A062318 -id:A062318 - cp's OEIS frontend

A114482 Let S(1)=1, S(2)=10; S(2n)=concatenation of S(2n-1), S(2n-2) and 0; and S(2n+1)=concatenation of S(2n), S(2n) and 0. Sequence gives S(infinity).

1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0

Offset: 1

Leroy Quet, Nov 30 2005

Number of terms in S(n) is A062318(n).

Interpreting S(n) in binary and converting to decimal gives 1,2,20,164,84296,43159880,5792821120672400,...,.

			S(3) = {1,0,1,0,0}, S(4) = {1,0,1,0,0,1,0,0}, S(5) = {1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,0,0}, ...

Cf. A114483, A062318, A112361.

a[1] = {1}; a[2] = {1, 0}; a[n_] := a[n] = If[EvenQ[n], Join[a[n - 1], a[n - 2], {0}] // Flatten, Join[a[n - 1], a[n - 1], {0}] // Flatten]; a[8] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jan 01 2006

Edited by N. J. A. Sloane, Jan 03 2006

A138002 a(3n)=2n, a(3n+1)=a(n)+n, a(3n+2)=2*a(n), a(n)=0 for n<3.

Original entry on oeis.org

0, 0, 0, 2, 1, 0, 4, 2, 0, 6, 5, 4, 8, 5, 2, 10, 5, 0, 12, 10, 8, 14, 9, 4, 16, 8, 0, 18, 15, 12, 20, 15, 10, 22, 15, 8, 24, 20, 16, 26, 18, 10, 28, 16, 4, 30, 25, 20, 32, 21, 10, 34, 17, 0, 36, 30, 24, 38, 29, 20, 40, 28, 16, 42, 35, 28, 44, 31, 18, 46, 27, 8, 48, 40, 32, 50, 33, 16, 52

Offset: 0

Reinhard Zumkeller, Feb 26 2008

nonn

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

Cf. A025480.

a[0]=0;a[n_]:=Mod[n,3]*a[Floor[n/3]]+(2-Mod[n,3])*Floor[n/3];Array[a,79,0] (* James C. McMahon, Jun 05 2025 *)

a(n) = (n mod 3) * a(floor(n/3)) + (2 - n mod 3) * floor(n/3), a(n) = 0.

a(A062318(n)) = 0.

A319021 Next larger integer with same sum of digits in base 3 as n.

Original entry on oeis.org

3, 4, 9, 6, 7, 10, 11, 14, 27, 12, 13, 18, 15, 16, 19, 20, 23, 28, 21, 22, 29, 24, 25, 32, 35, 44, 81, 30, 31, 36, 33, 34, 37, 38, 41, 54, 39, 40, 45, 42, 43, 46, 47, 50, 55, 48, 49, 56, 51, 52, 59, 62, 71, 82, 57, 58, 63, 60, 61, 64, 65, 68, 83, 66, 67, 72

Offset: 1

Rémy Sigrist, Sep 08 2018

This sequence is the base-3 variant of A057168 (base-2) and of A228915 (base-10).

All integers except those in A062318 appear in this sequence.

			The first terms, alongside the ternary representations of n and of a(n), are:
  n   a(n)  ter(n)  ter(a(n))
  --  ----  ------  ---------
   1     3      1     10
   2     4      2     11
   3     9     10    100
   4     6     11     20
   5     7     12     21
   6    10     20    101
   7    11     21    102
   8    14     22    112
   9    27    100   1000
  10    12    101    110
  11    13    102    111
  12    18    110    200
  13    15    111    120
  14    16    112    121
  15    19    120    201

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A053735, A057168, A062318, A228915.

nli3[n_]:=Module[{nd3=Total[IntegerDigits[n,3]],k=n+1},While[Total[IntegerDigits[k,3]]!=nd3,k++];k]; Array[nli3,70] (* Harvey P. Dale, Jun 27 2023 *)

a(n, base=3) = my (c=0); for (w=0, oo, my (d=n % base); if (d+1 < base && c, return ((n+1)*base^w + ((c-1)%(base-1) + 1)*base^((c-1)\(base-1))-1), c += d; n \= base))

def a(n, base=3):
    c, b, w = 0, base, 0
    while True:
        d = n%b
        if d+1 < b and c:
            return (n+1)*b**w + ((c-1)%(b-1)+1)*b**((c-1)//(b-1))-1
        c += d; n //= b; w += 1
print([a(n) for n in range(1, 67)]) # Michael S. Branicky, Jul 10 2022 after Rémy Sigrist

a(3^k) = 3^(k+1) for any k >= 0.

