A007089 -id:A007089 - cp's OEIS frontend

A060236 If n mod 3 = 0 then a(n) = a(n/3), otherwise a(n) = n mod 3.

1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2

Offset: 1

Henry Bottomley, Mar 21 2001

A cubefree word. Start with 1, apply the morphisms 1 -> 121, 2 -> 122, take limit. See A080846 for another version.
Ultimate modulo 3: n-th digit of terms in "Ana sequence" (see A060032 for definition).
Equals A005148(n) reduced mod 3. In "On a sequence Arising in Series for Pi" Morris Newman and Daniel Shanks conjectured that 3 never divides A005148(n) and D. Zagier proved it. - Benoit Cloitre, Jun 22 2002
Also equals A038502(n) mod 3.
Last nonzero digit in ternary representation of n. - Franklin T. Adams-Watters, Apr 01 2006
a(2*n) = length of n-th run of twos. - Reinhard Zumkeller, Mar 13 2015

			a(10)=1 since 10=3^0*10 and 10 mod 3=1;
a(72)=2 since 24=3^3*8 and 8 mod 3=2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Jean Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull. EATCS, #79 (2003), pp. 178-228.
Index entries for sequences related to final digits of numbers
Index entries for sequences that are fixed points of mappings

Cf. A026225 (indices of 1's), A026179 (indices of 2's).
Cf. A060032 (concatenate 3^n terms).
Cf. A030341, A007089.

following Franklin T. Adams-Watters's comment.
a060236 = head . dropWhile (== 0) . a030341_row
-- Reinhard Zumkeller, Mar 13 2015

[(Floor(n/3^Valuation(n, 3)) mod 3): n in [1..120]]; // G. C. Greubel, Nov 05 2024

Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {1, 2, 2}}] &, {1}, 5] (* Robert G. Wilson v, Mar 04 2005 *)
Table[Mod[n/3^IntegerExponent[n, 3], 3], {n, 1, 120}] (* Clark Kimberling, Oct 19 2016 *)
lnzd[m_]:=Module[{s=Split[m]},If[FreeQ[Last[s],0],s[[-1,1]],s[[-2,1]]]]; lnzd/@Table[IntegerDigits[n,3],{n,120}] (* Harvey P. Dale, Oct 19 2018 *)

a(n)=if(n<1, 0, n/3^valuation(n,3)%3) /* Michael Somos, Nov 10 2005 */

[n/3^valuation(n, 3)%3 for n in range(1,121)] # G. C. Greubel, Nov 05 2024

a(3*n) = a(n), a(3*n + 1) = 1, a(3*n + 2) = 2. - Michael Somos, Jul 29 2009

a(n) = 1 + A080846(n). - Joerg Arndt, Jan 21 2013

A132141 Numbers whose ternary representation begins with 1.

Original entry on oeis.org

1, 3, 4, 5, 9, 10, 11, 12, 13, 14, 15, 16, 17, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108

Offset: 1

Reinhard Zumkeller, Aug 20 2007

The lower and upper asymptotic densities of this sequence are 1/2 and 3/4, respectively. - Amiram Eldar, Feb 28 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Bryan Brown, Michael Dairyko, Stephan Ramon Garcia, Bob Lutz and Michael Someck, Four quotient set gems, The American Mathematical Monthly, Vol. 121, No. 7 (2014), pp. 590-598; arXiv preprint, arXiv:1312.1036 [math.NT], 2013.
Christian Mauduit, Propriétés arithmétiques des substitutions, in Séminaire de Théorie des Nombres, Paris, 1989-90, pp. 177-190 (in French).
Index entries for 3-automatic sequences.

