A060867 -id:A060867 - cp's OEIS frontend

A256530 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).

0, 1, 9, 21, 49, 61, 97, 157, 225, 237, 273, 333, 417, 525, 657, 813, 961, 973, 1009, 1069, 1153, 1261, 1393, 1549, 1729, 1933, 2161, 2413, 2689, 2989, 3313, 3661, 3969, 3981, 4017, 4077, 4161, 4269, 4401, 4557, 4737, 4941, 5169, 5421, 5697, 5997, 6321, 6669, 7041, 7437, 7857, 8301, 8769, 9261, 9777, 10317, 10881, 11469

Offset: 0

Omar E. Pol, Apr 21 2015

On the infinite square grid at stage 0 there are no ON cells, so a(0) = 0.
At stage 1, only one cell is turned ON, so a(1) = 1.
If n is a power of 2 so the structure is a square of side length 2n - 1 that contains (2n-1)^2 ON cells.
The structure grows by the four corners as square waves forming layers of ON cells up the next square structure, and so on (see example).
Note that a(24) = 1729 is also the Hardy-Ramanujan number (see A001235).
Has the same rules as A256534 but here a(1) = 1 not 4.
Has a smoother behavior than A160414 with which shares infinitely many terms (see example).
A256531, the first differences, gives the number of cells turned ON at n-th stage.

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
9;
21,    49;
61,    97,  157,  225;
237,  273,  333,  417,  525,  657,  813,  961;
...
Right border gives A060867.
This triangle T(n,k) shares with the triangle A160414 the terms of the column k, if k is a power of 2, for example both triangles share the following terms: 1, 9, 21, 49, 61, 97, 225, 237, 273, 417, 961, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                       _ _ _ _
.      |  _ _  |                     |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _ _ _ _ _   _ _ _ _ _ _  |_| |
.      |_ _| |  _ _ _ _  | |  _ _ _ _  | |_ _|
.          | | |  _ _  | | | |  _ _  | | |
.          | | | |  _|_|_|_|_|_|_  | | | |
.          | | | |_|  _ _   _ _  |_| | | |
.          | | |_ _| |  _|_|_  | |_ _| | |
.          | |_ _ _| |_|  _  |_| |_ _ _| |
.          |  _ _ _|  _| |_| |_  |_ _ _  |
.          | |  _ _| | |_ _ _| | |_ _  | |
.          | | |  _| |_ _| |_ _| |_  | | |
.          | | | | |_ _ _ _ _ _ _| | | | |
.          | | | |_ _| | | | | |_ _| | | |
.       _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
.      |  _| |_ _ _ _ _ _| |_ _ _ _ _ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                     | |_ _| |
.      |_ _ _ _|                     |_ _ _ _|
.
After 10 generations there are 273 ON cells, so a(10) = 273.

Paolo Xausa, Table of n, a(n) for n = 0..8191
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000225, A011782, A001235, A060867, A139250, A147562, A160410, A160414, A256531, A256534.

With[{z=7},Join[{0},Flatten[Array[(2^#-1)^2+12Range[0,2^(#-1)-1]^2&,z]]]] (* Generates 2^z terms *) (* Paolo Xausa, Nov 15 2023, after Omar E. Pol *)

For i = 1 to z: for j = 0 to 2^(i-1)-1: n = n+1: a(n) = (2^i-1)^2 + 3*(2*j)^2: next j: next i

A286331 Triangle read by rows: T(n,k) is the number of n X n matrices of rank k over F_2.

