A001743 -id:A001743 - cp's OEIS frontend

A007932 Numbers that contain only 1's, 2's and 3's.

1, 2, 3, 11, 12, 13, 21, 22, 23, 31, 32, 33, 111, 112, 113, 121, 122, 123, 131, 132, 133, 211, 212, 213, 221, 222, 223, 231, 232, 233, 311, 312, 313, 321, 322, 323, 331, 332, 333, 1111, 1112, 1113, 1121, 1122, 1123, 1131, 1132, 1133, 1211, 1212, 1213, 1221

Offset: 1

R. Muller

This sequence is the alternate number system in base 3. - Robert R. Forslund (forslund(AT)tbaytel.net), Jun 27 2003
a(n) is the "bijective base-k numeration" or "k-adic notation" for k=3. - Chris Gaconnet (gaconnet(AT)gmail.com), May 27 2009
a(n) = n written in base 3 where zeros are not allowed but threes are. The three distinct digits used are 1, 2 and 3 instead of 0, 1 and 2. To obtain this sequence from the "canonical" base 3 sequence with zeros allowed, just replace any 0 with a 3 and then subtract one from the group of digits situated on the left: (20-->13; 100-->23; 110-->33; 1000-->223; 1010-->233). This can be done in any integer positive base b, replacing zeros with positive b's and subtracting one from the group of digits situated on the left. And zero is the only digit that can be replaced, since there is always a more significant digit greater than 0, on the left, from which to subtract one. - Robin Garcia, Jan 07 2014

			a(100)  = 3131.
a(10^3) = 323231.
a(10^4) = 111123331.
a(10^5) = 11231311131.
a(10^6) = 1212133131231.
a(10^7) = 123133223331331.
a(10^8) = 13221311111312131.
a(10^9) = 2113123122313232231.
- _Hieronymus Fischer_, Jun 06 2012

K. Atanassov, On the 97th, 98th and the 99th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 3, 89-93.
A. Salomaa, Formal Languages, Academic Press, 1973. pages 90-91. [From Chris Gaconnet (gaconnet(AT)gmail.com), May 27 2009]

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
K. Atanassov, On Some of Smarandache's Problems
EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics
Robert R. Forslund, A Logical Alternative to the Existing Positional Number System, Southwest Journal of Pure and Applied Mathematics. Vol. 1 1995 pp. 27-29.
James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.
F. Smarandache, Only Problems, Not Solutions!.
Wikipedia, Bijective numeration
Index entries for 10-automatic sequences.

Cf. A007931, A052382, A084544, A084545, A046034, A089581, A084984, A001742, A001743, A001744, A202267, A202268, A014261, A014263.

NextNbr[n_] := Block[{d = IntegerDigits[n + 1], l}, l = Length[d]; While[l != 1, If[ d[[l]] > 3, d[[l - 1]]++; d[[l]] = 1]; l-- ]; If[ d[[1]] > 3, d[[1]] = 11]; FromDigits[d]]; NestList[ NextNbr, 1, 51]
Table[FromDigits/@Tuples[{1,2,3},n],{n,4}]//Flatten (* Harvey P. Dale, Mar 29 2018 *)

a(n) = my (w=3); while (n>w, n -= w; w *= 3); my (d=digits(w+n-1, 3)); d[1] = 0; fromdigits(d) + (10^(#d-1)-1)/9 \\ Rémy Sigrist, Aug 28 2018

