A030386 -id:A030386 - cp's OEIS frontend

A030397 a(n)=(# 1's)-(# 2's) in first n terms of A030386.

1, 0, 0, 0, 1, 2, 3, 2, 3, 3, 4, 4, 3, 4, 3, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 2, 2, 3, 3, 3, 4, 4, 5, 6, 7, 8, 9, 8, 9, 10, 10, 11, 12, 12, 11, 12, 13, 12, 13, 12, 11, 12, 12, 11, 12, 12, 12, 13, 14, 14, 15, 14, 14, 15, 15, 15, 16, 16

Offset: 1

Clark Kimberling

nonn

A030398 a(n)=(# 2's)-(# 3's) in first n terms of A030386.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

A030308 Triangle T(n, k): Write n in base 2, reverse order of digits, to get the n-th row.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

This is the quite common, so-called "bittest" function, see PARI code. - M. F. Hasler, Jul 21 2013
For a given number m and a digit position k the corresponding sequence index n can be calculated by n(m, k) = m*(1 + floor(log_2(m))) - 2^(1 + floor(log_2(m))) + k + 1. For example: counted from right to left, the second digit of m = 13 (binary 1101) is '0'. Hence the sequence index is n = n(13, 2) = 39. - Hieronymus Fischer, May 05 2007
A070939(n) is the length of n-th row; A000120(n) is the sum of n-th row; A030101(n) is the n-th row seen as binary number; A000035(n) = T(n, 0). - Reinhard Zumkeller, Jun 17 2012

			Triangle begins :
0
1
0, 1
1, 1
0, 0, 1
1, 0, 1
0, 1, 1
1, 1, 1
0, 0, 0, 1
1, 0, 0, 1 - _Philippe Deléham_, Oct 12 2011

Reinhard Zumkeller, Rows n = 0..1023 of triangle, flattened
Index entries for sequences related to binary expansion of n

Cf. A030190.

Cf. A030341, A030386, A031235, A030567, A031007, A031045, A031087, A031298 for the base-3 to base-10 analogs.

a030308 n k = a030308_tabf !! n !! k
a030308_row n = a030308_tabf !! n
a030308_tabf = iterate bSucc [0] where
   bSucc []       = [1]
   bSucc (0 : bs) = 1 : bs
   bSucc (1 : bs) = 0 : bSucc bs
-- Reinhard Zumkeller, Jun 17 2012

A030308_row := n -> op(convert(n,base, 2)):
seq(A030308_row(n), n=0..23); # Peter Luschny, Nov 28 2017

Flatten[Table[Reverse[IntegerDigits[n, 2]], {n, 0, 23}]] (* T. D. Noe, Oct 12 2011 *)

A030308(n,k)=bittest(n,k) \\ Assuming that columns are numbered starting with k=0, as suggested by the formula from R. Zumkeller. - M. F. Hasler, Jul 21 2013

for n in range(20): print([int(z) for z in str(bin(n)[2:])[::-1]]) # Indranil Ghosh, Mar 31 2017

A030308_row = lambda n: n.bits() if n > 0 else [0]
for n in (0..23): print(A030308_row(n)) # Peter Luschny, Nov 28 2017

(0 to 31).map(Integer.toString(, 2).reverse).mkString.split("").map(Integer.parseInt()).toList // Alonso del Arte, Feb 10 2020

a(n) = floor(m/2^(k - 1)) mod 2, where m = max(j|A001855(j) < n) and k = n - A001855(m). - Hieronymus Fischer, May 05 2007, Sep 10 2007

T(n, k) = (n // 2^k) mod 2, for 0 <= k <= log[2](n) and n > 0; T(0, 0) = 0. ('//' denotes integer division). - Peter Luschny, Apr 20 2023

Initial 0 and better name by Philippe Deléham, Oct 12 2011

A053737 Sum of digits of (n written in base 4).