A053735(a(n)) = A053735(n).

A331789 T(b,n) is the smallest m such that for any N, at least one of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N in base b. Square array read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 2, 7, 1, 3, 5, 15, 1, 2, 3, 8, 31, 1, 3, 5, 7, 17, 63, 1, 2, 5, 4, 15, 26, 127, 1, 3, 3, 7, 9, 15, 53, 255, 1, 2, 5, 6, 5, 14, 31, 80, 511, 1, 3, 5, 7, 9, 11, 29, 63, 161, 1023, 1, 2, 3, 4, 9, 6, 23, 24, 63, 242, 2047, 1, 3, 5, 7, 9, 11, 13, 35, 49, 127, 485, 4095

Offset: 2

Jianing Song, Jan 25 2020

The main sequence is A331787; this is added because some people may search for this.

			Table begins
  b\n  1  2  3   4   5   6    7    8    9    10
   2   1  3  7  15  31  63  127  255  511  1023
   3   1  2  5   8  17  26   53   80  161   242
   4   1  3  3   7  15  15   31   63   63   127
   5   1  2  5   4   9  14   29   24   49    74
   6   1  3  5   7   5  11   23   35   47    35
   7   1  2  3   6   9   6   13   20   27    48
   8   1  3  5   7   9  11    7   15   31    47
   9   1  2  5   4   9  10   13    8   17    26
  10   1  3  3   7   9   9   13   15    9    19

Jianing Song, Table of n, a(n) for n = 2..7627 (Note: T(b,n) occurs at the ((n+b-2)*(n+b-1)/2-b+3)-th place)

Cf. A331787.

Cf. A000225 (row 2), A062318 (row 3 with an offset shift), A331788 (row 10).

T(b,n) = my(s=(n-1)\(b-1), t=(n-1)%(b-1)+1); b^s*(2*t-gcd(t,b-1)+1)-1

If n = (b-1)*s + t, 1 <= t <= b-1, then T(b,n) = b^s*(2*t-gcd(t,b-1)+1) - 1. See A331787 for a proof of the formula in base b.
T(b,k) = A331787(b,k) + 1.
T(b,n) = T(b,n-1) + b*T(b,n-b+1) - b*T(b,n-b) for b >= 2, n >= b+1.
T(b,n) = O(b^(n/(b-1))).

A112361 Number of terms in s(n), where s(n) is defined in A114483.

Original entry on oeis.org

1, 2, 5, 8, 17, 35, 53, 107, 215, 323, 647, 971, 1943, 3887, 5831, 11663, 23327, 46655, 69983, 139967, 209951, 419903, 839807, 1259711, 2519423, 5038847, 7558271, 15116543, 22674815, 45349631, 90699263, 136048895, 272097791, 544195583

Offset: 1

Leroy Quet, Nov 30 2005

			As defined in A114483, s(4) = {1,0,1,0,1,1,0,1}; so a(4) = 8.

Cf. A114482, A114483, A062318.

Corrected and extended by Joshua Zucker, Jul 27 2006

A114483 s(1)={1}. s(2)={1,0}. If a(n) = 0, s(n+2) = s(n+1) U s(n) U {1}. If a(n) = 1, s(n+2) = s(n+1) U s(n+1) U {1}. (U represents concatenation of finite sequences.) {a(n)} is the limit of {s(n)} as n -> infinity.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1

Offset: 1

Leroy Quet, Nov 30 2005

Number of terms in s(n) is A112361(n).

			s(3) = {1,0,1,0,1}, s(4) = {1,0,1,0,1,1,0,1}, s(5) = {1,0,1,0,1,1,0,1,1,0,1,0,1,1,0,1,1}

Cf. A114482, A062318, A112361.

