Dec 02 2009

A082985 Coefficient table for Chebyshev's U(2n,x) polynomial expanded in decreasing powers of (-4(1-x^2)).

Original entry on oeis.org

1, 1, 3, 1, 5, 5, 1, 7, 14, 7, 1, 9, 27, 30, 9, 1, 11, 44, 77, 55, 11, 1, 13, 65, 156, 182, 91, 13, 1, 15, 90, 275, 450, 378, 140, 15, 1, 17, 119, 442, 935, 1122, 714, 204, 17, 1, 19, 152, 665, 1729, 2717, 2508, 1254, 285, 19

Offset: 0

Gary W. Adamson, May 29 2003

Sum of row #n = A000204(2n+1), i.e., A002878(n).
Row #n has the unsigned coefficients of a polynomial whose roots are 2 sin(2*Pi*k/(2n+1)) [for k=1 to 2n].
The positive roots are the diagonal lengths of a regular (2n+1)-gon, inscribed in the unit circle.
Polynomial of row #n = Sum_{m=0..n} (-1)^m T(n,m) x^(2n-2m).
This is also the unsigned coefficient table of Chebyshev's 2*T(2*n+1,x) polynomials expanded in decreasing odd powers of 2*x. - Wolfdieter Lang, Mar 07 2007
The n-th row are the coefficients of the polynomial S(n) where S(0)=1, S(1)=x+3, and S(n) = (x+2)*S(n-1) - S(n-2) (see Sun link). - Michel Marcus, Mar 07 2016

			Expansion of polynomials:
  x^0;
  x^2  -  3*x^0;
  x^4  -  5*x^2 +  5*x^0;
  x^6  -  7*x^4 + 14*x^2 -  7*x^0;
  x^8  -  9*x^6 + 27*x^4 - 30*x^2 +  9*x^0;
  x^10 - 11*x^8 + 44*x^6 - 77*x^4 + 55*x^2 - 11*x^0; ...
Polynomial #4 has 8 roots: 2*sin(2*Pi*k/9) for k=1 to 8.
Coefficients (with signs removed) are
  1;
  1,  3;
  1,  5,  5;
  1,  7, 14,  7;
  1,  9, 27, 30,  9;
  1, 11, 44, 77, 55, 11;
  ...

J. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, 1992; see pp. 151,175.
Stephen Eberhart, "Mathematical-Physical Correspondence," Number 37-38, Jan 08, 1982.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

Vincenzo Librandi, Rows n = 0..100 of triangle, flattened
K. Dilcher and K. B. Stolarsky, A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112 (2005), 673-681.
Zhi-Hong Sun, Expansions and identities concerning Lucas sequences, Fibonacci Quart. 44 (2006), no. 2, 145-153. See Theorem 3.1

Cf. companion triangle A084534.
Cf. diagonals: A005408, A000330, A005585, A050486, A054333, A057788.
Cf. A054142, A172431.

Flat(List([0..10], n-> List([0..n], k-> Binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1) ))); # G. C. Greubel, Dec 30 2019

[Binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 30 2019

A082985 := proc(n,m)
    binomial(2*n-m,m)*(2*n+1)/(2*n-2*m+1) ;
end proc: # R. J. Mathar, Sep 08 2013

T[n_, m_]:= Binomial[2*n-m, m]*(2*n+1)/(2*n-2*m+1); Table[T[n, m], {n, 0, 9}, {m, 0, n}]//Flatten (* Jean-François Alcover, Oct 08 2013, after R. J. Mathar *)

T(n,k)=binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1); \\ G. C. Greubel, Dec 30 2019