Original entry on oeis.org

1, 1, 1, 1, 9, 6, 1, 49, 294, 168, 1, 225, 7350, 37800, 20160, 1, 961, 144150, 4036200, 19373760, 9999360, 1, 3969, 2542806, 326932200, 8543828160, 39687459840, 20158709760, 1, 16129, 42677334, 23435953128, 2812314375360, 71124337751040, 325139829719040, 163849992929280

Offset: 0

Geoffrey Critzer, May 07 2017

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   9,      6;
  1,  49,    294,     168;
  1, 225,   7350,   37800,    20160;
  1, 961, 144150, 4036200, 19373760, 9999360;
  ...
T(2,1) = 9 because there are 9, 2 X 2 matrices in F_2 that have rank 1: {{0, 0}, {0, 1}}, {{0, 0}, {1, 0}}, {{0, 0}, {1, 1}}, {{0, 1}, {0, 0}}, {{0, 1}, {0, 1}}, {{1, 0},  {0, 0}}, {{1, 0}, {1, 0}}, {{1,1}, {0, 0}}, {{1, 1}, {1, 1}}.

Robert Israel, Table of n, a(n) for n = 0..1710 (rows 0 to 57, flattened)
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Wikipedia, q-binomial

Main diagonal is A002884.
Column for k = 1 is A060867.
Row sums are A002416.

T:= (n,k) -> mul((2^n-2^j)^2/(2^k-2^j),j=0..k-1):
seq(seq(T(n,k),k=0..n),n=0..10); # Robert Israel, May 15 2017

q = 2; Table[Table[Product[(q^n - q^i)^2/(q^k - q^i), {i, 0, k - 1}], {k, 0, n}], {n, 0, 6}] // Grid

T(n,k) = Product_{j=0..k-1} (2^n - 2^j)^2/(2^k - 2^j) = A022166(n,k) * Product_{j=0..k-1} (2^n - 2^j).

A165665 a(n) = (32^n - 2) 2^n.

Original entry on oeis.org

1, 8, 40, 176, 736, 3008, 12160, 48896, 196096, 785408, 3143680, 12578816, 50323456, 201310208, 805273600, 3221159936, 12884770816, 51539345408, 206157905920, 824632672256, 3298532786176, 13194135339008, 52776549744640

Offset: 0

Klaus Brockhaus, Sep 24 2009

Binomial transform of A058481. Second binomial transform of (A082505 without initial term 0). Third binomial transform of A010686.
Partial sums are in A060867.
a(n) is the sum of the odd numbers taken progressively by moving through them by 2^n-tuples. a(0)=1; a(1) = 3+5=8; a(2) = 7+9+11+13 = 40; a(3) = 15+17+19+21+23+25+27+29 = 176; a(n) = sum_{k=0,1,..,A000225(n)} (A000225(n+1)+2*k).  - J. M. Bergot, Dec 06 2014
The number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A058481, A082505, A010686 (repeat 1, 5), A060867, A010036, A124647.

Magma
```
[ (3*2^n-2)*2^n: n in [0..23] ];
```

Table[(3*2^n-2)2^n,{n,0,30}] (* or  *) LinearRecurrence[{6,-8},{1,8},30] (* Harvey P. Dale, Nov 18 2020 *)

a(n)=(3*2^n-2)*2^n \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 6*a(n-1)-8*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
a(n) = 8*A010036(n-1) for n > 0.
G.f.: (2*x+1)/((1-2*x)*(1-4*x)).
E.g.f.: 3*e^(4*x) - 2*e^(2*x). - Robert Israel, Dec 15 2014

A224195 Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.

Original entry on oeis.org

3, 5, 7, 9, 13, 15, 17, 25, 29, 31, 33, 49, 57, 61, 63, 65, 97, 113, 121, 125, 127, 129, 193, 225, 241, 249, 253, 255, 257, 385, 449, 481, 497, 505, 509, 511, 513, 769, 897, 961, 993, 1009, 1017, 1021, 1023, 1025, 1537, 1793, 1921, 1985, 2017, 2033, 2041, 2045, 2047

