A248375 -id:A248375 - cp's OEIS frontend

A037477 a(n) = Sum{d(i)9^i: i=0,1,...,m}, where Sum{d(i)8^i: i=0,1,...,m} is the base 8 representation of n.

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 81, 82, 83, 84, 85

Offset: 0

Clark Kimberling

Numbers that do not contain the digit 8 in their base 9 expansion. - M. F. Hasler, Oct 05 2014

			a(63) = 7*9+7 = 70 since 63 = 77[8], i.e., "77" when written in base 8;
a(64) = 1*9^2 = 81 since 64 = 100[8]. - _M. F. Hasler_, Oct 05 2014

François Marques, Table of n, a(n) for n = 0..10000 (first 1000 terms from Clark Kimberling)

Cf. A248375.
Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), A011539 (b=10), A095778 (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), this sequence (b=9), A007095 (b=10), A171397 (b=11).

Table[FromDigits[RealDigits[n, 8], 9], {n, 0, 100}]
Select[Range[0,100],DigitCount[#,9,8]==0&] (* Harvey P. Dale, Aug 06 2024 *)

a(n) = vector(#n=digits(n,8),i,9^(#n-i))*n~ \\ M. F. Hasler, Oct 05 2014

a(n) = fromdigits(digits(n, 8), 9); \\ François Marques, Oct 15 2020

def A037477(n): return int(oct(n)[2:],9) # Chai Wah Wu, Jan 27 2025

For n<64, a(n) = floor(9n/8) = A248375(n). - M. F. Hasler, Oct 05 2014

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A047226 Numbers that are congruent to {0, 1, 2, 3, 4} mod 6; a(n)=floor(6(n-1)/5).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 78, 79

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A032766, A004773, A001068, this, A047368, A004777, A248375.

[n: n in [0..100] | n mod 6 in [0..4]]; // Vincenzo Librandi, Jan 06 2013

A047226 := proc(n)
    option remember;
    if n <= 6 then
        op(n,[0,1,2,3,4,6]) ;
    else
        procname(n-1)+procname(n-5)-procname(n-6) ;
    end if;
end proc: # R. J. Mathar, Jul 25 2013

Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 4}, Mod[#, 6]]&] (* Vincenzo Librandi, Jan 06 2013 *)

G.f.: x^2*(1+x+x^2+x^3+2*x^4) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Sep 17 2015, Jul 16 2013: (Start)
a(n) = floor( 6*(n-1)/5 ).
a(n) = a(n-1) + a(n-5) - a(n-6) for n>6.
a(n) = n - 1 + floor((n-1)/5). (End)
Sum_{n>=2} (-1)^n/a(n) = (9-4*sqrt(3))*Pi/36 + log(2+sqrt(3))/(2*sqrt(3)) + log(2)/6. - Amiram Eldar, Dec 17 2021

Explicit formula added to definition by M. F. Hasler, Oct 05 2014

A168183 Numbers that are not multiples of 9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Reinhard Zumkeller, Nov 30 2009

It seems that, for any n >= 1, there exists no positive integer z such that digit_sum(z) = digit_sum(a(n)+z). - Max Lacoma, Sep 19 2019. Giovanni Resta: this follows immediately from the well-known fact that sod(x) == x (mod 9).

Ivan Panchenko, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Complement of A008591.

Cf. A010887, A010888, A109012, A168182, A248375.

a168183 n = a168183_list !! (n-1)
a168183_list = [1..8] ++ map (+ 9) a168183_list
-- Reinhard Zumkeller, Mar 04 2014

[n+Floor((n-1)/8) : n in [1..100]]; // Wesley Ivan Hurt, Sep 12 2015

A168183:=n->n+floor((n-1)/8): seq(A168183(n), n=1..100); # Wesley Ivan Hurt, Sep 12 2015

Select[Table[n,{n,200}],Mod[#,9]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)
With[{nn=81},Complement[Range[nn],9Range[Floor[nn/9]]]] (* Harvey P. Dale, Sep 07 2011 *)

is(n)=!!(n%9) \\ Charles R Greathouse IV, Sep 02 2015

a(n)=(9*n-1)\8 \\ Charles R Greathouse IV, Sep 02 2015

from gmpy2 import f_mod
[n for n in range(100) if f_mod(n,9)] # Bruno Berselli, Dec 05 2016