From Hieronymus Fischer, May 30 2012 and Jun 08 2012: (Start)
The formulas are designed to calculate base-10 numbers only using the digits 1, 2, 3.
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 3)*10^j,
where m = floor(log_3(2*n+1)), b(j) = floor((2*n+1-3^m)/(2*3^j)).
Special values:
a(k*(3^n-1)/2) = k*(10^n-1)/9, k=1,2,3.
a((5*3^n-3)/2) = (4*10^n-1)/3 = 10^n + (10^n-1)/3.
a((3^n-1)/2 - 1) = (10^(n-1)-1)/3, n>1.
Inequalities:
a(n) <= (10^log_3(2*n+1)-1)/9, equality holds for n=(3^k-1)/2, k>0.
a(n) > (3/10)*(10^log_3(2*n+1)-1)/9, n>0.
Lower and upper limits:
lim inf a(n)/10^log_3(2*n) = 1/30, for n --> infinity.
lim sup a(n)/10^log_3(2*n) = 1/9, for n --> infinity.
G.f.: g(x) = (x^(1/2)*(1-x))^(-1) Sum_{j=>0} 10^j*(x^3^j)^(3/2) * (1-x^3^j)*(1 + 2x^3^j + 3x^(2*3^j))/(1 - x^3^(j+1)).
Also: g(x) = (1/(1-x)) Sum_{j>=0} (1 - 4(x^3^j)^3 + 3(x^3^j)^4)*x^3^j*f_j(x)/(1-x^3^j), where f_j(x) = 10^j*x^((3^j-1)/2)/(1-(x^3^j)^3). The f_j obey the recurrence f_0(x) = 1/(1-x^3), f_(j+1)(x) = 10x*f_j(x^3).
Also: g(x) = (1/(1-x))*(h_(3,0)(x) + h_(3,1)(x) + h_(3,2)(x) - 3*h_(3,3)(x)), where h_(3,k)(x) = Sum_{j>=0} 10^j*x^((3^(j+1)-1)/2) * (x^3^j)^k/(1-(x^3^j)^3).
(End)

Edited and extended by Robert G. Wilson v, Dec 14 2002

Crossrefs added by Hieronymus Fischer, Jun 06 2012

A084544 Alternate number system in base 4.

Original entry on oeis.org

1, 2, 3, 4, 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34, 41, 42, 43, 44, 111, 112, 113, 114, 121, 122, 123, 124, 131, 132, 133, 134, 141, 142, 143, 144, 211, 212, 213, 214, 221, 222, 223, 224, 231, 232, 233, 234, 241, 242, 243, 244, 311, 312, 313, 314, 321

Offset: 1

Robert R. Forslund (forslund(AT)tbaytel.net), Jun 27 2003

			From _Hieronymus Fischer_, Jun 06 2012: (Start)
a(100)  = 1144.
a(10^3) = 33214.
a(10^4) = 2123434.
a(10^5) = 114122134.
a(10^6) = 3243414334.
a(10^7) = 211421121334.
a(10^8) = 11331131343334.
a(10^9) = 323212224213334. (End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics
R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1 1995 pp. 27-29.
R. R. Forslund, Positive Integer Pages [Broken link]
James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.
Index entries for 10-automatic sequences.

Cf. A007931, A007932, A052382, A084545, A046034, A089581, A084984, A001742, A001743, A001744, A202267, A202268, A014261, A014263.

def A084544(n):
    m = (3*n+1).bit_length()-1>>1
    return int(''.join((str(((3*n+1-(1<<(m<<1)))//(3<<((m-1-j)<<1))&3)+1) for j in range(m)))) # Chai Wah Wu, Feb 08 2023

