A056830 -id:A056830 - cp's OEIS frontend

A265721 Decimal representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

1, 0, 4, 99, 16, 1935, 64, 32319, 256, 522495, 1024, 8381439, 4096, 134189055, 16384, 2147368959, 65536, 34359279615, 262144, 549753978879, 1048576, 8796085682175, 4194304, 140737458995199, 16777216, 2251799696244735, 67108864, 36028796549201919, 268435456

Offset: 0

Robert Price, Dec 14 2015

Rule 33 also generates this sequence.

			From _Michael De Vlieger_, Dec 14 2015: (Start)
First 8 rows, replacing leading zeros with ".", the row converted to its binary (A265720), then decimal equivalent at right:
              1                ->               1  =     1
            . . 0              ->               0  =     0
          . . 1 0 0            ->             100  =     4
        1 1 0 0 0 1 1          ->         1100011  =    99
      . . . . 1 0 0 0 0        ->           10000  =    16
    1 1 1 1 0 0 0 1 1 1 1      ->     11110001111  =  1935
  . . . . . . 1 0 0 0 0 0 0    ->         1000000  =    64
1 1 1 1 1 1 0 0 0 1 1 1 1 1 1  -> 111111000111111  = 32319
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (0,21,0,-84,0,64).

Cf. A265718, A265720, A059841, A056830, A000975, A265722, A128918, A265723, A265724.

rule = 1; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]],2], {k,1,rows}]

print([2*4**n  - 7*2**(n-1) - 1 if n%2 else 2**n for n in range(50)]) # Karl V. Keller, Jr., Aug 24 2021

From Colin Barker, Dec 14 2015 and Apr 16 2019: (Start)
a(n) = 21*a(n-2) - 84*a(n-4) + 64*a(n-6) for n>5.
G.f.: (1-17*x^2+99*x^3+16*x^4-144*x^5) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)).
(End)
a(n) = 2*4^n  - 7*2^(n-1) - 1 for odd n; a(n) = 2^n for even n. -  Karl V. Keller, Jr., Aug 24 2021

A059848 As a square table by antidiagonals, the n-digit number which in base k starts 1010101...

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 0, 0, 1, 3, 5, 2, 1, 0, 1, 4, 10, 10, 3, 0, 0, 1, 5, 17, 30, 21, 3, 1, 0, 1, 6, 26, 68, 91, 42, 4, 0, 0, 1, 7, 37, 130, 273, 273, 85, 4, 1, 0, 1, 8, 50, 222, 651, 1092, 820, 170, 5, 0, 0, 1, 9, 65, 350, 1333, 3255, 4369, 2460, 341, 5, 1, 0, 1, 10

Offset: 0

Henry Bottomley, Feb 26 2001

			T(5,3)=10101 base 3=81+9+1=91; T(4,6)=1010 base 6=216+6=222. Table starts {0,0,0,0,...}, {1,1,1,1,...}, {0,1,2,3,...}, {1,2,5,10,...}, ...

Rows include A000004, A000012, A001477, A002522, A034262, A059826, A059830, A059839. Columns include A000035, A008619, A000975, A033113, A033114, A033115, A033116, A033117, A033118, A033119, A056830.

T(n, k)=floor[k^(n+1)/(k^2-1)] =T(n-2, k)+k^(n-1) =k*T(n-1, k)-((-1)^n-1)/2

A089815 a(n) = floor((n+2)^(n+2)/((n+2)^2-1)).

Original entry on oeis.org

1, 3, 17, 130, 1333, 17157, 266305, 4842756, 101010101, 2377597255, 62350352785, 1802828015430, 56984650387477, 1954883439200265, 72340172838076673, 2872362020438669320, 121815504877079063701, 5495610154611982192011, 262801002506265664160401

Offset: 0

Paul Barry, Nov 12 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000975, A033113-A033119, A059848, A062864, A056830.

[Floor((n+2)^(n+2)/((n+2)^2-1)): n in [0..50]]; // G. C. Greubel, Oct 10 2017

A089815:=n->floor((n+2)^(n+2)/((n+2)^2-1)): seq(A089815(n), n=0..30); # Wesley Ivan Hurt, Apr 11 2017

Table[Floor[(n + 2)^(n + 2)/((n + 2)^2 - 1)], {n, 0, 50}] (* G. C. Greubel, Oct 10 2017 *)

for(n=0,50, print1(floor((n+2)^(n+2)/((n+2)^2-1)), ", ")) \\ G. C. Greubel, Oct 10 2017

A104720 Expansion of 1/((1-x)(1-x^2)(1-10x)).

Original entry on oeis.org

1, 11, 112, 1122, 11223, 112233, 1122334, 11223344, 112233445, 1122334455, 11223344556, 112233445566, 1122334455667, 11223344556677, 112233445566778, 1122334455667788, 11223344556677889, 112233445566778899, 1122334455667789000, 11223344556677890010, 112233445566778900111, 1122334455667789001121

Offset: 0

Paul Barry, Mar 20 2005

Partial sums of A056830(n+1).