A168182(a(n)) = 1.
A010888(a(n)) = A010887(n-1).
A109012(a(n)) < 9.
From Wesley Ivan Hurt, Sep 12 2015: (Start)
a(n) = a(n-1) + a(n-8) - a(n-9), n>9.
a(n) = n + floor((n-1)/8). (End)
From Philippe Deléham, Dec 05 2016: (Start)
a(n) = 1 + A248375(n-1).
G.f.: x*(1-x^9)/((1-x)^2*(1-x^8)). (End)
E.g.f.: 1 + (1/8)*(-cos(x) + (-5+9*x)*cosh(x) - 2*cos(x/sqrt(2))*cosh(x/sqrt(2)) + sin(x) + (-4+9*x)*sinh(x) + 2*sin(x/sqrt(2))*(sqrt(2)*cosh(x/sqrt(2)) + sinh(x/sqrt(2)))). - Stefano Spezia, Sep 20 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3) + 3*cosec(2*Pi/9) - 3*tan(Pi/18)) * Pi/27. - Amiram Eldar, May 11 2025

A281899 a(n) = n + 6*floor(n/3).

Original entry on oeis.org

0, 1, 2, 9, 10, 11, 18, 19, 20, 27, 28, 29, 36, 37, 38, 45, 46, 47, 54, 55, 56, 63, 64, 65, 72, 73, 74, 81, 82, 83, 90, 91, 92, 99, 100, 101, 108, 109, 110, 117, 118, 119, 126, 127, 128, 135, 136, 137, 144, 145, 146, 153, 154, 155, 162, 163, 164, 171, 172, 173, 180, 181, 182, 189

Offset: 0

Bruno Berselli, Feb 06 2017

Equivalently, numbers that are congruent to {0, 1, 2} mod 9.
Also numbers m such that floor(m/3) = 3*floor(m/9).
The n-th term is 3*n, 3*n-2 or 3*n-4.
For n > 0, numbers k such that 3 | floor(k/3). - Wesley Ivan Hurt, Dec 01 2020

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A002264.
Subsequence of A060464 and A248375.
The first differences are in A105395.
Cf. similar sequences with formula n+i*floor(n/3): A004773 (i=1), A047217 (i=2), A047240 (i=3), A047354 (i=4), A047469 (i=5), this sequence (i=6).
Cf. numbers that are congruent to {0, 1, 2} mod j: the sequences are listed in the previous row for j = 4..9, respectively.

Magma
```
[n+6*(n div 3): n in [0..70]];
```

A281899:=n->n+6*floor(n/3): seq(A281899(n), n=0..100); # Wesley Ivan Hurt, Feb 09 2017

Table[n + 6 Floor[n/3], {n, 0, 70}]
LinearRecurrence[{1,0,1,-1},{0,1,2,9},90] (* Harvey P. Dale, Feb 25 2018 *)

Maxima
```
makelist(n+6*floor(n/3), n, 0, 70);
```

a(n)=n\3*6 + n \\ Charles R Greathouse IV, Feb 07 2017

Python
```
[n+6*int(n/3) for n in range(70)]
```
Sage
```
[n+6*floor(n/3) for n in range(70)]
```

G.f.: x*(1 + x + 7*x^2)/((1 - x)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4).
a(n) = 3*n - 2*(n mod 3). In general, n + 3*h*floor(n/3) = (h+1)*n - h*(n mod 3).
a(n) + a(n+s) = a(2*n+s-1) + 1, where s is nonnegative and not divisible by 3. Example: for s=14, a(n) + a(n+14) = a(2*n+13) + 1; for n=3, a(3) + a(17) = a(19) + 1 = 9 + 47 = 55 + 1 = 56.
a(6*k+r) = 18*k + a(r), where 0 <= r <= 5.
a(n) = 7*A002264(n) + A002264(n+1) + A002264(n+2).

A047368 Numbers that are congruent to {0, 1, 2, 3, 4, 5} mod 7; a(n)=floor(7(n-1)/6).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A001068, A004773, A004777, A032766, A047226, A248375.

