A160717 -id:A160717 - cp's OEIS frontend

A134808 Cyclops numbers.

0, 101, 102, 103, 104, 105, 106, 107, 108, 109, 201, 202, 203, 204, 205, 206, 207, 208, 209, 301, 302, 303, 304, 305, 306, 307, 308, 309, 401, 402, 403, 404, 405, 406, 407, 408, 409, 501, 502, 503, 504, 505, 506, 507, 508, 509, 601, 602, 603, 604, 605, 606

Offset: 1

Omar E. Pol, Nov 21 2007

Numbers with middle digit 0, that have only one digit 0, and the total number of digits is odd; the digit 0 represents the eye of a cyclops.

			109 is a cyclops number because 109 has only one digit 0 and this 0 is the middle digit.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Cf. A134809, A138131, A138148, A160717, A182809.

cyclopsQ[n_Integer, b_:10] := Module[{digitList = IntegerDigits[n, b], len, pos0s, flag}, len = Length[digitList]; pos0s = Select[Range[len], digitList[[#]] == 0 &]; flag = OddQ[len] && (Length[pos0s] == 1) && (pos0s == {(len + 1)/2}); Return[flag]]; Select[Range[0,999],cyclopsQ] (* Alonso del Arte, Dec 16 2010 *)
Reap[Do[id=IntegerDigits[n];If[Position[id,0]=={{(Length[id]+1)/2}},Sow[n]],{n,0,10^3}]][[2,1]] (* Zak Seidov, Dec 17 2010 *)
cycQ[n_]:=Module[{idn=IntegerDigits[n],len=IntegerLength[n]},OddQ[len] && DigitCount[ n,10,0]==1&&idn[[(len+1)/2]]==0]; Join[{0},Select[Range[ 0,700],cycQ]] (* Harvey P. Dale, Mar 07 2020 *)

a(n, {base=10}) = my (l=0); my (r=n-1); while (r >= (base-1)^(2*l), r -= (base-1)^(2*l); l++); return (base^(l+1) * ( (base^l-1)/(base-1) + if (base>2, fromdigits(digits(r \ ((base-1)^l), (base-1)), base)) ) + ( (base^l-1)/(base-1) + if (base>2, fromdigits(digits(r % ((base-1)^l), (base-1)), base)))) \\ Rémy Sigrist, Apr 29 2017

from itertools import product
def cyclops(upto=float('inf'), upton=float('inf')): # generator
  yield 0
  c, n, half_digits, pow10 = 0, 1, 0, 10
  while 100**(half_digits+1) < upto and n < upton:
    half_digits += 1
    pow10 *= 10
    for left in product("123456789", repeat=half_digits):
      left_plus_eye = int("".join(left))*pow10
      for right in product("123456789", repeat=half_digits):
        c, n = left_plus_eye + int("".join(right)), n+1
        if c <= upto and n <= upton: yield c
print([c for c in cyclops(upto=606)])
print([c for c in cyclops(upton=52)]) # Michael S. Branicky, Jan 05 2021

def is_cyclops(n, base=10):
    dg = n.digits(base) if n > 0 else [0]
    return len(dg) % 2 == 1 and dg[len(dg)//2] == 0 and dg.count(0) == 1
is_A134808 = lambda n: is_cyclops(n)
# D. S. McNeil, Dec 17 2010

A160410 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 4, 16, 28, 64, 76, 112, 148, 256, 268, 304, 340, 448, 484, 592, 700, 1024, 1036, 1072, 1108, 1216, 1252, 1360, 1468, 1792, 1828, 1936, 2044, 2368, 2476, 2800, 3124, 4096, 4108, 4144, 4180, 4288, 4324, 4432, 4540, 4864, 4900, 5008, 5116, 5440, 5548, 5872, 6196