From Hieronymus Fischer, Jun 06 and Jun 08 2012: (Start)
The formulas are designed to calculate base-10 numbers only using the digits 1..4.
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 4)*10^j,
where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).
Special values:
a(k*(4^n-1)/3) = k*(10^n-1)/9, k = 1,2,3,4.
a((7*4^n-4)/3) = (13*10^n-4)/9 = 10^n + 4*(10^n-1)/9.
a((4^n-1)/3 - 1) = 4*(10^(n-1)-1)/9, n > 1.
Inequalities:
a(n) <= (10^log_4(3*n+1)-1)/9, equality holds for n=(4^k-1)/3, k>0.
a(n) > (4/10)*(10^log_4(3*n+1)-1)/9, n > 0.
Lower and upper limits:
lim inf a(n)/10^log_4(3*n) = 2/45, for n --> infinity.
lim sup a(n)/10^log_4(3*n) = 1/9, for n --> infinity.
G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1 - 5z(j)^4 + 4z(j)^5)/((1-z(j))(1-z(j)^4)), where z(j) = x^4^j.
Also: g(x) = (1/(1-x)) Sum_{j>=0} (1-5(x^4^j)^4 + 4(x^4^j)^5)*x^4^j*f_j(x)/(1-x^4^j), where f_j(x) = 10^j*x^((4^j-1)/3)/(1-(x^4^j)^4). The f_j obey the recurrence f_0(x) = 1/(1-x^4), f_(j+1)(x) = 10x*f_j(x^4).
Also: g(x) = (1/(1-x))* (h_(4,0)(x) + h_(4,1)(x) + h_(4,2)(x) + h_(4,3)(x) - 4*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*x^((4^(j+1)-1)/3) * (x^4^j)^k/(1-(x^4^j)^4).
(End)
a(n) = A045926(n) / 2. - Reinhard Zumkeller, Jan 01 2013

Offset set to 1 according to A007931, A007932 by Hieronymus Fischer, Jun 06 2012

A084545 Alternate number system in base 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 11, 12, 13, 14, 15, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 41, 42, 43, 44, 45, 51, 52, 53, 54, 55, 111, 112, 113, 114, 115, 121, 122, 123, 124, 125, 131, 132, 133, 134, 135, 141, 142, 143, 144, 145, 151, 152, 153, 154, 155, 211, 212, 213, 214, 215, 221, 222

Offset: 1

Robert R. Forslund (forslund(AT)tbaytel.net), Jun 27 2003

			From _Hieronymus Fischer_, Jun 06 2012: (Start)
a(100)  = 345.
a(10^3) = 12445.
a(10^4) = 254445.
a(10^5) = 11144445.
a(10^6) = 223444445.
a(10^7) = 4524444445.
a(10^8) = 145544444445.
a(10^9) = 3521444444445. (End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics
R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1 1995 pp. 27-29.
R. R. Forslund, Positive Integer Pages [Wayback Machine link]
James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.
Index entries for 10-automatic sequences.

Cf. A007931, A007932, A052382, A084544, A046034, A089581, A084984, A001742, A001743, A001744, A202267, A202268, A014261, A014263.

a(n) = my (w=5); while (n>w, n -= w; w *= 5); my (d=digits(w+n-1, 5)); d[1] = 0; fromdigits(d) + (10^(#d-1)-1)/9 \\ Rémy Sigrist, Dec 04 2019

From Hieronymus Fischer, Jun 06 and Jun 08 2012: (Start)
The formulas are designed to calculate base-10 numbers only using the digits 1..5.
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 5)*10^j, where m = floor(log_5(4*n+1)), b(j) = floor((4*n+1-5^m)/(4*5^j)).
a(k*(5^n-1)/4) = k*(10^n-1)/9, for k = 1,2,3,4,5.
a((9*5^n-5)/4) = (14*10^n-5)/9 = 10^n + 5*(10^n-1)/9.
a((5^n-1)/4 - 1) = 5*(10^(n-1)-1)/9, n>1.
a(n) <= (10^log_5(4*n+1)-1)/9, equality holds for n=(5^k-1)/4, k>0.
a(n) > (5/10)*(10^log_5(4*n+1)-1)/9, n>0.
lim inf a(n)/10^log_5(4*n) = 1/18, for n --> infinity.
lim sup a(n)/10^log_5(4*n) = 1/9, for n --> infinity.
G.f.: g(x) = (x^(1/4)*(1-x))^(-1) sum_{j>=0} 10^j*z(j)^(5/4)*(1 - 6z(j)^5 + 5z(j)^6)/((1-z(j))(1-z(j)^5)), where z(j) = x^5^j.
Also: g(x) = (1/(1-x)) sum_{j>=0} (1-6(x^5^j)^5+5(x^5^j)^6)*x^5^j*f_j(x)/(1-x^5^j), where f_j(x) = 10^j*x^((5^j-1)/4)/(1-(x^5^j)^5). The f_j obey the recurrence f_0(x) = 1/(1-x^5), f_(j+1)(x) = 10x*f_j(x^5).
Also: g(x) = 1/(1-x))*(h_(5,0)(x) + h_(5,1)(x) + h_(5,2)(x) + h_(4,1)(x) + h_(5,4)(x) - 5*h_(5,5)(x)), where h_(5,k)(x) = sum_{j>=0} 10^j*x^((5^(j+1)-1)/4) * (x^5^j)^k/(1-(x^5^j)^5).
(End)

