A085601 -id:A085601 - cp's OEIS frontend

A267816 Decimal representation of the n-th iteration of the "Rule 221" elementary cellular automaton starting with a single ON (black) cell.

1, 3, 23, 111, 479, 1983, 8063, 32511, 130559, 523263, 2095103, 8384511, 33546239, 134201343, 536838143, 2147418111, 8589803519, 34359476223, 137438429183, 549754765311, 2199021158399, 8796088827903, 35184363700223, 140737471578111, 562949919866879

Offset: 0

Robert Price, Jan 20 2016

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

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A267814.

Similar entries: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A281481 (2^(n - 1) * (2^n + 1) + 1), A281482 (2^(n + 1) * (2^n + 1) - 1). - Jaroslav Krizek, Jan 22 2017

rule=221; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Conjectures from Colin Barker, Jan 22 2016 and Apr 16 2019: (Start)
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.
G.f.: (1-4*x+16*x^2-16*x^3) / ((1-x)*(1-2*x)*(1-4*x)).
(End)
a(n) = 2^(n + 1) * (2^n - 1) - 1, for n > 0. - Jaroslav Krizek, Jan 22 2017

A281481 a(n) = 2^(n - 1) * (2^n + 1) + 1.

Original entry on oeis.org

2, 4, 11, 37, 137, 529, 2081, 8257, 32897, 131329, 524801, 2098177, 8390657, 33558529, 134225921, 536887297, 2147516417, 8590000129, 34359869441, 137439215617, 549756338177, 2199024304129, 8796095119361, 35184376283137, 140737496743937, 562949970198529

Offset: 0

Jaroslav Krizek, Jan 22 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Similar sequences: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A267816 (2^(n + 1) * (2^n - 1) - 1), A281482 (2^(n + 1) * (2^n + 1) - 1).

Cf. A278930.

[2^(n - 1) * (2^n + 1) + 1: n in [0..200]];

Vec((2 - 10*x + 11*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 22 2017

From Colin Barker, Jan 22 2017: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
G.f.: (2 - 10*x + 11*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)
a(n) = A278930(n - 2) for n >= 7. - Georg Fischer, Mar 26 2019

A296807 Take a prime, convert it to base 2. Consider it as a string of digits and delete its leftmost and rightmost digit. Leading zeros are kept. Repeat the process. a(n) is the least prime that, in the first n steps of this process, generates a string that is a prime read in base 2.

Original entry on oeis.org

2, 13, 43, 151, 2143, 2143, 12479, 57727, 246527, 4267455487, 276009615632383, 4469780781584383, 576406542684520447

Offset: 0

Paolo P. Lava, Paolo Iachia, Dec 21 2017

a(n) >= 2*4^n + 3*2^n - 1 = A297928(n) >= 2*(4^n + 2^n) + 1 = A085601(n), n > 0. - Iain Fox, Dec 29 2017 (edited by Iain Fox, Jan 08 2018)

a(17) <= 2^163 + 361736822347711983585853439 (probably much smaller), building on a Cunningham chain of length 17 found by Jaroslaw Wroblewski. a(n) exists for n <= 17, and probably for all n. - Jens Kruse Andersen, Jan 21 2018

			a(1) = 13 because 13 in base 2 is 1101 and 10 is 2 and 13 is the least number with this property;
a(2) = 43 because 43 in base 2 is 101011 while 0101 is 5 and 10 is 2 and 43 is the least number with this property;
a(3) = 151 because 151 in base 2 is 10010111 while 001011 is 11, 0101 is 5 and 10 is 2 and 151 is the least number with this property.

Cf. A000030, A010879, A004676, A296806.

with(numtheory): P:=proc(q) local a,b,c,i,j,k,n,ok,x; x:=5; for k from 1 to q do for n from x to q do a:=convert(ithprime(n),base,2); ok:=1; for i from 1 to k do b:=nops(a)-i; while a[b]=0 do b:=b-1; od;
c:=0; for j from b by -1 to i+1 do c:=2*c+a[j]; od;if not isprime(c) then ok:=0; break; fi; od;if ok=1 then x:=n; print(ithprime(n)); break; fi; od; od; end: P(10^20);

Table[SelectFirst[Prime@ Range[#, # + 10^5] &@ PrimePi[2 (4^n + 2^n) + 1], AllTrue[Map[FromDigits[#, 2] &, Rest@ NestWhileList[Most@ Rest@ # &, IntegerDigits[#, 2], Length@ # > 2 &]], PrimeQ] &], {n, 8}] (* Michael De Vlieger, Dec 29 2017 *)

a(n) = if(!n, return(2)); forprime(p=2*4^n + 3*2^n - 1, , my(b=p); for(x=1, n, b = (b - (b>=4*2^(logint(p, 2) - 2*x))*4*2^(logint(p, 2) - 2*x) - 1)/2; if(!isprime(b) || (b==2 && x!=n), next(2))); return(p)) \\ Iain Fox, Dec 29 2017 (corrected by Iain Fox, Oct 26 2019)

Definition corrected, a(10)-a(12) by Jens Kruse Andersen, Jan 21 2018

A220980 a(n) = 5^(4n+2) + 5^(3n+2) + 3 * 5^(2n+1) + 5^(n+1) + 1: the right Aurifeuillian factor of 5^(10n+5) - 1.