Offset: 0

Omar E. Pol, May 20 2009

On the infinite square grid, we consider cells to be the squares, and we start at round 0 with all cells in the OFF state, so a(0) = 0.
At round 1, we turn ON four cells, forming a square.
The rule for n > 1: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.
Therefore:
At Round 2, we turn ON twelve cells around the square.
At round 3, we turn ON twelve other cells. Three cells around of every corner of the square.
And so on.
For the first differences see the entry A161411.
Shows a fractal behavior similar to the toothpick sequence A139250.
A very similar sequence is A160414, which uses the same rule but with a(1) = 1, not 4.
When n=2^k then the polygon formed by ON cells is a square with side length 2^(k+1).
a(n) is also the area of the figure of A147562 after n generations if A147562 is drawn as overlapping squares. - Omar E. Pol, Nov 08 2009
From Omar E. Pol, Mar 28 2011: (Start)
Also, toothpick sequence starting with four toothpicks centered at (0,0) as a cross.
Rule: Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of three toothpicks of new generation. (Note that these three toothpicks looks like a T-toothpick, see A160172.)
The sequence gives the number of toothpicks after n stages. A161411 gives the number of toothpicks added at the n-th stage.
(End)

			From _Omar E. Pol_, Sep 24 2015: (Start)
With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
    4;
   16;
   28,  64;
   76, 112, 148, 256;
  268, 304, 340, 448, 484, 592, 700, 1024;
  ...
Right border gives the elements of A000302 greater than 1.
This triangle T(n,k) shares with the triangle A256534 the terms of the column k, if k is a power of 2, for example, both triangles share the following terms: 4, 16, 28, 64, 76, 112, 256, 268, 304, 448, 1024, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                         _ _ _ _
.      |  _ _  |                       |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _     _ _     _ _     _ _  |_| |
.      |_ _| |  _|_ _|_  |   |  _|_ _|_  | |_ _|
.          | |_|  _ _  |_|   |_|  _ _  |_| |
.          |   | |  _|_|_ _ _ _|_|_  | |   |
.          |  _| |_|  _ _     _ _  |_| |_  |
.          | | |_ _| |  _|_ _|_  | |_ _| | |
.          | |_ _| | |_|  _ _  |_| | |_ _| |
.          |       |   | |   | |   |       |
.          |  _ _  |  _| |_ _| |_  |  _ _  |
.          | |  _|_| | |_ _ _ _| | |_|_  | |
.          | |_|  _| |_ _|   |_ _| |_  |_| |
.          |   | | |_ _ _ _ _ _ _ _| | |   |
.          |  _| |_ _| |_     _| |_ _| |_  |
.       _ _| | |_ _ _ _| |   | |_ _ _ _| | |_ _
.      |  _| |_ _|   |_ _|   |_ _|   |_ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                       | |_ _| |
.      |_ _ _ _|                       |_ _ _ _|
.
After 10 generations there are 304 ON cells, so a(10) = 304.
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..1000
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.],
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 31.
Omar E. Pol, Illustration of initial terms (2009)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000079, A000302, A048883, A139250, A147562, A160118, A160412, A160414, A161411, A160717, A160720, A160727, A256530, A256534.

RasterGraphics[state_?MatrixQ,colors_Integer:2,opts___]:=
Graphics[Raster[Reverse[1-state/(colors -1)]],
AspectRatio ->(AspectRatio/.{opts}/.AspectRatio ->Automatic),
Frame ->True, FrameTicks ->None, GridLines ->None];
rule=1340761804646523638425234105559798690663900360577570370705802859623\
705267234688669629039040624964794287326910250673678735142700520276191850\
5902735959769690
Show[GraphicsArray[Map[RasterGraphics,CellularAutomaton[{rule, {2,
{{4,2,1}, {32,16,8}, {256,128,64}}}, {1,1}}, {{{1,1}, {1,1}}, 0}, 9,-10]]]];
ca=CellularAutomaton[{rule,{2,{{4,2,1},{32,16,8},{256,128,64}}},{1,
1}},{{{1,1},{1,1}},0},99,-100];
Table[Total[ca[[i]],2],{i,1,Length[ca]}]
(* John W. Layman, Sep 01 2009; Sep 02 2009 *)
a[n_] := 4*Sum[3^DigitCount[k, 2, 1], {k, 0, n-1}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 17 2017, after N. J. A. Sloane *)

A160410(n)=sum(i=0,n-1,3^norml2(binary(i)))<<2 \\ M. F. Hasler, Dec 04 2012

Equals 4*A130665. This provides an explicit formula for a(n). - N. J. A. Sloane, Jul 13 2009

a(2^k) = (2*(2^k))^2 for k>=0.