Offset set to 1 according to A007931, A007932 and more terms added by Hieronymus Fischer, Jun 06 2012

A029581 Numbers in which all digits are composite.

Original entry on oeis.org

4, 6, 8, 9, 44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 89, 94, 96, 98, 99, 444, 446, 448, 449, 464, 466, 468, 469, 484, 486, 488, 489, 494, 496, 498, 499, 644, 646, 648, 649, 664, 666, 668, 669, 684, 686, 688, 689, 694, 696, 698, 699, 844, 846, 848

Offset: 1

N. J. A. Sloane

If n is represented as a zerofree base-4 number (see A084544) according to n=d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n) = Sum_{j=0..m} c(d(j))*10^j, where c(k)=4,6,8,9 for k=1..4. - Hieronymus Fischer, May 30 2012

			From _Hieronymus Fischer_, May 30 2012: (Start)
a(1000) = 88649.
a(10^4) = 6468989
a(10^5) = 449466489. (End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 10-automatic sequences.

Cf. A002808, A001744, A046034, A084544, A084984, A017042, A001743, A014261, A014263, A202267, A202268.

[n: n in [1..1000] | Set(Intseq(n)) subset [4, 6, 8, 9]]; // Vincenzo Librandi, Dec 17 2018

Table[FromDigits/@Tuples[{4, 6, 8, 9}, n], {n, 3}] // Flatten (* Vincenzo Librandi, Dec 17 2018 *)

From Hieronymus Fischer, May 30 and Jun 25 2012: (Start)
a(n) = Sum_{j=0..m-1} (2*b(j) mod 8 + 4 + floor(b(j)/4) - floor((b(j)+1)/4))*10^j, where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).
Also: a(n) = Sum_{j=0..m-1} (A010877(2*b(j)) + 4 + A002265(b(j)) - A002265(b(j)+1))*10^j.
Special values:
a(1*(4^n-1)/3) = 4*(10^n-1)/9.
a(2*(4^n-1)/3) = 2*(10^n-1)/3.
a(3*(4^n-1)/3) = 8*(10^n-1)/9.
a(4*(4^n-1)/3) = 10^n-1.
a(n) < 4*(10^log_4(3*n+1)-1)/9, equality holds for n=(4^k-1)/3, k > 0.
a(n) < 4*A084544(n), equality holds iff all digits of A084544(n) are 1.
a(n) > 2*A084544(n).
Lower and upper limits:
lim inf a(n)/10^log_4(n) = 1/10*10^log_4(3) = 0.62127870, for n --> inf.
lim sup a(n)/10^log_4(n) = 4/9*10^log_4(3) = 2.756123868970, for n --> inf.
where 10^log_4(n) = n^1.66096404744...
G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1-z(j))*(4 + 6z(j) + 8*z(j)^2 + 9*z(j)^3)/(1-z(j)^4), where z(j) = x^4^j.
Also: g(x) = (1/(1-x))*(4*h_(4,0)(x) + 2*h_(4,1)(x) + 2*h_(4,2)(x) + h_(4,3)(x) - 9*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*x^((4^(j+1)-1)/3)*(x^(k*4^j)/(1-x^4^(j+1)). (End)
Sum_{n>=1} 1/a(n) = 1.039691381254753739202528087006945643166147087095114911673083135126969046250... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

Offset corrected by Arkadiusz Wesolowski, Oct 03 2011

A250256 Least positive integer whose decimal digits divide the plane into n regions (A249572 variant).

