A031045 -id:A031045 - cp's OEIS frontend

A030341 Triangle T(n,k): write n in base 3, reverse order of digits.

0, 1, 2, 0, 1, 1, 1, 2, 1, 0, 2, 1, 2, 2, 2, 0, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 1, 2, 1, 2, 2, 1, 0, 0, 2, 1, 0, 2, 2, 0, 2, 0, 1, 2, 1, 1, 2, 2, 1, 2, 0, 2, 2, 1, 2, 2, 2, 2, 2, 0, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1

Offset: 0

Clark Kimberling

			Triangle begins :
0
1
2
0, 1
1, 1
2, 1
0, 2
1, 2
2, 2
0, 0, 1
1, 0, 1
2, 0, 1
0, 1, 1
1, 1, 1
2, 1, 1 ...

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system

Cf. A081604 (row lengths), A053735 (row sums), A007089, A003137.

Cf. A030308, A030386, A031235, A030567, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

a030341 n k = a030341_tabf !! n !! k
a030341_row n = a030341_tabf !! n
a030341_tabf = iterate succ [0] where
   succ []     = [1]
   succ (2:ts) = 0 : succ ts
   succ (t:ts) = (t + 1) : ts
-- Reinhard Zumkeller, Feb 21 2013

A030341_row := n -> op(convert(n, base, 3)):
seq(A030341_row(n), n=0..32); # Peter Luschny, Nov 28 2017

Flatten[Table[Reverse[IntegerDigits[n,3]],{n,0,40}]] (* Harvey P. Dale, Oct 20 2014 *)

A030341(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\3^k%3 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030567 and others. - M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

A031298 Triangle T(n,k): write n in base 10, reverse order of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 4, 0

Offset: 0

Clark Kimberling

The length of n-th row is given in A055642(n). - Reinhard Zumkeller, Jul 04 2012

According to the formula for T(n,1), columns are numbered starting with 1. One might also number columns starting with the offset 0, as to have the coefficient of 10^k in column k. - M. F. Hasler, Jul 21 2013

Reinhard Zumkeller, Rows n = 0..2500 of triangle, flattened

Cf. A030308, A030341, A030386, A031235, A030567, A031007, A031045, A031087 for the base-2 to base-9 analogs.

a031298 n k = a031298_tabf !! n !! k
a031298_row n = a031298_tabf !! n
a031298_tabf = iterate succ [0] where
   succ []     = [1]
   succ (9:ds) = 0 : succ ds
   succ (d:ds) = (d + 1) : ds
-- Reinhard Zumkeller, Jul 04 2012

Table[Reverse[IntegerDigits[n]],{n,0,50}]//Flatten (* Harvey P. Dale, Mar 07 2023 *)

T(n,k)=n\10^(k-1)%10 \\ M. F. Hasler, Jul 21 2013

T(n,1) = A010879(n); T(n,A055642(n)) = A000030(n). - Reinhard Zumkeller, Jul 04 2012

Initial 0 and better name by Philippe Deléham, Oct 20 2011

Edited by M. F. Hasler, Jul 21 2013

A030567 Triangle T(n,k): Write n in base 6 and reverse order of digits to get row n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0, 1, 0, 1, 1, 1, 1, 1, 2

Offset: 0

Clark Kimberling

If columns are numbered starting with k=0, then T(n,k) contains the coefficient of 6^k in n's base-6 expansion. - M. F. Hasler, Jul 21 2013

R. J. Mathar, Table of n, a(n) for n = 0..5950

See A030548 for a quite complete list of crossreferences.
Cf. A030568 - A030573 for positions of a given digit.
Cf. A030575 - A030580 for run lengths, A030581 - A030585 for more.
Row sums (same as those of A030548) are in A053827.
Cf. A030308, A030341, A030386, A031235, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Flatten[Table[Reverse[IntegerDigits[n,6]],{n,0,50}]] (* Harvey P. Dale, Sep 27 2015 *)

A030567(n,k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\6^k%6 \\ Assuming that columns start with k=0, cf. comment. TO DO: implement flattened sequence, such that A030567(n)=a(n). - M. F. Hasler, Jul 21 2013

Initial 0 and better name from Philippe Deléham, Oct 20 2011

Edited and crossrefs added by M. F. Hasler, Jul 21 2013

A053829 Sum of digits of (n written in base 8).

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7}, 1->{1,2,3,4,5,6,7,8}, 2->{2,3,4,5,6,7,8,9}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20)=2+4=6 because 20 is written as 24 base 8.
From _Omar E. Pol_, Feb 21 2010: (Start)
It appears that this can be written as a triangle (See the conjecture in the entry A000120):
0,
1,2,3,4,5,6,7,
1,2,3,4,5,6,7,8,2,3,4,5,6,7,8,9,3,4,5,6,7,8,9,10,4,5,6,7,8,9,10,11,5,6,7,8,9,10,11,12,6,7,8,9,10,11,12,13,7,8,9,10,11,12,13,14,
1,2,3,4,5,6,7,8,2,3,4,5,6,7,8,9,3,4,5,6,7,8,9,10,4,5,6,7,8,9,10...
where the rows converge to A173528. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Robert Walker, Self Similar Sloth Canon Number Sequences
Eric Weisstein's World of Mathematics, Digit Sum.
Eric Weisstein's World of Mathematics, Octal.

Cf. A000120, A007953, A231680, A231681, A231682, A231683.