Edited by David Applegate and N. J. A. Sloane, Jul 13 2009

A160711 Cyclops squares: squares (A000290) that are also cyclops numbers (A134808).

Original entry on oeis.org

0, 11025, 19044, 21025, 24025, 32041, 38025, 42025, 47089, 51076, 58081, 59049, 65025, 66049, 67081, 75076, 87025, 93025, 1110916, 1140624, 1170724, 1190281, 1240996, 1270129, 1290496, 1340964, 1350244, 1380625, 1420864, 1430416

Offset: 1

Omar E. Pol, Jun 08 2009

			19044 is in the sequence because it is a square (138^2) and is also a cyclops number (odd number of digits, middle digit is the only zero).
11025 is in the sequence because it is a square (105^2) and is also a cyclops number (odd number of digits, middle digit is the only zero). - _Michael B. Porter_, Jul 09 2016

Kenny Lau, Table of n, a(n) for n = 1..20001

Cf. A000290, A134808, A134809, A134148, A138131, A153806, A160712, A160717, A239828.

Select[Range[0, 1200]^2, And[OddQ@ Length@ #, #[[Ceiling[Length[#]/2]]] == 0, Count[#, 0] == 1] &@ IntegerDigits@ # &] (* Michael De Vlieger, Jul 08 2016 *)
cnQ[n_]:=Module[{len=IntegerLength[n]},OddQ[len]&&DigitCount[n,10,0]==1 && IntegerDigits[n][[(len+1)/2]]==0]; Join[{0},Select[Range[1200]^2,cnQ]] (* Harvey P. Dale, Mar 19 2018 *)

A160712 Composite cyclops numbers (A134808).

Original entry on oeis.org

102, 104, 105, 106, 108, 201, 202, 203, 204, 205, 206, 207, 208, 209, 301, 302, 303, 304, 305, 306, 308, 309, 402, 403, 404, 405, 406, 407, 408, 501, 502, 504, 505, 506, 507, 508, 602, 603, 604, 605, 606, 608, 609, 702, 703, 704

Offset: 1

Omar E. Pol, Jun 08 2009

Cf. A134808, A134809, A134148, A138131, A153806, A160711, A160717.

Edited by Omar E. Pol, Jul 04 2009

A182809 Cyclops-Fibonacci numbers.

Original entry on oeis.org

0, 75025, 6557470319842, 14472334024676221, 99194853094755497

Offset: 1

Omar E. Pol, Dec 16 2010

Note that a(5) = 99194853094755497 is the only known Cyclops-Fibonacci prime.
Next term, if it exists, is > Fibonacci(2359000). - Lars Blomberg, May 10 2011
This sequence is similar to A182811 in the sense that both have four positive terms and the only known prime is also the largest known term. - Omar E. Pol, Feb 18 2018
Indices in A000045 are 0, 25, 63, 79, 83. - Michel Marcus and Omar E. Pol, Feb 18 2018

			a(2) = 75025 is in the sequence because 75025 is a Fibonacci number A000045 and 75025 is also a cyclops number A134808.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 99194853094755497

Intersection of A000045 and A134808.

Cf. A134809, A138148, A160717, A182811.

(* First run the program given for A134808 *) Select[Fibonacci[Range[10^3]], cyclopsQ] (* Alonso del Arte, Dec 16 2010 *)

a(4) inserted by Alois P. Heinz, Dec 16 2010

A160725 Cyclops semiprimes.

Original entry on oeis.org

106, 201, 202, 203, 205, 206, 209, 301, 302, 303, 305, 309, 403, 407, 501, 502, 505, 703, 706, 707, 802, 803, 807, 901, 905, 11013, 11014, 11015, 11017, 11019, 11021, 11023, 11029, 11031, 11035, 11038, 11041, 11042, 11051, 11053

Offset: 1

Omar E. Pol, Jun 12 2009

Cyclops numbers (A134808) that are also semiprimes (A001358).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A134808, A134809, A138131, A138148, A153806, A153807, A160711, A160712, A160717.