Offset: 1

Brad Clardy, Apr 01 2013

The table is constructed so that row labels are 2^n - 1, and column labels are 2^n. The body of the table is the row*col + 1. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the antidiagonals starting from the top left corner.
All of these numbers have the following property:
   let m be a member of A(n),
   if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then
   the differences between successive members of B(n) is a repeating series
   of 1's with the last difference in the pattern m. The number of ones in
   the pattern is 2^j - 1, where j is the column index.
As an example consider A(4) which is 9,
   the sequence B(n) where i XOR 8 = i - 8 starts as:
   8, 9, 10, 11, 12, 13, 14, 15, 24... (A115419)
   with successive differences of:
   1, 1, 1, 1, 1, 1, 1, 9.
The main diagonal is the 6th cyclotomic polynomial evaluated at powers of two (A020515).
The formula for diagonals above the main diagonal
  2^(2*n+1) - 2^(n + (a+1)/2) + 1 n>=(a+1)/2 a=odd number above diagonal
  2^(2*n) - 2^(n  + (b/2)) + 1    n>=(b/2)+1 b=even number above diagonal
The formulas for diagonals below the main diagonal
  2^(2*n+1) - 2^(n + 1 -(a+1)/2) + 1 n>=(a+1)/2 a=odd number below diagonal
  2^(2*n) - 2^(n - (b/2)) + 1       n>=(b/2)+1 b=even number below diagonal
Primes of this sequence are in A152449.

			Using the lexicographic ordering of A057555 the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
  +1  |    2    4     8    16    32     64    128    256     512    1024 ...
  ----|-----------------------------------------------------------------
  1   |    3    5     9    17    33     65    129    257     513    1025
  3   |    7   13    25    49    97    193    385    769    1537    3073
  7   |   15   29    57   113   225    449    897   1793    3585    7169
  15  |   31   61   121   241   481    961   1921   3841    7681   15361
  31  |   63  125   249   497   993   1985   3969   7937   15873   31745
  63  |  127  253   505  1009  2017   4033   8065  16129   32257   64513
  127 |  255  509  1017  2033  4065   8129  16257  32513   65025  130049
  255 |  511 1021  2041  4081  8161  16321  32641  65281  130561  261121
  511 | 1023 2045  4089  8177 16353  32705  65409 130817  261633  523265
  1023| 2047 4093  8185 16369 32737  65473 130945 261889  523777 1047553
  ...

Brad Clardy, Table of n, a(n) for n = 1..1000

Cf. A081118, A152449 (primes), A057555 (lexicographic ordering), A115419 (example).
Rows: A000051(i=1), A181565(2), A083686(3), A195744(4), A206371(5), A196657(6).
Cols: A000225(j=1), A036563(2), A048490(3), A176303 (7 offset of 8).
Diagonals: A020515 (main), A092440, A060867 (above), A134169 (below).

//program generates values in a table form
for i:=1 to 10 do
    m:=2^i - 1;
    m,[ m*2^n +1 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along antidiagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
    for j:=1 to i-1 do
       m:=2^j -1;
       k:=m*2^(i-j) + 1;
       if IsPrime(k) then k,"*";
          else k;
       end if;;
    end for;
end for;

Table[(2^j-1)*2^(i-j+1) + 1, {i, 10}, {j, i}] (* Paolo Xausa, Apr 02 2024 *)

a(n) = (2^(A057555(2*n-1)) - 1)*2^(A057555(2*n)) + 1 for n>=1. [corrected by Jason Yuen, Feb 22 2025]

a(n) = A081118(n)+2; a(n)=(2^i-1)*2^j+1, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 04 2013

A060868 Number of n X n matrices over GF(3) with rank 1.

Original entry on oeis.org

2, 32, 338, 3200, 29282, 264992, 2389298, 21516800, 193690562, 1743333152, 15690352658, 141214236800, 1270931319842, 11438391444512, 102945551698418, 926510051379200, 8338590720693122, 75047317261079072, 675425857674234578, 6078832726041680000, 54709494555295826402

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), May 04 2001

			a(2) = 32 because there are 33 (the second element in sequence A060705) singular 2 X 2 matrices over GF(3), that have rank <= 1 of which only the zero matrix has rank zero so a(2) = 33 - 1 = 32.