			From _Seiichi Manyama_, Sep 29 2018: (Start)
   1                  * 8 + 0  = 8;
   11                 * 8 + 1  = 89;
   112                * 8 + 1  = 897;
   1122               * 8 + 2  = 8978;
   11223              * 8 + 2  = 89786;
   112233             * 8 + 3  = 897867;
   1122334            * 8 + 3  = 8978675;
   11223344           * 8 + 4  = 89786756;
   112233445          * 8 + 4  = 897867564;
   1122334455         * 8 + 5  = 8978675645;
   11223344556        * 8 + 5  = 89786756453;
   112233445566       * 8 + 6  = 897867564534;
   1122334455667      * 8 + 6  = 8978675645342;
   11223344556677     * 8 + 7  = 89786756453423;
   112233445566778    * 8 + 7  = 897767564534231;
   1122334455667788   * 8 + 8  = 8978675645342312;
   11223344556677889  * 8 + 8  = 89786756453423120;
   112233445566778899 * 8 + 9  = 897867564534231201.
   1                  * 9 + 1  = 10;
   11                 * 9 + 2  = 101;
   112                * 9 + 2  = 1010;
   1122               * 9 + 3  = 10101;
   11223              * 9 + 3  = 101010;
   112233             * 9 + 4  = 1010101;
   1122334            * 9 + 4  = 10101010;
   11223344           * 9 + 5  = 101010101;
   112233445          * 9 + 5  = 1010101010;
   1122334455         * 9 + 6  = 10101010101;
   11223344556        * 9 + 6  = 101010101010;
   112233445566       * 9 + 7  = 1010101010101;
   1122334455667      * 9 + 7  = 10101010101010;
   11223344556677     * 9 + 8  = 101010101010101;
   112233445566778    * 9 + 8  = 1010101010101010;
   1122334455667788   * 9 + 9  = 10101010101010101;
   11223344556677889  * 9 + 9  = 101010101010101010;
   112233445566778899 * 9 + 10 = 1010101010101010101. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (11,-9,-11,10).

Cf. A056830, A294328, A294344.

List([0..25],n->1000*10^n/891+(-1)^n/44-(18*n+47)/324); # Muniru A Asiru, Sep 29 2018

seq(coeff(series(((1-x)*(1-x^2)*(1-10*x))^(-1),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Sep 29 2018

a[n_]:=1000*10^n/891 + (-1)^n/44 - (18*n + 47)/324 ; Array[a,50,0] (* or *)
a[n_]:=Floor[(2*10^(n + 3) - 99*n)/1782]; Array[a,50,0] (* Stefano Spezia, Sep 01 2018 *)
LinearRecurrence[{11,-9,-11,10},{1,11,112,1122},30] (* Harvey P. Dale, Jun 20 2021 *)

a(n) = 1000*10^n/891 + (-1)^n/44 - (18n+47)/324.
a(n) = floor((2*10^(n+3) - 99n)/1782). - Hieronymus Fischer, Dec 05 2006
a(n) = 10*a(n-1) + (2*n + 3 + (-1)^n)/4, a(0)=1, a(1)=11. - Vincenzo Librandi, Mar 22 2011

A377919 a(0) = 0; thereafter a(n) is the lexicographically earliest missing nonnegative integer such that the digits in the sequence (ignoring the commas) alternate in parity.

Original entry on oeis.org

0, 1, 2, 10, 101, 21, 210, 1010, 10101, 2101, 21010, 101010, 1010101, 210101, 2101010, 10101010, 101010101, 21010101, 210101010, 1010101010, 10101010101, 2101010101, 21010101010, 101010101010, 1010101010101, 210101010101, 2101010101010, 10101010101010, 101010101010101, 21010101010101, 210101010101010, 1010101010101010, 10101010101010101, 2101010101010101, 21010101010101010, 101010101010101010

Offset: 0

N. J. A. Sloane, Dec 08 2024

A "lexicographically earliest" variant of A098951.

			We begin by arranging the nonnegative integers whose digits alternate in parity in lexicographic order. First, 0; then the numbers with first digit 1: 1, 10, 101, 1010, 10101, 101010, ...; then the numbers with first digit 2: 2, 20, 201, 2010, 20101, 201010, ...; then the numbers with first digit 3, and so on.
The sequence begins with 0 and from then on we choose the first unused number from the above list which preserves alternating parity of the digits in the sequence.
(The above list does not have an OEIS entry, since there are uncountably many terms before the number 2 appears. In fact there are uncountably many terms before 12 appears. The beginning of the list coincides with A056830.)

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,100,-100,100,-100).

Cf. A056830, A098951.

LinearRecurrence[{1, -1, 1, 100, -100, 100, -100}, {0, 1, 2, 10, 101, 21, 210, 1010, 10101}, 50] (* Paolo Xausa, Dec 09 2024 *)

a(0) - a(6) are 0, 1, 2, 10, 101, 21, 210. Thereafter, for k >= 2,
  a(4*k-1) = a(4*k-5) || 10,
  a(4*k)   = a(4*k-4) || 01,
  a(4*k+1) = a(4*k-3) || 01,
  a(4*k+2) = a(4*k-2) || 10,
where || denotes concatenation.