[[binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 30 2019

Triangle read by rows: row #n has n+1 terms. T(n,0)=1, T(n,n)=2n+1, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k, m-k).
T(k, s) = ((2k+1)/(2s+1))*binomial(k+s, 2s), 0 <= s <= k; then transpose the triangle. - Gary W. Adamson, May 29 2003
From Wolfdieter Lang, Mar 07 2007: (Start)
Signed version: a(n,m)=0 if n < m, otherwise a(n,m) = ((-1)^m)*binomial(2*n+1-m,m)*(2*n+1)/(2*n+1-m). From the Rivlin reference, p. 37, eq.(1.92), using the differential eq. for T(2*n+1,x). Also from Waring's formula.
Signed version: a(n,m)=0 if n < m, otherwise a(n,m) = ((-1)^m)*(Sum_{k=0..n-m} binomial(m+k,k)*binomial(2*n+1,2*(k+m)))/2^(2*(n-m)). Proof: De Moivre's formula for cos((2*n+1)*phi) rewritten in terms of odd powers of cos(phi). Cf. Rivlin reference p. 4, eq.(1.10).
Signed version: a(n,m) = A084930(n,n-m)/2^(2*(n-m)) (scaled coefficients of Chebyshev's T(2*n+1,x), decreasing odd powers).
Unsigned version: a(n,m)=0 if n < m, otherwise a(n,m) = binomial(2*n-m,m)*(2*n+1)/(2*(n-m)+1). From the differential eq. for U(2*n,x). (End)
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2). - Philippe Deléham, Feb 24 2012
Sum_{i>=0} T(n-i,n-2*i) = A003945(n). - Philippe Deléham, Feb 24 2012
Sum_{i>=0} T(n-i, n-2*i)*4^i = 3^n = A000244(n). - Philippe Deléham, Feb 24 2012
From Paul Weisenhorn Nov 25 2019: (Start)
T(r,k) = binomial(2*r-k,k-1) + binomial(2*r-1-k,k-2) with 1 <= r and 1 <= k <= r.
For a given n, one gets r = floor((1+sqrt(8*n))/2), k = n-(r^2-r)/2, a(n) = binomial(2*r-k,k-1) + binomial(2*r-1-k,k-2). (End)

Edited by Anne Donovan (anned3005(AT)aol.com), Jun 11 2003

Re-edited by Don Reble, Nov 12 2005

A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

Original entry on oeis.org

6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 1

Enoch Haga, Jan 24 2004

Sequence arising in Farideh Firoozbakht's solution to Prime Puzzle 251 - 23 is the only pointer prime (A089823) not containing digit "1".

The monotonic increasing value of successive product of digits strongly suggests that in successive n the digit 1 must be present.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Tanya Khovanova, Recursive Sequences
Carlos Rivera, Puzzle 251, Pointer primes, The Prime Puzzles and Problems Connection.
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).
Cf. A089823, A091628, A091630, A091631, A091632.
Similar to A003945, A042950, A058764, A087009, A081808.

[3*2^n : n in [1..40]]; // Wesley Ivan Hurt, Jul 17 2020

3*2^Range[1, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

[3*2^n for n in range(1,51)] # G. C. Greubel, Jan 05 2023

a(n) = 3 * 2^n = product of digits of A091628(n).
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 6*2^(n-1).
a(n) = 2*a(n-1), with a(1) = 6.
G.f.: 6*x/(1-2*x). (End)
E.g.f.: 3*(exp(2*x) - 1). - G. C. Greubel, Jan 05 2023

Edited and extended by Ray Chandler, Feb 07 2004

A275401 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-2,0) or (-1,1) and new values introduced in order 0..2.