Harry J. Smith, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (13,-39,27).

Column k=3 of A379587.

Cf. A060705, A060867, A060869, A238976.

a(n) = { (3^n - 1)^2 / 2 } \\ Harry J. Smith, Jul 13 2009

a(n) = 1/2 * (3^n - 1)^2.
G.f.: -2*x*(3*x+1) / ((x-1)*(3*x-1)*(9*x-1)). - Colin Barker, Dec 23 2012
E.g.f.: exp(x)*(1 - 2*exp(2*x) + exp(8*x))/2. - Stefano Spezia, Dec 26 2024

More terms from Jason Earls, May 05 2001

A258844 Numbers k with the property that it is possible to write the base 2 expansion of k as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have (a+b)^2 = k.

Original entry on oeis.org

9, 36, 49, 100, 225, 784, 961, 1296, 2601, 3969, 7225, 8281, 14400, 16129, 18496, 21609, 29241, 34969, 42025, 65025, 116964, 123201, 133225, 246016, 261121, 278784, 465124, 508369, 672400, 700569, 828100, 1046529, 1368900, 1590121, 1782225, 4064256, 4190209, 4326400

Offset: 1

Paolo P. Lava, Jun 12 2015

Obviously all terms are squares.

The terms that have b=1 are: 9, 49, 225, 961, 3969, 16129, 65025, ...; see A060867 ((2^n-1)^2). - Michel Marcus, Jun 13 2015

			9 in base 2 is 1001. If we take 1001 = concat(10,01) then 10 and 01 converted to base 10 are 2 and 1. Finally (2 + 1)^2 = 3^2 = 9;
36 in base 2 is 100100. If we take 100100 = concat(10,0100) then 10 and 0100 converted to base 10 are 2 and 4. Finally (2 + 4)^2 = 6^2 = 36.

Cf. A258813, A258843.

with(numtheory): P:=proc(q) local a,b,c,k,n;
for n from 1 to q do c:=convert(n,binary,decimal);
for k from 1 to ilog10(c) do
a:=convert(trunc(c/10^k),decimal,binary);
b:=convert((c mod 10^k),decimal,binary);
if a*b>0 then if (a+b)^2=n then print(n); break;
fi; fi; od; od; end: P(10^6);

isok(n) = {b = binary(n); if (#b > 1, for (k=1, #b-1, vba = Vecrev(vector(k, i, b[i])); vbb = Vecrev(vector(#b-k, i, b[k+i])); da = sum(i=1, #vba, vba[i]*2^(i-1)); db = sum(i=1, #vbb, vbb[i]*2^(i-1)); if ((da+ db)^2 == n, return(1));););} \\ Michel Marcus, Jun 13 2015

A085903 Expansion of (1 + 2x^2)/((1 + x)(1 - 2x)(1 - 2*x^2)).

Original entry on oeis.org

1, 1, 7, 9, 31, 49, 127, 225, 511, 961, 2047, 3969, 8191, 16129, 32767, 65025, 131071, 261121, 524287, 1046529, 2097151, 4190209, 8388607, 16769025, 33554431, 67092481, 134217727, 268402689, 536870911, 1073676289, 2147483647

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 16 2003

Resultant of the polynomial x^n - 1 and the Chebyshev polynomial of the first kind T_2(x).

This sequence is the case P1 = 1, P2 = 0, Q = -2 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (1,4,-2,-4).

Cf. A083420, A028400, A062510, A088037, A107663.

Cf. A060867. A100047.