More terms from Joshua Zucker, Jul 27 2006

A319023 Let S be the sequence generated by these rules: 1 is in S, and if k is in S, then A057168(k) and A319021(k) are in S, and duplicates are deleted as they occur; a(n) is the n-th term of S.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 8, 6, 7, 10, 27, 16, 14, 11, 12, 29, 81, 32, 20, 19, 13, 17, 18, 30, 31, 82, 243, 64, 34, 24, 22, 21, 15, 23, 28, 39, 36, 47, 33, 84, 245, 729, 128, 66, 38, 25, 35, 43, 45, 40, 54, 55, 49, 37, 88, 90, 246, 247, 730, 2187, 256, 130, 68, 72, 41

Offset: 1

Rémy Sigrist, Sep 08 2018

This sequence is a permutation of the natural numbers (with inverse A319024):
- this sequence is injective,
- this sequence is surjective: by contradiction:
- let m be the least integer missing from the sequence,
- as a(1) = 1, we have m > 1,
- also, m belongs to A000225 and to A062318,
- however the only positive integer belonging to both sequences is 1,
- hence a contradiction, QED.

			The first terms, alongside b(n) = A057168(a(n)) and t(n) = A319021(a(n)), are:
  n   a(n)  b(n)  t(n)
  --  ----  ----  ----
   1     1     2     3
   2     2     4     4
   3     3     5     9
   4     4     8     6
   5     5     6     7
   6     9    10    27
   7     8    16    14
   8     6     9    10
   9     7    11    11
  10    10    12    12
  11    27    29    81
  12    16    32    20
  13    14    19    16
  14    11    13    13
  15    12    17    18
  16    29    30    31
  17    81    82   243
  18    32    64    34
  19    20    24    22
  20    19    21    21

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A319023
Index entries for sequences that are permutations of the natural numbers

Cf. A000225, A057168, A062318, A319021, A319024 (inverse).

PARI
```
See Links section.
```

A356517 Square array A(n, k), n >= 2, k >= 0, read by antidiagonals upwards; A(n, k) is the least integer with sum of digits k in base n.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 2, 7, 0, 1, 2, 5, 15, 0, 1, 2, 3, 8, 31, 0, 1, 2, 3, 7, 17, 63, 0, 1, 2, 3, 4, 11, 26, 127, 0, 1, 2, 3, 4, 9, 15, 53, 255, 0, 1, 2, 3, 4, 5, 14, 31, 80, 511, 0, 1, 2, 3, 4, 5, 11, 19, 47, 161, 1023, 0, 1, 2, 3, 4, 5, 6, 17, 24, 63, 242, 2047

Offset: 2

Rémy Sigrist, Aug 10 2022

The expansion of A(n, k) in base n is:
       q  n-1  ...  n-1
          <- p times ->
where q = k mod (n-1) and p = floor(k / (n-1)).

			Array A(n, k) begins:
  n\k|  0  1  2  3   4   5   6    7    8    9    10    11    12
  ---+---------------------------------------------------------
    2|  0  1  3  7  15  31  63  127  255  511  1023  2047  4095
    3|  0  1  2  5   8  17  26   53   80  161   242   485   728
    4|  0  1  2  3   7  11  15   31   47   63   127   191   255
    5|  0  1  2  3   4   9  14   19   24   49    74    99   124
    6|  0  1  2  3   4   5  11   17   23   29    35    71   107
    7|  0  1  2  3   4   5   6   13   20   27    34    41    48
    8|  0  1  2  3   4   5   6    7   15   23    31    39    47
    9|  0  1  2  3   4   5   6    7    8   17    26    35    44
   10|  0  1  2  3   4   5   6    7    8    9    19    29    39
Array A(n, k) begins (with values given in base n):
  n\k|  0  1   2    3     4      5       6        7         8          9
  ---+------------------------------------------------------------------
    2|  0  1  11  111  1111  11111  111111  1111111  11111111  111111111
    3|  0  1   2   12    22    122     222     1222      2222      12222
    4|  0  1   2    3    13     23      33      133       233        333
    5|  0  1   2    3     4     14      24       34        44        144
    6|  0  1   2    3     4      5      15       25        35         45
    7|  0  1   2    3     4      5       6       16        26         36
    8|  0  1   2    3     4      5       6        7        17         27
    9|  0  1   2    3     4      5       6        7         8         18
   10|  0  1   2    3     4      5       6        7         8          9

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 antidiagonals)

Cf. A000225, A051885, A062318, A140576, A165804, A180516, A181287, A181288, A181303, A138530, A240236.