g:= proc(x,n)
  local L,i;
  L:= convert(x+9^(2*n),base,9);
  add((L[i]+1)*10^(i-1),i=1..n)+add((L[i]+1)*10^i,i=n+1..2*n)
end proc:
select(t -> numtheory:-bigomega(t)=2,[seq(seq(g(i,n),i=0..9^(2*n)-1),n=1..2)]); # Robert Israel, Jan 20 2019

Select[Range@ 12000, And[OddQ@ #2, #3[[Ceiling[#2/2] ]] == 0, Count[#3, 0] == 1, PrimeOmega@ #1 == 2] & @@ {#, IntegerLength@ #, IntegerDigits@ #} &] (* or *)
Select[Flatten@ Table[a (10^(d + 1)) + b, {d, 2}, {a, FromDigits /@ Tuples[Range@ 9, {d}]}, {b, FromDigits /@ Tuples[Range@ 9, {d}]}], PrimeOmega@ # == 2 &] (* Michael De Vlieger, Jan 20 2019 *)

A183056 Cyclops partition numbers.

Original entry on oeis.org

101, 26015, 483502844, 27517052599, 2814570987591, 269232701252579, 12269218019229465, 477535459708164115593, 55271949286085137715955, 98175979536033971312388, 28848173767368633057992125893483779

Offset: 1

Omar E. Pol, Dec 21 2010

			a(1) = 101 is in the sequence because 101 is a partition number A000041 and it is also a cyclops number A134808.

Cf. A000041, A134808, A134809, A160717.

cnQ[n_]:=Module[{idn=IntegerDigits[n],len},len=Length[idn];DigitCount[ n,10,0]==1&&OddQ[len]&&idn[[(len+1)/2]]==0]; Select[PartitionsP[ Range[ 2000]],cnQ] (* Harvey P. Dale, Apr 10 2019 *)

Intersection of A000041 and A134808.

a(11) from Alois P. Heinz, Dec 23 2010

A285767 Cyclops octagonal numbers: a(n) = n(3n-2) with one "zero" digit in the middle.

Original entry on oeis.org

0, 408, 11041, 18096, 22016, 23056, 28033, 38081, 56033, 61061, 1140833, 1170625, 1250656, 1410416, 1460216, 1540833, 2120161, 2130261, 2140385, 2150533, 2310896, 2390561, 2460696, 2520833, 2570576, 2780181, 2920533, 3230256, 3280256, 3490565, 3660865, 3680776

Offset: 1

K. D. Bajpai, Apr 25 2017

The n-th octagonal number x(n) = n*(3*n - 2).
Subset of A000567.
All the terms have the number of digits odd with only one "zero" digit in the middle.

			For n = 12; x(12) = 12*(3*12 - 2) = 408 that is 12th octagonal number with one zero digit in the middle, hence appears in the sequence.
For n = 61; x(61) = 61*(3*61 - 2) = 11041 that is 61st octagonal number with one zero digit in the middle, hence appears in the sequence.

Intersection of A000567 and A134808.

Cf. A000567, A134808, A134809, A160717.

iscyclops:= proc(n) local L,t;
t:= ilog10(n);
if t::odd then return false fi;
L:= convert(n,base,10);
L[1+t/2] = 0 and numboccur(0,L) = 1
end proc:
iscyclops(0):= true:
select(iscyclops, [seq(n*(3*n-2),n=0..1000)]);

Select[Table[n (3 n - 2), {n, 0, 1110}], And[OddQ@ Length@ #, Count[#, 0] == 1, Take[#, {Ceiling[Length[#]/2]}] == {0}] &@ IntegerDigits@ # &] (* Michael De Vlieger, Apr 26 2017 *)

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A134808 Cyclops numbers.

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A160410 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

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A160711 Cyclops squares: squares (A000290) that are also cyclops numbers (A134808).

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A160712 Composite cyclops numbers (A134808).

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A182809 Cyclops-Fibonacci numbers.

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A160725 Cyclops semiprimes.

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A183056 Cyclops partition numbers.

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A285767 Cyclops octagonal numbers: a(n) = n*(3*n-2) with one "zero" digit in the middle.

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A285767 Cyclops octagonal numbers: a(n) = n(3n-2) with one "zero" digit in the middle.