Cf. A007089, A132140, A157671, A131835.

a132141 n = a132141_list !! (n-1)
a132141_list = filter ((== 1) . until (< 3) (flip div 3)) [1..]
-- Reinhard Zumkeller, Feb 06 2015

Flatten[(Range[3^#,2 3^#-1])&/@Range[0,4]] (* Zak Seidov, Mar 03 2009 *)

s=[];for(n=0,4,for(x=3^n,2*3^n-1,s=concat(s,x)));s \\ Zak Seidov, Mar 03 2009

a(n) = n + 3^logint(n<<1,3) >> 1; \\ Kevin Ryde, Feb 19 2022

A number n is a term iff 3^m <= n < 2*3^m -1, for m=0,1,2,... - Zak Seidov, Mar 03 2009

a(n) = n + (3^floor(log_3(2*n)) - 1)/2. - Kevin Ryde, Feb 19 2022

A048787 Write n in base 3 then rotate left one place.

Original entry on oeis.org

1, 2, 1, 4, 7, 2, 5, 8, 1, 4, 7, 10, 13, 16, 19, 22, 25, 2, 5, 8, 11, 14, 17, 20, 23, 26, 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68

Offset: 1

John W. Layman and Anthony C. Hill (hilla(AT)hotmail.com)

A={a(n)} is self-similar in the sense that the subsequence remaining after deleting the first occurrence of each integer is identical to the original sequence A (Kimberling's "upper-trimming" operation).

			a(33)=19 since 33 = 1020(base 3) -> 0201(base 3) = 19.

Indranil Ghosh, Table of n, a(n) for n = 1..59049

Cf. A255588-A255594, A255689-A255693.

Table[ FromDigits[ RotateLeft[ IntegerDigits[n, 3]], 3], {n, 1, 76}] (* Zerinvary Lajos, Mar 25 2007 *)

def A048787(n):
    x=A007089(n)
    return int (x[1:]+x[0],3) # Indranil Ghosh, Feb 08 2017

A060109 Numbers in Morse code, with 1 for a dot, 2 for a dash and 0 between digits/letters.

Original entry on oeis.org

22222, 12222, 11222, 11122, 11112, 11111, 21111, 22111, 22211, 22221, 12222022222, 12222012222, 12222011222, 12222011122, 12222011112, 12222011111, 12222021111, 12222022111, 12222022211, 12222022221, 11222022222, 11222012222, 11222011222, 11222011122, 11222011112, 11222011111

Offset: 0

Henry Bottomley, Feb 28 2001

			a(10) = 12222022222 since 1 is ".----" and 0 is "-----".

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Wikipedia, Morse code

Cf. A059852 (Morse code for letters), A008777 (number of dots and dashes).

Cf. A060110 (these base-3 numbers converted to decimal), A321332 (duration of the code for n).

import Data.List (inits, tails)
a060109 n = if n == 0 then 22222 else read (conv n) :: Integer where
   conv 0 = []
   conv x = conv x' ++ mCode !! d where (x', d) = divMod x 10
   mCode = map ('0' :) (mc ++ (reverse $ init $ tail $ map reverse mc))
   mc = zipWith (++) (inits "111111") (tails "22222")
-- Reinhard Zumkeller, Feb 20 2015

With[{a = Association@ Array[# -> If[# < 6, PadRight[ConstantArray[1, #], 5, 2], PadRight[ConstantArray[2, # - 5], 5, 1]] &, 10, 0]}, Array[FromDigits@ Flatten@ Riffle[Map[Lookup[a, #] &, IntegerDigits[#]], 0] &, 25]] (* Michael De Vlieger, Nov 02 2020 *)

apply( {A060109(n)=if(n>9,self()(n\10)*10^6)+fromdigits([1+(abs(k-n%10)>2)|k<-[3..7]])}, [0..39]) \\ M. F. Hasler, Jun 22 2020

a(n) A007089(A060110(n)) = a(floor(n/10))*10^6 + a(n%10) for n > 9 and a(n) = 33333 - a(n-5) for n%10 > 4, where % is the modulo (remainder) operator. - M. F. Hasler, Jun 22 2020

A073785 Numbers in base -3.