PARI
```
A(n,k) = { (1+k%(n-1))*n^(k\(n-1))-1 }
```

def A(n,k): return (1+(k % (n-1)))*n**(k//(n-1))-1

A(2, k) = 2^k - 1.
A(3, k) = A062318(k+1).
A(4, k) = A180516(k+1).
A(5, k) = A181287(k+1).
A(6, k) = A181288(k+1).
A(7, k) = A181303(k+1).
A(8, k) = A165804(k+1).
A(9, k) = A140576(k+1).
A(10, k) = A051885(k).
A(n, 0) = 0.
A(n, 1) = 1.
A(n, k) = k iff k < n.
A(n, n) = 2*n - 1.
A(n, n+1) = 3*n - 1 for any n > 2.

A372783 In the ternary expansion of n: keep the initial digit, and then replace 0's with 1's and 1's with 0's.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, May 13 2024

This sequence is a self-inverse permutation of the nonnegative integers.

			For n = 42: the ternary expansion of 42 is "1120", so the ternary expansion of a(42) is "1021", and a(42) = 34.

Cf. A004488, A062318 (fixed points), A115303.

Table[t3=IntegerDigits[n,3];fd=First[t3];tr=Rest[t3]/.{1->0,0->1};PrependTo[tr,fd];FromDigits[tr,3],{n,0,67}] (* James C. McMahon, May 31 2024 *)

a(n) = { my (t = digits(n, 3)); for (i = 2, #t, t[i] = [1,0,2][1+t[i]];); fromdigits(t, 3); }

a(n) = n iff n belongs to A062318.

A358027 Expansion of g.f.: (1 + x - 2x^2 + 2x^4)/((1-x)(1-3x^2)).

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 35, 62, 107, 188, 323, 566, 971, 1700, 2915, 5102, 8747, 15308, 26243, 45926, 78731, 137780, 236195, 413342, 708587, 1240028, 2125763, 3720086, 6377291, 11160260, 19131875, 33480782, 57395627

Offset: 0

G. C. Greubel, Oct 31 2022

Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A052993, A062318, A164123, A254006.

I:=[3,6,11]; [1,2] cat [n le 3 select I[n] else Self(n-1) +3*Self(n-2) -3*Self(n-3): n in [1..60]];

LinearRecurrence[{1,3,-3}, {1,2,3,6,11}, 61]

def A254006(n): return 3^(n/2)*(1 + (-1)^n)/2
def A358027(n): return (1/3)*( 4*A254006(n) + 7*A254006(n-1) +2*int(n==0) + 2*int(n==1) - 3 )
[A358027(n) for n in (0..60)]

a(n) = (1/3)*(2*[n=0] + 2*[n=1] - 3 + 4*A254006(n) + 7*A254006(n-1)).
a(n) = a(n-1) - 3*a(n-2) + 3*a(n-3), for n >= 5.
E.g.f.: (1/3)*( 2 + 2*x - 3*exp(x) + 4*cosh(sqrt(3)*x) + (7/sqrt(3))*sinh(sqrt(3)*x) ).
G.f.: (1 +x -2*x^2 +2*x^4)/((1-x)*(1-3*x^2)). - Clark Kimberling, Oct 31 2022

cp's OEIS Frontend

A114482 Let S(1)=1, S(2)=10; S(2n)=concatenation of S(2n-1), S(2n-2) and 0; and S(2n+1)=concatenation of S(2n), S(2n) and 0. Sequence gives S(infinity).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A138002 a(3*n)=2*n, a(3*n+1)=a(n)+n, a(3*n+2)=2*a(n), a(n)=0 for n<3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A319021 Next larger integer with same sum of digits in base 3 as n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A331789 T(b,n) is the smallest m such that for any N, at least one of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N in base b. Square array read by ascending antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A112361 Number of terms in s(n), where s(n) is defined in A114483.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A114483 s(1)={1}. s(2)={1,0}. If a(n) = 0, s(n+2) = s(n+1) U s(n) U {1}. If a(n) = 1, s(n+2) = s(n+1) U s(n+1) U {1}. (U represents concatenation of finite sequences.) {a(n)} is the limit of {s(n)} as n -> infinity.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A319023 Let S be the sequence generated by these rules: 1 is in S, and if k is in S, then A057168(k) and A319021(k) are in S, and duplicates are deleted as they occur; a(n) is the n-th term of S.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A356517 Square array A(n, k), n >= 2, k >= 0, read by antidiagonals upwards; A(n, k) is the least integer with sum of digits k in base n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A138002 a(3n)=2n, a(3n+1)=a(n)+n, a(3n+2)=2*a(n), a(n)=0 for n<3.

A358027 Expansion of g.f.: (1 + x - 2x^2 + 2x^4)/((1-x)(1-3x^2)).