Original entry on oeis.org

1, 6, 8, 68, 88, 688, 888, 6888, 8888, 68888, 88888, 688888, 888888, 6888888, 8888888, 68888888, 88888888, 688888888, 888888888, 6888888888, 8888888888, 68888888888, 88888888888, 688888888888, 888888888888, 6888888888888, 8888888888888, 68888888888888

Offset: 1

Rick L. Shepherd, Nov 15 2014

Equivalently, with offset 0, least positive integer with n holes in its decimal digits. Leading zeros are not permitted. Variation of A249572 with the numeral "4" considered open at the top, as it is often handwritten. See also the comments in A249572.

For n > 2, a(n) + a(n+1) divides the plane into 2 regions. For n > 1, a(2n) - a(2n-1) divides the plane into n+1 regions. For n >= 1, a(2n+1) - a(2n) divides the plane into n regions. - Ivan N. Ianakiev, Feb 23 2015

			The integer 68, whose decimal digits have 3 holes, divides the plane into 4 regions. No smaller positive integer does this, so a(4) = 68.

Brady Haran and N. J. A. Sloane, What Number Comes Next? (2018), Numberphile video
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A249572, A250257, A250258, A001743, A001744, A001745, A001746, A002282.

I:=[1,6,8,68]; [n le 4 select I[n] else 10*Self(n-2)+8: n in [1..30]]; // Vincenzo Librandi, Nov 15 2014

Join[{1, 6, 8}, RecurrenceTable[{a[1]==68, a[2]==88, a[n]==10 a[n-2] + 8}, a, {n, 20}]] (* Vincenzo Librandi, Nov 16 2014 *)

a(n) = 10*a(n-2) + 8 for n >= 4.
From Chai Wah Wu, Jul 12 2016: (Start)
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n > 4.
G.f.: x*(10*x^3 - 8*x^2 + 5*x + 1)/((x - 1)*(10*x^2 - 1)). (End)
E.g.f.: (9 + 45*x - 40*cosh(x) + 31*cosh(sqrt(10)*x) - 40*sinh(x) + 4*sqrt(10)*sinh(sqrt(10)*x))/45. - Stefano Spezia, Aug 11 2025

A085557 Numbers that have more prime digits than nonprime digits.

Original entry on oeis.org

2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 122, 123, 125, 127, 132, 133, 135, 137, 152, 153, 155, 157, 172, 173, 175, 177, 202, 203, 205, 207, 212, 213, 215, 217, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232

Offset: 1

Jason Earls, Jul 04 2003

Begins to differ from A046034 at the 21st term (which is the first 3-digit term).

			133 is in the sequence as the prime digits are 3 and 3 (those are two digits; counted with multiplicity) and one nonprime digit 1 and so there are more prime digits than nonprime digits. - _David A. Corneth_, Sep 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A193238, A046034, A046035, A118950, A019546, A203263, A035232, A039996, A085823, A052382, A084544, A084984, A017042, A001743, A001744, A014261, A014263, A202267, A202268, A211681.

is(n) = my(d = digits(n), c = 0); for(i = 1, #d, if(isprime(d[i]), c++)); c<<1 > #d \\ David A. Corneth, Sep 06 2020

from itertools import count, islice
def A085557_gen(startvalue=1): # generator of terms
    return filter(lambda n:len(s:=str(n))<(sum(1 for d in s if d in {'2','3','5','7'})<<1),count(max(startvalue,1)))
A085557_list = list(islice(A085557_gen(),20)) # Chai Wah Wu, Feb 08 2023

A160561 Cyclops primes with circular digits {0,6,8,9}.