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Mar 28 2000

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

			a(20) = 1+1+0 = 2 because 20 is written as 110 base 4.
From _Omar E. Pol_, Feb 21 2010: (Start)
This can be written as a triangle (cf. A000120):
  0,
  1,2,3,
  1,2,3,4,2,3,4,5,3,4,5,6,
  1,2,3,4,2,3,4,5,3,4,5,6,4,5,6,7,2,3,4,5,3,4,5,6,4,5,6,7,5,6,7,8,3,4,5,6,4,5,6,7,5,6,7,8,6,7,8,9,
  1,2,3,4,2,3,4,5,3,4,5,6,4,5,6,7,2,3,4,5,3,4,5,6,4,5,6,7,5,6,7,8,3,4,5,6,4,...
where the rows converge to A173524.
(End)

Donovan Johnson, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS Vol. 12 (2009), Article 09.5.6.
Steve Butler and Ron L. Graham, Shuffling with ordered cards, arXiv 1003:4422 [math.CO], 2010.
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.

Cf. A231664-A231667, A138530.
Cf. A173524. - Omar E. Pol, Feb 21 2010
Sum of digits of n written in bases 2-16: A000120, A053735, this sequence, A053824, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1).
Cf. A007090, A239690.

a053737 n = if n == 0 then 0 else a053737 m + r where (m, r) = divMod n 4
-- Reinhard Zumkeller, Mar 19 2015

for u=0:104; sol(u+1)=sum(dec2base(u,4)-'0');end
sol % Marius A. Burtea, Jan 17 2019

[&+Intseq(n,4):n in [0..104]]; // Marius A. Burtea, Jan 17 2019

A053737 := proc(n)
    add(d,d=convert(n,base,4)) ;
end proc: # R. J. Mathar, Oct 31 2012

Table[Plus @@ IntegerDigits[n, 4], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> {a, a+1, a+2, a+3}] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)
DigitSum[Range[0, 100], 4] (* Paolo Xausa, Aug 01 2024 *)

PARI
```
a(n)=if(n<1,0,if(n%4,a(n-1)+1,a(n/4)))
```

a(n) = sumdigits(n, 4); \\ Michel Marcus, Aug 24 2019

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(4n+i) = a(n)+i for 0 <= i <= 3.
a(n) = n - 3*Sum_{k>0} floor(n/4^k) = n - 3*A054893(n). (End)
G.f.: (Sum_{k>=0} (x^(4^k) + 2*x^(2*4^k) + 3*x^(3*4^k))/(1 + x^(4^k) + x^(2*4^k) + x^(3*4^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005
a(n) = A138530(n,4) for n > 3. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_{k>=0} A030386(n,k). - Philippe Deléham, Oct 21 2011
a(n) = A007953(A007090(n)). - Reinhard Zumkeller, Mar 19 2015
a(0) = 0; a(n) = a(n - 4^floor(log_4(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 4*log(4)/3 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

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

Original entry on oeis.org

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

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

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

A031045 Triangle T(n,k): write n in base 8, reverse order of digits.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

Robert Israel, Table of n, a(n) for n = 0..10003 (rows 0 to 3646, flattened)

Cf. A030308, A030341, A030386, A031235, A030567, A031007, A031087, A031298 for the base-2 to base-10 analogs.

seq(op(convert(n,base,8)),n=0..100); # Robert Israel, Jul 22 2019

Flatten[Table[Reverse[IntegerDigits[n,8]],{n,80}]] (* Harvey P. Dale, Aug 08 2011 *)

A031045(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.");*/n\8^k%8 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030341, ... Note: The operation could be done using bitwise arithmetic, bitand(n>>(3*k),7), but this is not significantly faster in PARI. - M. F. Hasler, Jul 21 2013

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

cp's OEIS Frontend

A030397 a(n)=(# 1's)-(# 2's) in first n terms of A030386.

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A030398 a(n)=(# 2's)-(# 3's) in first n terms of A030386.

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A030308 Triangle T(n, k): Write n in base 2, reverse order of digits, to get the n-th row.

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A053737 Sum of digits of (n written in base 4).

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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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