Original entry on oeis.org

71, 19151, 10165751, 6152578751, 3820806643751, 2384948876968751, 1490211490478593751, 931334495635986718751, 582078099253082277343751, 363798066973743438730468751, 227373698726297855377246093751, 142108550062403118610382324218751

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding left Aurifeuillian factor is A220979.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (781,-101530,2538250,-12203125,9765625).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[5^(4n+2) + 5^(3n+2) + 3 * 5^(2n+1) + 5^(n+1) + 1, {n, 0, 20}]

Aurifeuillian factorization: 5^(10n+5) - 1 = (5^(2n+1) - 1) * A220979(n) * a(n).

G.f.: -(27734375*x^4-22687500*x^3+2417450*x^2-36300*x+71) / ((x-1)*(5*x-1)*(25*x-1)*(125*x-1)*(625*x-1)). [Colin Barker, Jan 03 2013]

A220981 a(n) = 6^(4n+2) - 6^(3n+2) + 3 * 6^(2n+1) - 6^(n+1) + 1: the left Aurifeuillian factor of 6^(12n+6) + 1.

Original entry on oeis.org

13, 39493, 58809673, 78002205553, 101481622729633, 131604778271166913, 170578072060319947393, 221073129991920857571073, 286511629376393032228157953, 371319255900007820952456748033, 481229795439713382306649129101313

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A220982.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (1555,-345210,12427560,-72550080,60466176).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[6^(4n+2) - 6^(3n+2) + 3 * 6^(2n+1) - 6^(n+1) + 1, {n, 0, 20}]
LinearRecurrence[{1555,-345210,12427560,-72550080,60466176},{13,39493,58809673,78002205553,101481622729633},20] (* Harvey P. Dale, Oct 01 2021 *)

Aurifeuillian factorization: 6^(12n+6) + 1 = (6^(4n+2) + 1) * a(n) * A220982(n).

G.f.: -(21835008*x^4+24984288*x^3+1885788*x^2+19278*x+13) / ((x-1)*(6*x-1)*(36*x-1)*(216*x-1)*(1296*x-1)). [Colin Barker, Jan 03 2013]

A220982 a(n) = 6^(4n+2) + 6^(3n+2) + 3 * 6^(2n+1) + 6^(n+1) + 1: the right Aurifeuillian factor of 6^(12n+6) + 1.

Original entry on oeis.org

97, 55117, 62169337, 78727802257, 101638351073377, 131638631590149697, 170585384377200633217, 221074709452366968135937, 286511970539849391404729857, 371319329591314394530363646977, 481229811357035602199451623479297

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding left Aurifeuillian factor is A220981.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (1555,-345210,12427560,-72550080,60466176).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[6^(4n+2) + 6^(3n+2) + 3 * 6^(2n+1) + 6^(n+1) + 1, {n, 0, 20}]

Aurifeuillian factorization: 6^(12n+6) + 1 = (6^(4n+2) + 1) * A220981(n) * a(n).

G.f.: -(162922752*x^4-124050528*x^3+9947772*x^2-95718*x+97) / ((x-1)*(6*x-1)*(36*x-1)*(216*x-1)*(1296*x-1)). [Colin Barker, Jan 03 2013]

A220987 The left Aurifeuillian factor of 11^(22n+11) + 1.

Original entry on oeis.org

58367, 3812903020530517, 107454987376543082369146967, 2808133028073215608147547774721982717, 72885505321551844061773948114862247606146502767, 1890579685660625069233746109183146734516524279847333062117

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A220988.

Wikipedia, Cunningham Project

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[z = 11^n; 161051 z^10 - 161051 z^9 + 73205 z^8 - 14641 z^7 - 1331 z^6 + 1331 z^5 - 121 z^4 - 121 z^3 + 55 z^2 - 11 z + 1, {n, 0, 10}]

a(n) = 161051 z^10 - 161051 z^9 + 73205 z^8 - 14641 z^7 - 1331 z^6 + 1331 z^5 - 121 z^4 - 121 z^3 + 55 z^2 - 11 z + 1 with z = 11^n.

Aurifeuillian factorization: 11^(22n+11) + 1 = (11^(2n+1) + 1) * a(n) * A220988(n).

A220988 The right Aurifeuillian factor of 11^(22n+11) + 1.

Original entry on oeis.org

407353, 4572972882642803, 109245858982819139102535553, 2812355783638980226466572392952970603, 72895462357781065526518523423275265184080402953, 1890603163831201090586603020695655490130990020251181357603

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding left Aurifeuillian factor is A220987.