Original entry on oeis.org

809, 66089, 68099, 86069, 88069, 89069, 99089, 6680689, 6680699, 6680969, 6690689, 6690899, 6690989, 6860869, 6860989, 6860999, 6890699, 6890969, 6960869, 6980669, 6980899, 6980969, 6990889, 8660689, 8660699, 8660969

Offset: 1

Ki Punches, May 19 2009

The sequence is probably infinite.
The sequence A134809 restricted to cases with digits 6, 8 or 9 (see A001743) at the off-center positions.
Primes in A274765. - Omar E. Pol, Jul 06 2016
Each term is equal to 9 mod 10. - Harvey P. Dale, Feb 02 2021

Kenny Lau, Table of n, a(n) for n = 1..20000

Intersection of A000040 (primes), A001743 (numbers with circular digits) and A134808 (cyclops numbers).
Also intersection of A043580 (primes with circular digits) and A134809 (cyclops primes).
Cf. A134808, A134809, A153807, A156200, A274765.

Select[Prime@ Range[10^6], And[OddQ@ Length@ #, Times @@ Boole@ Map[MemberQ[{0, 6, 8, 9}, #] &, Union@ #] == 1, Part[#, Ceiling[Length[#]/2]] == 0, Count[#, 0] == 1] &@ IntegerDigits@ # &] (* Michael De Vlieger, Jul 05 2016 *)
Table[Select[FromDigits/@(Flatten[Join[{Take[#,Length[#]/2],0,Take[#,-Length[#]/2]}]]&/@Tuples[{6,8,9},n]),PrimeQ],{n,2,6,2}]//Flatten (* Harvey P. Dale, Feb 02 2021 *)

Edited and corrected by Ray Chandler and R. J. Mathar, May 20 2009

Definition simplified by Omar E. Pol, Jun 05 2009

A246880 6((10^n-1)/9)(10^(n+1))+9*(10^n-1)/9.

Original entry on oeis.org

0, 609, 66099, 6660999, 666609999, 66666099999, 6666660999999, 666666609999999, 66666666099999999, 6666666660999999999, 666666666609999999999, 66666666666099999999999, 6666666666660999999999999, 666666666666609999999999999, 66666666666666099999999999999

Offset: 0

Felix Fröhlich, Sep 06 2014

Numbers of the form 6...609...9 (i.e., consisting of an odd number of digits with the middle digit 0, all digits to the left of the middle digit 6 and all digits to the right of the middle digit 9).

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Subsequence of A001743, A039434, A111065, A111156, A153806, A169731 and A227793.

[(6*((10^n - 1)/9))*(10^(n + 1)) + (9*(10^n - 1)/9) : n in [0..15]]; // Wesley Ivan Hurt, Sep 15 2014

A246880:=n->(6*((10^n - 1)/9))*(10^(n + 1)) + (9*(10^n - 1)/9): seq(A246880(n),n=0..15); # Wesley Ivan Hurt, Sep 15 2014

Table[(6*((10^n - 1)/9))*(10^(n + 1)) + (9*(10^n - 1)/9), {n, 15}] (* Wesley Ivan Hurt, Sep 15 2014 *)
Join[{0}, CoefficientList[Series[20/(3 x - 300 x^2) + 1/(x^2 - x) + 17/(30 x^2 - 3 x), {x, 0, 30}], x]] (* Wesley Ivan Hurt, Sep 15 2014 *)

a(n)=6*((10^n-1)/9)*(10^(n+1))+9*(10^n-1)/9

G.f.: 20/(3-300*x)+1/(x-1)+17/(30*x-3); a(n) = 111*a(n-1)-1110*a(n-2)+1000*a(n-3). - Wesley Ivan Hurt, Sep 15 2014

A250257 Least nonnegative integer whose decimal digits divide the plane into n regions.