Original entry on oeis.org

1, 2, 2, 5, 9, 3, 14, 54, 16, 6, 41, 324, 84, 31, 12, 122, 1944, 444, 178, 63, 24, 365, 11664, 2344, 1011, 394, 129, 48, 1094, 69984, 12376, 5758, 2404, 1017, 260, 96, 3281, 419904, 65344, 32771, 14884, 8122, 2645, 534, 192, 9842, 2519424, 345008, 186538, 91849

Offset: 1

R. H. Hardin, Jul 26 2016

Table starts
...1....2.....5......14.......41.......122.........365.........1094
...2....9....54.....324.....1944.....11664.......69984.......419904
...3...16....84.....444.....2344.....12376.......65344.......345008
...6...31...178....1011.....5758.....32771......186538......1061795
..12...63...394....2404....14884.....91849......566331......3495079
..24..129..1017....8122....65045....518049.....4142271.....33102260
..48..260..2645...27373...283169...2940064....30551417....317215507
..96..534..6980...94065..1266905..17159883...232200615...3146508594
.192.1083.18464..323306..5656712.100110527..1766745035..31272546310
.384.2210.48959.1109170.25396322.587134035.13518543301.312946900074

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

R. H. Hardin, Table of n, a(n) for n = 1..243

Column 1 is A003945(n-2).

Row 1 is A007051(n-1).

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>3
k=2: [order 9] for n>10
k=3: [order 24] for n>28
k=4: [order 65] for n>69
Empirical for row n:
n=1: a(n) = 4*a(n-1) -3*a(n-2)
n=2: a(n) = 6*a(n-1) for n>2
n=3: a(n) = 4*a(n-1) +6*a(n-2) +4*a(n-3)
n=4: a(n) = 3*a(n-1) +11*a(n-2) +25*a(n-3) +2*a(n-4) -24*a(n-5) for n>6
n=5: [order 9] for n>10
n=6: [order 16] for n>17
n=7: [order 21] for n>22

A161645 First differences of A161644: number of new ON cells at generation n of the triangular cellular automaton described in A161644.

Original entry on oeis.org

0, 1, 3, 6, 6, 6, 12, 18, 12, 6, 12, 24, 30, 24, 30, 42, 24, 6, 12, 24, 30, 30, 42, 66, 66, 36, 30, 60, 84, 72, 78, 96, 48, 6, 12, 24, 30, 30, 42, 66, 66, 42, 42, 78, 114, 114, 114, 150, 138, 60, 30, 60, 84, 90, 114, 174, 198, 132, 90, 144, 210, 192, 192, 210, 96, 6, 12, 24

Offset: 0

David Applegate and N. J. A. Sloane, Jun 15 2009

See the comments in A161644.

It appears that a(n) is also the number of V-toothpicks or Y-toothpicks added at the n-th stage in a toothpick structure on hexagonal net, starting with a single Y-toothpick in stage 1 and adding only V-toothpicks in stages >=2 (see A161206, A160120, A182633). - Omar E. Pol, Dec 07 2010

			From _Omar E. Pol_, Apr 08 2015: (Start)
The positive terms written as an irregular triangle in which the row lengths are the terms of A011782:
1;
3;
6,6;
6,12,18,12;
6,12,24,30,24,30,42,24;
6,12,24,30,30,42,66,66,36,30,60,84,72,78,96,48;
6,12,24,30,30,42,66,66,42,42,78,114,114,114,150,138,60,30,60,84,90,114,174,198,132,90,144,210,192,192,210,96;
...
It appears that the right border gives A003945.
(End)

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Describes the dual structure where new triangles are joined at vertices rather than edges.]

Rémy Sigrist, Table of n, a(n) for n = 0..10000
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
N. J. A. Sloane, Illustration of first 7 generations of A161644 and A295560 (edge-to-edge version)
N. J. A. Sloane, Illustration of first 11 generations of A161644 and A295560 (vertex-to-vertex version) [Include the 6 cells marked x to get A161644(11), exclude them to get A295560(11).]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A161644, A139251, A160161, A161207, A182633, A295559, A295560.

See A342271 for a(n)/3.

A275037 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (0,-1) (-1,-2) or (-2,0) and new values introduced in order 0..2.