A089816 a(n) = floor((n+3)^(n+2)/((n+3)^2-1)).

Original entry on oeis.org

1, 4, 26, 222, 2451, 33288, 538084, 10101010, 216145205, 5195862732, 138679078110, 4070332170534, 130325562613351, 4521260802379792, 168962471790509960, 6767528048726614650, 289242639716420115369

Offset: 0

Paul Barry, Nov 12 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000975, A033113-A033119, A059848, A062864, A056830.

[Floor((n+3)^(n+2)/((n+3)^2-1)): n in [0..25]]; // G. C. Greubel, Oct 11 2017

Table[Floor[(n + 3)^(n + 2)/((n + 3)^2 - 1)], {n, 0, 50}] (* G. C. Greubel, Oct 11 2017 *)

for(n=0, 25, print1(floor((n+3)^(n+2)/((n+3)^2-1)), ", ")) \\ G. C. Greubel, Oct 11 2017

A141116 Smallest n-digit prime with no identical adjacent digits (or 0 if no such prime exists).

Original entry on oeis.org

2, 13, 101, 1013, 10103, 101021, 1010129, 10101023, 101010157, 1010101039, 10101010163, 101010101063, 1010101010131, 10101010101019, 101010101010131, 1010101010101037, 10101010101010141, 101010101010101083

Offset: 1

Rick L. Shepherd, Jun 05 2008

For n >= 1, a(n) >= A056830(n), the least n-digit positive integer with no identical adjacent digits (also the least positive integer whose digits occur in n runs). Conjecture: For all n, a(n) <> 0.
If the conjecture is true, then this sequence and the following two sequences are equivalent: i) Smallest prime with exactly n runs of digits and ii) Smallest prime with at least n runs of digits. For each n <= 625, a(n) is an n-digit prime (provided that each probable prime shown in the link is indeed a prime -- or at least one of very many (slightly) larger probable prime candidates is prime).
As each a(n) shown is very near A056830(n), I believe it is extremely unlikely that a randomly-given n would yield a 0 term (but I don't have a proof for arbitrary n).

			a(4) = 1013 because 1013 is the smallest 4-digit prime having no identical adjacent digits; the only smaller 4-digit prime, 1009, is disqualified by the "00", identical adjacent digits (of run length 2). Also each digit, 1, 0, 1, 3, occurs in a run of identical digits of length 1 for a total of 4 runs with 1013 being the smallest prime of any length with 4 runs of digits.

Rick L. Shepherd, Table of n, a(n) for n = 1..625

Cf. A003617, A007809, A056830.

A144863 Start with 1, then at each step prepend 10 and append 01.

Original entry on oeis.org

1, 10101, 101010101, 1010101010101, 10101010101010101, 101010101010101010101, 1010101010101010101010101, 10101010101010101010101010101, 101010101010101010101010101010101

Offset: 1

Artur Jasinski, Sep 23 2008, Sep 25 2008

Bisection of A094028. - Omar E. Pol, Nov 12 2008
a(n) is also A144864(n) written in base 2. - Omar E. Pol, Nov 13 2008
Quadrisection of A147759. - Omar E. Pol, Nov 16 2008

Index entries for linear recurrences with constant coefficients, signature (10001,-10000).

Cf. A056830, A094028, A135576, A144864, A147759.

a = {}; k = {1}; Do[x = FromDigits[k, 2]; AppendTo[a, FromDigits[RealDigits[x, 2]]]; AppendTo[k, 0]; AppendTo[k, 1]; PrependTo[k, 0]; PrependTo[k, 1], {n, 1, 100}];
Table[FromDigits[RealDigits[1/12 (-4 + 16^n), 2]], {n, 1, 10}]
a = {}; k = 1; Do[AppendTo[a, k]; k = 10000 k + 101, {n, 1, 10}]; a
Table[1/99 (-1 + 100^(-1 + 2 n)), {n, 1, 20}]
LinearRecurrence[{10001,-10000},{1,10101},20] (* Harvey P. Dale, Aug 22 2014 *)

a(n) = (-1+100^(-1+2*n))/99.
If a(n) is interpreted as binary number, (-4+16^n)/12 gives the decimal representation of a(n).
a(n) = 10000*a(n-1)+101, n>1.
G.f.: x*(1+100*x) / ( (10000*x-1)*(x-1) ).

cp's OEIS Frontend

A265721 Decimal representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

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A059848 As a square table by antidiagonals, the n-digit number which in base k starts 1010101...

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A089815 a(n) = floor((n+2)^(n+2)/((n+2)^2-1)).

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A104720 Expansion of 1/((1-x)(1-x^2)(1-10x)).

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A377919 a(0) = 0; thereafter a(n) is the lexicographically earliest missing nonnegative integer such that the digits in the sequence (ignoring the commas) alternate in parity.

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A089816 a(n) = floor((n+3)^(n+2)/((n+3)^2-1)).

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Magma

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PARI

A141116 Smallest n-digit prime with no identical adjacent digits (or 0 if no such prime exists).

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A144863 Start with 1, then at each step prepend 10 and append 01.

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