Wikipedia, Cunningham Project

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[z = 11^n; 161051 z^10 + 161051 z^9 + 73205 z^8 + 14641 z^7 - 1331 z^6 - 1331 z^5 - 121 z^4 + 121 z^3 + 55 z^2 + 11 z + 1, {n, 0, 10}]

a(n) = 161051 z^10 + 161051 z^9 + 73205 z^8 + 14641 z^7 - 1331 z^6 - 1331 z^5 - 121 z^4 + 121 z^3 + 55 z^2 + 11 z + 1 with z = 11^n.

Aurifeuillian factorization: 11^(22n+11) + 1 = (11^(2n+1) + 1) * A220987(n) * a(n).

A220989 a(n) = 12^(2n+1) - 6 * 12^n + 1: the left Aurifeuillian factor of 12^(6n+3) + 1.

Original entry on oeis.org

7, 1657, 247969, 35821441, 5159655937, 743006877697, 106993187463169, 15407021359595521, 2218611104160546817, 319479999339664244737, 46005119908998197280769, 6624737266944778960896001, 953962166440636632998608897

Offset: 0

Stuart Clary, Dec 27 2012

The corresponding right Aurifeuillian factor is A220990.

Wikipedia, Cunningham Project
Index entries for linear recurrences with constant coefficients, signature (157,-1884,1728).

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Table[12^(2n+1) - 6 * 12^n + 1, {n, 0, 20}]

Aurifeuillian factorization: 12^(6n+3) + 1 = (12^(2n+1) + 1) * a(n) * A220990(n).

G.f.: -(1008*x^2+558*x+7) / ((x-1)*(12*x-1)*(144*x-1)). [Colin Barker, Jan 03 2013]

A297928 a(n) = 24^n + 32^n - 1.

Original entry on oeis.org

4, 13, 43, 151, 559, 2143, 8383, 33151, 131839, 525823, 2100223, 8394751, 33566719, 134242303, 536920063, 2147581951, 8590131199, 34360131583, 137439739903, 549757386751, 2199026401279, 8796099313663, 35184384671743, 140737513521151, 562950003752959, 2251799914348543

Offset: 0

Iain Fox, Jan 08 2018

For n > 0, in binary, this is a 1 followed by n-1 0's followed by 10 followed by n 1's.

			a(0) = 2*4^0 + 3*2^0 - 1 = 4;   in binary, 100.
a(1) = 2*4^1 + 3*2^1 - 1 = 13;  in binary, 1101.
a(2) = 2*4^2 + 3*2^2 - 1 = 43;  in binary, 101011.
a(3) = 2*4^3 + 3*2^3 - 1 = 151; in binary, 10010111.
a(4) = 2*4^4 + 3*2^4 - 1 = 559; in binary, 1000101111.
...

Index entries for linear recurrences with constant coefficients, signature (7,-14,8)

A lower bound for A296807.

Cf. A000918, A085601.

Table[2 4^n+3 2^n-1,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{4,13,43},30] (* Harvey P. Dale, Apr 22 2018 *)

PARI
```
a(n) = 2*4^n + 3*2^n - 1
```

first(n) = Vec((4 - 15*x + 8*x^2)/((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^n))

G.f.: (4 - 15*x + 8*x^2)/((1 - x)*(1 - 2*x)*(1 - 4*x)).
E.g.f.: 2*e^(4*x) + 3*e^(2*x) - e^x.
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3), n > 2.
a(n) = A000918(n) + A085601(n).

cp's OEIS Frontend

A267816 Decimal representation of the n-th iteration of the "Rule 221" elementary cellular automaton starting with a single ON (black) cell.

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A281481 a(n) = 2^(n - 1) * (2^n + 1) + 1.

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A296807 Take a prime, convert it to base 2. Consider it as a string of digits and delete its leftmost and rightmost digit. Leading zeros are kept. Repeat the process. a(n) is the least prime that, in the first n steps of this process, generates a string that is a prime read in base 2.

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A220980 a(n) = 5^(4n+2) + 5^(3n+2) + 3 * 5^(2n+1) + 5^(n+1) + 1: the right Aurifeuillian factor of 5^(10n+5) - 1.

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A220981 a(n) = 6^(4n+2) - 6^(3n+2) + 3 * 6^(2n+1) - 6^(n+1) + 1: the left Aurifeuillian factor of 6^(12n+6) + 1.

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A220982 a(n) = 6^(4n+2) + 6^(3n+2) + 3 * 6^(2n+1) + 6^(n+1) + 1: the right Aurifeuillian factor of 6^(12n+6) + 1.

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A220987 The left Aurifeuillian factor of 11^(22n+11) + 1.

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A220988 The right Aurifeuillian factor of 11^(22n+11) + 1.

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A297928 a(n) = 24^n + 32^n - 1.