[Round((Sqrt(2)^n - 1)*(Sqrt(2)^n - (-1)^n)): n in [1..40]]; // Vincenzo Librandi, Apr 28 2014

seq(simplify((sqrt(2)^n - 1)*(sqrt(2)^n - (-1)^n)), n = 1..30); # Peter Bala, Apr 27 2014

CoefficientList[ Series[(1 + 2x^2)/(1 - x - 4x^2 + 2x^3 + 4x^4), {x, 0, 30}], x] (* Robert G. Wilson v, May 04 2013 *)
LinearRecurrence[{1,4,-2,-4},{1,1,7,9},40] (* Harvey P. Dale, Jul 25 2016 *)

a(n) = polresultant(x^n - 1, 2*x^2 - 1) \\ David Wasserman, Feb 10 2005

def A085903(n): return (1<>1))-1)**2 # Chai Wah Wu, Jun 19 2024

a(2*n) = 2*4^n - 1, a(2*n + 1) = (2^n - 1)^2; interlaces A083420 with A060867 (squares of Mersenne numbers A000225). - Creighton Dement, May 19 2005
A107663(2*n) = a(2*n) = A083420(n). - Creighton Dement, May 19 2005
From Peter Bala, Apr 27 2014: (Start)
a(n) = (sqrt(2)^n - 1)*(sqrt(2)^n - (-1)^n).
a(n) = Product_{k = 1..n} ( 2 - exp(4*k*Pi*i/n) ). (End)
E.g.f.: exp(-x) + exp(2*x) - 2*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Jun 16 2016

More terms from David Wasserman, Feb 10 2005

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007

A086699 Number of n X n matrices over GF(2) with rank n-1.

Original entry on oeis.org

1, 9, 294, 37800, 19373760, 39687459840, 325139829719040, 10654345790226432000, 1396491759480328106803200, 732164571206732295657278668800, 1535460761275478347250381697633484800, 12880379193826999985837000446453418557440000

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 28 2003

nonn

a(n)/2^(n^2) is the probability that a random linear operator T on an n dimensional vector space over the field with two elements is such that the dimension of the range of T equals n-1. This probability is Product{j>=2} 1 - 1/2^j which is 2 times the probability that the dimension of the range of T equals n. Cf. A048651. - Geoffrey Critzer, Jun 28 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..58

Cf. A002884, A060867.

Table[Product[(q^n - q^i)^2/(q^(n - 1) - q^i), {i, 0, n - 2}] /.  q -> 2, {n, 0, 15}] (* Geoffrey Critzer, Jun 28 2017 *)

a(n) = prod(j=0, n-2, (2^n - 2^j)^2 / (2^(n-1)- 2^j)); \\ Michel Marcus, Jun 28 2017

for n>=2 : a(n) = product j=0...n-2 (2^n - 2^j)^2 / (2^(n-1)- 2^j).

More terms from David Wasserman, Mar 28 2005

A128831 Number of n-tuples where each entry is chosen from the subsets of {1,2,3} such that the intersection of all n entries is empty.

Original entry on oeis.org

1, 27, 343, 3375, 29791, 250047, 2048383, 16581375, 133432831, 1070599167, 8577357823, 68669157375, 549554511871, 4397241253887, 35181150961663, 281462092005375, 2251748274470911, 18014192351838207, 144114363443707903

Offset: 1

Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007

The general formula where each entry is chosen from the subsets of {1,..,k} is (2^n-1)^k. This may be shown by exhibiting a bijection to a set whose cardinality is obviously (2^n-1)^k, namely the set of all k-tuples with each entry chosen from the 2^n-1 proper subsets of {1,..,n}, i.e. for of the k entries {1,..,n} is forbidden. The bijection is given by (X_1,..,X_n) |-> (Y_1,..,Y_k) where for each j in {1,..,k} and each i in {1,..,n}, i is in Y_j if and only if j is in X_i. Sequence A060867 is the case where the entries are chosen from subsets of {1,2}.