Cf. A173528. - Omar E. Pol, Feb 21 2010

a053829 n = q 0 $ divMod n 8 where
   q r (0, d) = r + d
   q r (m, d) = q (r + d) $ divMod m 8
-- Reinhard Zumkeller, May 15 2011

Table[Plus @@ IntegerDigits[n, 8], {n, 0, 95}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 7}]] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)

PARI
```
a(n)=if(n<1,0,if(n%8,a(n-1)+1,a(n/8)))
```

a(n) = sumdigits(n, 8); \\ Michel Marcus, Jul 10 2022

def A053829(n): return sum(int(d) for d in oct(n)[2:]) # Chai Wah Wu, Jul 09 2022

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(8n+i) = a(n)+i for 0 <= i <= 7.
a(n) = n-7*(Sum_{k>0} floor(n/8^k)) = n-7*A054897(n). (End)
a(n) = A138530(n,8) for n > 7. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_k>=0 {A031045(n,k)}. - Philippe Deléham, Oct 21 2011
a(0) = 0; a(n) = a(n - 8^floor(log_8(n))) + 1. - Ilya Gutkovskiy, Aug 24 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 8*log(8)/7 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A031235 Triangle T(n,k): write n in base 5, reverse order of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 0, 2, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 4, 2, 1, 0

Offset: 0

Clark Kimberling

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened

Cf. A030308, A030341, A030386, A030567, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Cf. A007091.

a031235 n k = a031235_tabf !! n !! k
a031235_row n = a031235_tabf !! n
a031235_tabf = iterate succ [0] where
   succ []     = [1]
   succ (4:ts) = 0 : succ ts
   succ (t:ts) = (t + 1) : ts
-- Reinhard Zumkeller, Sep 18 2015

Reverse[IntegerDigits[#,5]]&/@Range[0,40]//Flatten (* Harvey P. Dale, Aug 02 2016 *)

A031235(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\5^k%5 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030341, ... - M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

A030386 Triangle T(n,k): write n in base 4, reverse order of digits.

Original entry on oeis.org

0, 1, 2, 3, 0, 1, 1, 1, 2, 1, 3, 1, 0, 2, 1, 2, 2, 2, 3, 2, 0, 3, 1, 3, 2, 3, 3, 3, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 0, 2, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 0, 3, 1, 1, 3, 1, 2, 3, 1, 3, 3, 1, 0, 0, 2, 1, 0, 2, 2, 0, 2, 3, 0, 2, 0, 1, 2

Offset: 0

Clark Kimberling

			Triangle begins:
0
1
2
3
0, 1
1, 1
2, 1
3, 1
0, 2
1, 2
2, 2
3, 2
0, 3
1, 3
2, 3
3, 3
0, 0, 1
1, 0, 1 ... - _Philippe Deléham_, Oct 20 2011

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened

Cf. A030308, A030341, A031235, A030567, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Cf. A007090.

a030386 n k = a030386_tabf !! n !! k
a030386_row n = a030386_tabf !! n
a030386_tabf = iterate succ [0] where
   succ []     = [1]
   succ (3:ts) = 0 : succ ts
   succ (t:ts) = (t + 1) : ts
-- Reinhard Zumkeller, Sep 18 2015

A030386_row := n -> op(convert(n, base, 4)):
seq(A030386_row(n), n=0..36); # Peter Luschny, Nov 28 2017

Flatten[Table[Reverse[IntegerDigits[n,4]],{n,0,50}]] (* Harvey P. Dale, Oct 13 2012 *)

A030386(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\4^k%4 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030341, ... \\ M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

A031087 Triangle T(n,k): write n in base 9, reverse order of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5

Offset: 0

Clark Kimberling

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened

Cf. A030308, A030341, A030386, A031235, A030567, A031007, A031045, A031298 for the base-2 to base-10 analogs.

Cf. A031076, A007095.

a031087 n k = a031087_row n !! (k-1)
a031087_row n | n < 9     = [n]
              | otherwise = m : a031087_row n' where (n',m) = divMod n 9
a031087_tabf = map a031087_row [0..]
-- Reinhard Zumkeller, Jul 07 2015

A031087(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\9^k%9 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030567 and others. - M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

A031007 Triangle T(n,k): Write n in base 7, reverse order of digits, to get row n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 6, 4, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 6, 5, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6, 6

Offset: 0

Clark Kimberling

Cf. A030308, A030341, A030386, A031235, A030567, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Flatten[Table[Reverse[IntegerDigits[n,7]],{n,0,50}]] (* Harvey P. Dale, Feb 25 2014 *)

A031007(n, k=-1)={k<0&&error("Flattened sequence not yet implemented.");n\7^k%7} \\ Assuming that columns start with k=0 as in A030308, A030341, ... TO DO: implement flattened sequence, such that A030567(n)=a(n). - M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

cp's OEIS Frontend

A030341 Triangle T(n,k): write n in base 3, reverse order of digits.

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A031298 Triangle T(n,k): write n in base 10, reverse order of digits.

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A030567 Triangle T(n,k): Write n in base 6 and reverse order of digits to get row n.

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A053829 Sum of digits of (n written in base 8).

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A031235 Triangle T(n,k): write n in base 5, reverse order of digits.

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A030386 Triangle T(n,k): write n in base 4, reverse order of digits.

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Haskell

Maple

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A031087 Triangle T(n,k): write n in base 9, reverse order of digits.

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