Original entry on oeis.org

1, 0, 8, 48, 88, 488, 888, 4888, 8888, 48888, 88888, 488888, 888888, 4888888, 8888888, 48888888, 88888888, 488888888, 888888888, 4888888888, 8888888888, 48888888888, 88888888888, 488888888888, 888888888888, 4888888888888, 8888888888888, 48888888888888

Offset: 1

Rick L. Shepherd, Nov 15 2014

Equivalently, with offset 0, least nonnegative integer with n holes in its decimal digits. Leading zeros are not permitted. Identical to A249572 except that a(2) = 0, not 4. See also the comments in A249572.

			The integer 48, whose decimal digits have 3 holes, divides the plane into 4 regions. No smaller nonnegative integer does this, so a(4) = 48.

Index entries for linear recurrences with constant coefficients, signature (1, 10, -10).

Cf. A249572, A250256, A250258, A001743, A001744, A001745, A001746, A002282.

I:=[1,0,8,48,88]; [n le 5 select I[n] else 10*Self(n-2)+8: n in [1..30]]; // Vincenzo Librandi, Nov 15 2014

Join[{1, 0, 8}, RecurrenceTable[{a[1]==48, a[2]==88, a[n]==10 a[n-2] + 8}, a, {n, 20}]] (* Vincenzo Librandi, Nov 16 2014 *)

a(n) = 10*a(n-2) + 8 for n >= 5.
From Chai Wah Wu, Jul 12 2016: (Start)
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n > 5.
G.f.: x*(-40*x^4 + 50*x^3 - 2*x^2 - x + 1)/((x - 1)*(10*x^2 - 1)). (End)

A250258 Least nonnegative integer whose decimal digits divide the plane into n regions (A250257 variant).

Original entry on oeis.org

1, 0, 8, 68, 88, 688, 888, 6888, 8888, 68888, 88888, 688888, 888888, 6888888, 8888888, 68888888, 88888888, 688888888, 888888888, 6888888888, 8888888888, 68888888888, 88888888888, 688888888888, 888888888888, 6888888888888, 8888888888888, 68888888888888

Offset: 1

Rick L. Shepherd, Nov 15 2014

Equivalently, with offset 0, least nonnegative integer with n holes in its decimal digits. Leading zeros are not permitted. Variation of A250257 with the numeral "4" considered open at the top, as it is often handwritten. See also the comments in A249572.

			The integer 68, whose decimal digits have 3 holes, divides the plane into 4 regions. No smaller nonnegative integer does this, so a(4) = 68.

Index entries for linear recurrences with constant coefficients, signature (1, 10, -10).

Cf. A249572, A250256, A250257, A001743, A001744, A001745, A001746, A002282.

I:=[1,0,8,68,88]; [n le 5 select I[n] else 10*Self(n-2)+8: n in [1..40]]; // Vincenzo Librandi, Nov 16 2014

Join[{1, 0, 8}, RecurrenceTable[{a[1]==68, a[2]==88, a[n]==10 a[n-2] + 8}, a, {n, 20}]] (* Vincenzo Librandi, Nov 16 2014 *)

a(n) = 10*a(n-2) + 8 for n >= 5.
a(n) = A250256(n), n<>2.
From Chai Wah Wu, Jul 12 2016: (Start)
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n > 5.
G.f.: x*(-60*x^4 + 70*x^3 - 2*x^2 - x + 1)/((x - 1)*(10*x^2 - 1)). (End)

cp's OEIS Frontend

A007932 Numbers that contain only 1's, 2's and 3's.

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A084544 Alternate number system in base 4.

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A084545 Alternate number system in base 5.

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A029581 Numbers in which all digits are composite.

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A250256 Least positive integer whose decimal digits divide the plane into n regions (A249572 variant).

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A085557 Numbers that have more prime digits than nonprime digits.

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A160561 Cyclops primes with circular digits {0,6,8,9}.

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A246880 6((10^n-1)/9)(10^(n+1))+9*(10^n-1)/9.