			a(1)=(2^1-1)^3=1 because only one tuple of length one, namely ({}) has an empty intersection of its sole entry.
a(2)=27 because the valid 2-tuples are: ({},{}), ({},{1}), ({},{2}), ({},{3}), ({},{1,2}), ({},{1,3}), ({},{2,3}), ({},{1,2,3}), ({1},{}), ({2},{}), ({3},{}), ({1,2},{}), ({1,3},{}), ({2,3},{}), ({1,2,3},{}), ({1},{2}), ({1},{3}), ({1},{2,3}), ({2},{1}), ({2},{3}), ({2},{1,3}), ({3},{1}), ({3},{2}), ({3},{1,2}), ({1,2},{3}), ({1,3},{2}), ({2,3},{1})

R. P. Stanley, Enumerative Combinatorics, Volume 1, Wadsworth & Brooks 1986 p. 11.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (15,-70,120,-64).

Cf. A060867.

[(2^n-1)^3: n in [1..20]]; // Vincenzo Librandi, Mar 04 2018

Maple
```
for k from 1 to 20 do (2^k-1)^3; od;
```

Table[(2^n - 1)^3, {n, 30}] (* Vincenzo Librandi, Mar 04 2018 *)

a(n) = (2^n-1)^3; \\ Altug Alkan, Mar 04 2018

a(n) = (2^n-1)^3.

G.f.: x*(8*x^2+12*x+1)/((x-1)*(2*x-1)*(4*x-1)*(8*x-1)). [Colin Barker, Nov 17 2012]

A238976 a(n) = ((3^(n-1)-1)^2)/4.

Original entry on oeis.org

0, 1, 16, 169, 1600, 14641, 132496, 1194649, 10758400, 96845281, 871666576, 7845176329, 70607118400, 635465659921, 5719195722256, 51472775849209, 463255025689600, 4169295360346561, 37523658630539536, 337712928837117289, 3039416363020840000, 27354747277647913201, 246192725530212278416

Offset: 1

Kival Ngaokrajang, Mar 07 2014

If the Cantor square fractal is modified as shown in the illustration (see Links), then 4*a(n) is the total number of holes in the modified Cantor square fractal after n iterations. The total number of sides (outside) is 4*A171498(n-1). The total length of the sides (outside) converges to 20 when the initial total side length is 12 (starting with 5 unit squares).
For the Cantor square fractal, the total number of sides (outside) is 4*A168616(n+2). The total number of holes is 4*A060867(n-1) for n > 1. The total length of the sides (outside) converges to 12 with the same initial condition (i.e., 5 unit square); its maximum is 17.333... and is reached at n = 2, 3. The Cantor square fractal and modified one are not true fractals.
See illustrations in links.

Kival Ngaokrajang, Illustrations of initial terms
Eric Weisstein's World of Mathematics, Cantor Square Fractal
Index entries for linear recurrences with constant coefficients, signature (13,-39,27)

Cf. A024023, A060867, A060868, A168616, A171498.

a(n) = ((3^(n-1)-1)^2)/4; \\ Joerg Arndt, Mar 08 2014

a(n) = (A024023(n-1))^2/4.

G.f.: x*(3*x + 1)/((1-x)*(1-3*x)*(1-9*x)). - Ralf Stephan, Mar 14 2014

cp's OEIS Frontend

A256530 Number of ON cells at n-th stage in simple 2-dimensional cellular automaton (see Comments lines for definition).

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A286331 Triangle read by rows: T(n,k) is the number of n X n matrices of rank k over F_2.

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A165665 a(n) = (3*2^n - 2) * 2^n.

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A224195 Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.

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A060868 Number of n X n matrices over GF(3) with rank 1.

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A258844 Numbers k with the property that it is possible to write the base 2 expansion of k as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have (a+b)^2 = k.

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A085903 Expansion of (1 + 2*x^2)/((1 + x)*(1 - 2*x)*(1 - 2*x^2)).

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A165665 a(n) = (32^n - 2) 2^n.

A085903 Expansion of (1 + 2x^2)/((1 + x)(1 - 2x)(1 - 2*x^2)).