[n : n in [0..100] | n mod 7 in [0..5]]; // Wesley Ivan Hurt, Jun 15 2016

A047368:=n->(42*n-57-3*cos(Pi*n)-4*sqrt(3)*cos((4*n+1)*Pi/6)-12*sin((1-2*n)*Pi/6))/36: seq(A047368(n), n=1..100); # Wesley Ivan Hurt, Jun 15 2016

Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 4, 5}, Mod[#, 7]] &] (* Wesley Ivan Hurt, Jun 15 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 2, 3, 4, 5, 7}, 100] (* Vincenzo Librandi, Jun 16 2016 *)

a(n)=(n-1)*7\6 \\ M. F. Hasler, Oct 05 2014

G.f.: x^2*(1+x+x^2+x^3+x^4+2*x^5) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.
a(n) = (42*n-57-3*cos(Pi*n)-4*sqrt(3)*cos((4*n+1)*Pi/6)-12*sin((1-2*n)*Pi/6))/36.
a(6k) = 7k-2, a(6k-1) = 7k-3, a(6k-2) = 7k-4, a(6k-3) = 7k-5, a(6k-4) = 7k-6, a(6k-5) = 7k-7. (End)

Crossrefs and explicit formula in name added by M. F. Hasler, Oct 05 2014

A272574 a(n) = f(9, f(8, n)), where f(k,m) = floor(m*k/(k-1)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85, 86, 87, 90

Offset: 0

Bruno Berselli, May 03 2016

Also, numbers that are congruent to {0..6} mod 9.

The initial terms coincide with those of A037475 and A039111. First disagreement is after 60 (index 48): a(49) = 63, A037475(49) = 81 and A039111(50) = 71.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A248375: f(9,n).

Cf. similar sequences with the formula f(k, f(k-1, n)): A008585 (k=3), A042948 (k=4), A047217 (k=5), A047246 (k=6), A047337 (k=7), A047602 (k=8), this sequence (k=9), A272576 (k=10).

k:=9; f:=func; [f(k,f(k-1,n)): n in [0..70]];

f := (k, m) -> floor(m*k/(k-1)):
a := n -> f(9, f(8, n)):
seq(a(n), n = 0..70); # Peter Luschny, May 03 2016

f[k_, m_] := Floor[m*k/(k-1)];
a[n_] := f[9, f[8, n]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 09 2016 *)
CoefficientList[Series[x (1 + x + x^2 + x^3 + x^4 + x^5 + 3 x^6)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)), {x, 0, 70}], x] (* or *)
Table[(63 n - 12 - 12 Mod[n, 7] + 2 Mod[-n - 1, 7])/49, {n, 0, 70}] (* Michael De Vlieger, Dec 25 2016 *)
LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,1,2,3,4,5,6,9},90] (* Harvey P. Dale, May 08 2018 *)

f = lambda k, m: floor(m*k/(k-1))
a = lambda n: f(9, f(8, n))
[a(n) for n in range(71)] # Peter Luschny, May 03 2016

G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 + 3*x^6)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = a(n-1) + a(n-7) - a(n-8).
a(n) = (63*n - 12 - 12*(n mod 7) + 2*((-n-1) mod 7))/49. - Wesley Ivan Hurt, Dec 25 2016

cp's OEIS Frontend

A037477 a(n) = Sum{d(i)*9^i: i=0,1,...,m}, where Sum{d(i)*8^i: i=0,1,...,m} is the base 8 representation of n.

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A047226 Numbers that are congruent to {0, 1, 2, 3, 4} mod 6; a(n)=floor(6(n-1)/5).

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A168183 Numbers that are not multiples of 9.

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A281899 a(n) = n + 6*floor(n/3).

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A047368 Numbers that are congruent to {0, 1, 2, 3, 4, 5} mod 7; a(n)=floor(7(n-1)/6).

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Magma

Maple

Mathematica

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Extensions

A272574 a(n) = f(9, f(8, n)), where f(k,m) = floor(m*k/(k-1)).

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A037477 a(n) = Sum{d(i)9^i: i=0,1,...,m}, where Sum{d(i)8^i: i=0,1,...,m} is the base 8 representation of n.