Original entry on oeis.org

0, 1, 2, 120, 121, 122, 110, 111, 112, 100, 101, 102, 220, 221, 222, 210, 211, 212, 200, 201, 202, 12020, 12021, 12022, 12010, 12011, 12012, 12000, 12001, 12002, 12120, 12121, 12122, 12110, 12111, 12112, 12100, 12101, 12102, 12220, 12221, 12222

Offset: 0

Robert G. Wilson v, Aug 11 2002

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 189.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jaime Rangel-Mondragon, Negabinary Numbers to Decimal
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base

Cf. A007089.
Nonnegative numbers in negative bases: A039723 (b=-10), A039724 (b=-2), this sequence (b=-3), A007608 (b=-4), A073786 (b=-5), A073787 (b=-6), A073788 (b=-7), A073789 (b=-8), A073790 (b=-9).
Cf. A320636 (negative numbers in base -3).

a073785 0 = 0
a073785 n = a073785 n' * 10 + m where
   (n', m) = if r < 0 then (q + 1, r + 3) else (q, r)
             where (q, r) = quotRem n (negate 3)
-- Reinhard Zumkeller, Jul 07 2012

ToNegaBases[i_Integer, b_Integer] := FromDigits[ Rest[ Reverse[ Mod[ NestWhileList[(#1 - Mod[ #1, b])/-b &, i, #1 != 0 &], b]]]]; Table[ ToNegaBases[n, 3], {n, 0, 45}]

A073785 = base(n, b=-3) = if(n, base(n\b, b)*10 + n%b, 0) \\ Jianing Song, Oct 20 2018

def A073785(n):
    s, q = '', n
    while q >= 3 or q < 0:
        q, r = divmod(q, -3)
        if r < 0:
            q += 1
            r += 3
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

A081606 Numbers having at least one 1 in their ternary representation.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Mar 23 2003

Complement of A005823.
Integers m such that central Delannoy number A001850(m) == 0 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003
Integers m such that A026375(m) == 0 (mod 3). - Fabio Visonà, Aug 03 2023

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p.25.
Mathematics Stack Exchange, Proof that integers m in this sequence are such that A026375(m) == 0 (mod 3).

Cf. A007089, A062756, A081609, A081605, A074940.

Select[Range[100],DigitCount[#,3,1]>0&] (* Harvey P. Dale, Nov 26 2022 *)

from itertools import count, islice
def A081606_gen(): # generator of terms
    a = 0
    for n in count(1):
        b = int(bin(n)[2:],3)<<1
        yield from range(a+1,b)
        a = b
A081606_list = list(islice(A081606_gen(),30)) # Chai Wah Wu, Oct 13 2023

from gmpy2 import digits
def A081606(n):
    def f(x):
        s = digits(x>>1,3)
        for i in range(l:=len(s)):
            if s[i]>'1':
                break
        else:
            return n+int(s,2)
        return n-1+(int(s[:i] or '0',2)+1<Chai Wah Wu, Oct 29 2024

More terms from Emeric Deutsch and Bruce E. Sagan, Dec 04 2003

A163327 Self-inverse permutation of integers: swap the odd- and even-positioned digits in the ternary expansion of n, then convert back to decimal.

Original entry on oeis.org

0, 3, 6, 1, 4, 7, 2, 5, 8, 27, 30, 33, 28, 31, 34, 29, 32, 35, 54, 57, 60, 55, 58, 61, 56, 59, 62, 9, 12, 15, 10, 13, 16, 11, 14, 17, 36, 39, 42, 37, 40, 43, 38, 41, 44, 63, 66, 69, 64, 67, 70, 65, 68, 71, 18, 21, 24, 19, 22, 25, 20, 23, 26, 45, 48, 51, 46, 49, 52, 47, 50, 53

Offset: 0

Antti Karttunen, Jul 29 2009

			11 in ternary base (A007089) is written as '(000...)102' (... + 0*27 + 1*9 + 0*3 + 2), which results '1020' = 1*27 + 0*9 + 2*3 + 0 = 33, when the odd- and even-positioned digits are swapped, thus a(11) = 33.

Antti Karttunen, Table of n, a(n) for n = 0..728
Index entries for sequences that are permutations of the natural numbers

Cf. A007089, A163328-A163329.