Original entry on oeis.org

1, 1, 2, 2, 6, 3, 4, 16, 18, 6, 8, 48, 46, 54, 12, 16, 144, 137, 146, 162, 24, 32, 432, 401, 570, 450, 486, 48, 64, 1296, 1152, 2185, 2334, 1428, 1458, 96, 128, 3888, 3336, 8273, 11624, 9774, 4463, 4374, 192, 256, 11664, 9720, 31495, 57984, 64045, 40572, 14086

Offset: 1

R. H. Hardin, Jul 14 2016

Table starts
...1.....1......2.......4........8.........16..........32...........64
...2.....6.....16......48......144........432........1296.........3888
...3....18.....46.....137......401.......1152........3336.........9720
...6....54....146.....570.....2185.......8273.......31495.......120845
..12...162....450....2334....11624......57984......290927......1465435
..24...486...1428....9774....64045.....421793.....2788663.....18527905
..48..1458...4463...40572...351197....3056856....26631796....233894638
..96..4374..14086..169348..1931619...22207559...255796423...2976710318
.192.13122..44195..704704.10581148..160767947..2451632155..37779884891
.384.39366.139130.2937764.58059378.1166126087.23548292980.480461388679

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

R. H. Hardin, Table of n, a(n) for n = 1..338

Column 1 is A003945(n-2).
Column 2 is A008776(n-1).
Row 1 is A000079(n-2).

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>3
k=2: a(n) = 3*a(n-1) for n>2
k=3: [order 16] for n>17
k=4: [order 35] for n>37
k=5: [order 70] for n>74
Empirical for row n:
n=1: a(n) = 2*a(n-1) for n>2
n=2: a(n) = 3*a(n-1) for n>3
n=3: [order 9] for n>10
n=4: [order 17] for n>19
n=5: [order 28] for n>32
n=6: [order 67] for n>72

A275090 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (0,-2) (-1,-2) or (-2,-1) and new values introduced in order 0..2.

Original entry on oeis.org

1, 2, 2, 3, 14, 5, 6, 36, 81, 14, 12, 96, 192, 486, 41, 24, 288, 508, 1024, 2916, 122, 48, 864, 1680, 3088, 5440, 17496, 365, 96, 2592, 5304, 12816, 18440, 29120, 104976, 1094, 192, 7776, 17184, 53132, 87924, 111900, 155904, 629856, 3281, 384, 23328, 54484

Offset: 1

R. H. Hardin, Jul 16 2016

Table starts
....1........2........3.........6.........12..........24............48
....2.......14.......36........96........288.........864..........2592
....5.......81......192.......508.......1680........5304.........17184
...14......486.....1024......3088......12816.......53132........232156
...41.....2916.....5440.....18440......87924......510680.......2934756
..122....17496....29120....111900.....647648.....5038428......38872396
..365...104976...155904....675600....4706400....48675088.....512150588
.1094...629856...834176...4094240...34281472...477786368....6767371752
.3281..3779136..4463424..24794560..248629316..4669603380...89090043684
.9842.22674816.23883904.150165020.1807849760.45596095332.1174911360788

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

R. H. Hardin, Table of n, a(n) for n = 1..336

Column 1 is A007051(n-1).

Row 1 is A003945(n-2).

Empirical for column k:
k=1: a(n) = 4*a(n-1) -3*a(n-2)
k=2: a(n) = 6*a(n-1) for n>3
k=3: [order 9] for n>10
k=4: [order 14] for n>15
k=5: [order 27] for n>28
k=6: [order 64] for n>66
Empirical for row n:
n=1: a(n) = 2*a(n-1) for n>3
n=2: a(n) = 3*a(n-1) for n>4
n=3: [order 36] for n>38
n=4: [order 84] for n>88

cp's OEIS Frontend

A252753 Tree of Eratosthenes: a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).

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A072405 Triangle T(n, k) = C(n,k) - C(n-2,k-1) for n >= 3 and T(n, k) = 1 otherwise, read by rows.

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A287825 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 1.

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A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

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A082985 Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).

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A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

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A082985 Coefficient table for Chebyshev's U(2n,x) polynomial expanded in decreasing powers of (-4(1-x^2)).