Other bases: A057300, A126006, A217558.

from sympy.ntheory import digits
def a(n):
    d = digits(n, 3)[1:]
    return sum(3**(i+(1-2*(i&1)))*di for i, di in enumerate(d[::-1]))
print([a(n) for n in range(72)]) # Michael S. Branicky, Aug 05 2022

(define (A163327 n) (+ (A037314 (A163326 n)) (* 3 (A037314 (A163325 n)))))

a(n) = A037314(A163326(n)) + 3*A037314(A163325(n))

Edited by Charles R Greathouse IV, Nov 01 2009

A325820 Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 1, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 7, 12, 12, 7, 6, 0, 0, 7, 3, 15, 16, 15, 3, 7, 0, 0, 8, 5, 18, 11, 11, 18, 5, 8, 0, 0, 9, 4, 21, 24, 13, 24, 21, 4, 9, 0, 0, 10, 18, 24, 19, 21, 21, 19, 24, 18, 10, 0, 0, 11, 20, 27, 23, 26, 9, 26, 23, 27, 20, 11, 0, 0, 12, 19, 30, 36, 19, 15, 15, 19, 36, 30, 19, 12, 0

Offset: 0

Antti Karttunen, May 22 2019

			The array begins as:
  0,  0,  0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0, ...
  0,  1,  2,  3,  4,  5,  6,  7,  8,   9,  10,  11,  12, ...
  0,  2,  1,  6,  8,  7,  3,  5,  4,  18,  20,  19,  24, ...
  0,  3,  6,  9, 12, 15, 18, 21, 24,  27,  30,  33,  36, ...
  0,  4,  8, 12, 16, 11, 24, 19, 23,  36,  40,  44,  48, ...
  0,  5,  7, 15, 11, 13, 21, 26, 19,  45,  50,  52,  33, ...
  0,  6,  3, 18, 24, 21,  9, 15, 12,  54,  60,  57,  72, ...
  0,  7,  5, 21, 19, 26, 15, 13, 11,  63,  70,  68,  57, ...
  0,  8,  4, 24, 23, 19, 12, 11, 16,  72,  80,  76,  69, ...
  0,  9, 18, 27, 36, 45, 54, 63, 72,  81,  90,  99, 108, ...
  0, 10, 20, 30, 40, 50, 60, 70, 80,  90, 100,  83, 120, ...
  0, 11, 19, 33, 44, 52, 57, 68, 76,  99,  83,  91, 132, ...
  0, 12, 24, 36, 48, 33, 72, 57, 69, 108, 120, 132, 144, ...
  etc.
A(2,2) = 2*2 mod 3 = 1.

Cf. A169999 (the main diagonal).
Row/Column 0: A000004, Row/Column 1: A001477, Row/Column 2: A004488, Row/Column 3: A008585, Row/Column 4: A242399, Row/Column 9: A008591.
Cf. A007089, A207669, A207670, A263273, A325825.
Cf. A325821 (same table without the zero row and column).
Cf. A048720 (binary), A059692 (decimal), A004247 (full multiply).

up_to = 105;
A325820sq(b, c) = fromdigits(Vec(Pol(digits(b,3))*Pol(digits(c,3)))%3, 3);
A325820list(up_to) = { my(v = vector(up_to), i=0); for(a=0,oo, for(col=0,a, if(i++ > up_to, return(v)); v[i] = A325820sq(a-col,col))); (v); };
v325820 = A325820list(up_to);
A325820(n) = v325820[1+n];

A020915 Number of digits in base-3 representation of 2^n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 43, 44, 45, 45, 46, 47

Offset: 0

Clark Kimberling

For n > 0, first differences of A022331. - Michel Marcus, Oct 03 2013

William A. Tedeschi, Table of n, a(n) for n = 0..10000
Mike Winkler, The algorithmic structure of the finite stopping time behavior of the 3x+1 function, arXiv:1709.03385 [math.GM], 2017.

Cf. A007089, A020914, A076227, A081604, A000079.
Cf. A022331, A102525, A136409.
Cf. A022924 (first differences).

a020915 = a081604 . a000079  -- Reinhard Zumkeller, Mar 28 2015

[Round(1+Floor(n*(Log(2))/Log(3))): n in [0..80]]; // Vincenzo Librandi, Dec 05 2015

Table[Length[IntegerDigits[2^n,3]],{n,0,80}] (* Harvey P. Dale, May 02 2012 *)
Table[1 + Floor[n*Log[3, 2]], {n, 0, 73}] (* L. Edson Jeffery, Dec 04 2015 *)
IntegerLength[2^Range[0,80],3] (* Harvey P. Dale, Nov 17 2022 *)

a(n)=logint(2^n,3)+1 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = 1 + floor(n*log_3(2)) = 1 + floor(n*A102525) = 1 + A136409(n). - R. J. Mathar, May 23 2009
a(n) = A081604(A000079(n)). - Reinhard Zumkeller, Jul 12 2011
a(A020914(n)) = n + 1. - Reinhard Zumkeller, Mar 28 2015

More terms from James Sellers

A291770 A binary encoding of the zeros in ternary representation of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 7, 6, 6, 5, 4, 4, 5, 4, 4, 3, 2, 2, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 7, 6, 6, 5, 4, 4, 5, 4, 4, 3, 2, 2, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 15, 14, 14, 13, 12, 12, 13, 12, 12, 11, 10, 10, 9, 8, 8, 9, 8, 8, 11, 10, 10, 9, 8, 8, 9, 8, 8, 7, 6, 6

Offset: 1

Antti Karttunen, Sep 11 2017

The ones in the binary representation of a(n) correspond to the nonleading zeros in the ternary representation of n. For example: ternary(33) = 1020 and binary(a(33)) = 101 (a(33) = 5).

			   n      a(n)    ternary(n)  binary(a(n))
                  A007089(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        0            1           0
   2        0            2           0
   3        1           10           1
   4        0           11           0
   5        0           12           0
   6        1           20           1
   7        0           21           0
   8        0           22           0
   9        3          100          11
  10        2          101          10
  11        2          102          10
  12        1          110           1
  13        0          111           0
  14        0          112           0
  15        1          120           1
  16        0          121           0
  17        0          122           0
  18        3          200          11
  19        2          201          10
  20        2          202          10
  21        1          210           1
  22        0          211           0
  23        0          212           0
  24        1          220           1
  25        0          221           0
  26        0          222           0
  27        7         1000         111
  28        6         1001         110
  29        6         1002         110
  30        5         1010         101

Antti Karttunen, Table of n, a(n) for n = 1..59049

Cf. A007088, A007089, A077267, A289813, A289814, A291771.

Table[FromDigits[IntegerDigits[n, 3] /. k_ /; k < 3 :> If[k == 0, 1, 0], 2], {n, 110}] (* Michael De Vlieger, Sep 11 2017 *)

from sympy.ntheory.factor_ import digits
def a(n):
    k=digits(n, 3)[1:]
    return int("".join('1' if i==0 else '0' for i in k), 2)
print([a(n) for n in range(1, 111)]) # Indranil Ghosh, Sep 21 2017

(define (A291770 n) (if (zero? n) n (let loop ((n n) (b 1) (s 0)) (if (< n 3) s (let ((d (modulo n 3))) (if (zero? d) (loop (/ n 3) (+ b b) (+ s b)) (loop (/ (- n d) 3) (+ b b) s)))))))

For all n >= 0, a(A000244(n)) = A000225(n), that is, a(3^n) = (2^n) - 1. [The records in the sequence].
For all n >= 1, A000120(a(n)) = A077267(n).
For all n >= 1, A278222(a(n)) = A291771(n).

cp's OEIS Frontend

A060236 If n mod 3 = 0 then a(n) = a(n/3), otherwise a(n) = n mod 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

SageMath

Formula

A132141 Numbers whose ternary representation begins with 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

A048787 Write n in base 3 then rotate left one place.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A060109 Numbers in Morse code, with 1 for a dot, 2 for a dash and 0 between digits/letters.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A073785 Numbers in base -3.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

A081606 Numbers having at least one 1 in their ternary representation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Python

Extensions

A163327 Self-inverse permutation of integers: swap the odd- and even-positioned digits in the ternary expansion of n, then convert back to decimal.

Views

Author

Keywords