A038573 -id:A038573 - cp's OEIS frontend

A048967 Number of even entries in row n of Pascal's triangle (A007318).

0, 0, 1, 0, 3, 2, 3, 0, 7, 6, 7, 4, 9, 6, 7, 0, 15, 14, 15, 12, 17, 14, 15, 8, 21, 18, 19, 12, 21, 14, 15, 0, 31, 30, 31, 28, 33, 30, 31, 24, 37, 34, 35, 28, 37, 30, 31, 16, 45, 42, 43, 36, 45, 38, 39, 24, 49, 42, 43, 28, 45, 30, 31, 0, 63, 62, 63, 60, 65, 62, 63, 56, 69, 66, 67

Offset: 0

Brian Galebach

In rows 2^k - 1 all entries are odd.
a(n) = 0 (all the entries in the row are odd) iff n = 2^m - 1 for some m >= 0 and then n belongs to sequence A000225. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
Also number of zeros in n-th row of Sierpiński's triangle (cf. A047999): a(n) = A023416(A001317(n)). - Reinhard Zumkeller, Nov 24 2012

			Row 4 is 1 4 6 4 1 with 3 even entries so a(4)=3.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

Cf. A007318, A001316, A000120, A000225, A038573.

Cf. A249304.

import Data.List (transpose)
a048967 n = a048967_list !! n
a048967_list = 0 : xs where
   xs = 0 : concat (transpose [zipWith (+) [1..] xs, map (* 2) xs])
-- Reinhard Zumkeller, Nov 14 2014, Nov 24 2012

Table[n + 1 - Sum[ Mod[ Binomial[n, k], 2], {k, 0, n} ], {n, 0, 100} ]
a[n_] := n + 1 - 2^DigitCount[n, 2, 1]; Array[a, 100, 0] (* Amiram Eldar, Jul 27 2023 *)

a(n)=if(n<1,0,if(n%2==0,a(n/2)+n/2,2*a((n-1)/2)))

def A048967(n): return n+1-(1<Chai Wah Wu, May 03 2023

a(n) = n+1 - A001316(n) = n+1 - 2^A000120(n) = n+1 - Sum_{k=0..n} (C(n, k) mod 2) = Sum_{k=0..n} ((1 - C(n, k)) mod 2).
a(2n) = a(n) + n, a(2n+1) = 2a(n). - Ralf Stephan, Oct 07 2003
a(n) = row sums in A219463 = A000120(A219843(n)). - Reinhard Zumkeller, Nov 30 2012
A249304(n+1) = a(n+1) + a(n). - Reinhard Zumkeller, Nov 14 2014
G.f.: 1/(1 - x)^2 - Product_{k>=0} (1 + 2*x^(2^k)). - Ilya Gutkovskiy, Jul 19 2019

A038574 Write n in ternary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 5, 8, 1, 4, 5, 4, 13, 14, 5, 14, 17, 2, 5, 8, 5, 14, 17, 8, 17, 26, 1, 4, 5, 4, 13, 14, 5, 14, 17, 4, 13, 14, 13, 40, 41, 14, 41, 44, 5, 14, 17, 14, 41, 44, 17, 44, 53, 2, 5, 8, 5, 14, 17, 8, 17, 26, 5, 14, 17, 14, 41, 44, 17, 44, 53, 8, 17, 26, 17, 44, 53, 26, 53, 80

Offset: 0

N. J. A. Sloane

			15 = 120 -> 012 = 5, so a(15)=5.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A038573, A004185.

Table[FromDigits[Sort[IntegerDigits[n,3]],3],{n,0,90}] (* Harvey P. Dale, Dec 08 2015 *)

More terms from Erich Friedman.

A084471 Change 0 to 00 in binary representation of n.

Original entry on oeis.org

1, 4, 3, 16, 9, 12, 7, 64, 33, 36, 19, 48, 25, 28, 15, 256, 129, 132, 67, 144, 73, 76, 39, 192, 97, 100, 51, 112, 57, 60, 31, 1024, 513, 516, 259, 528, 265, 268, 135, 576, 289, 292, 147, 304, 153, 156, 79, 768, 385, 388, 195, 400, 201, 204, 103, 448, 225

Offset: 1

Reinhard Zumkeller, May 27 2003

a(n) = n iff n = 2^k - 1, k>0.

A023416(a(n))=A023416(n)*2; A000120(a(n))=A000120(n);

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A084472(n)=A007088(a(n)), A084473, A038573.
Ordered terms are in A060142.
Column k=2 of A340666.
Cf. A088698, A175047. - Robert G. Wilson v, Dec 10 2009

a084471 1 = 1
a084471 x = 2 * (2 - d) * a084471 x' + d  where (x',d) = divMod x 2
-- Reinhard Zumkeller, Jul 16 2012

a:= n-> Bits[Join](subs(0=[0$2][], Bits[Split](n))):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 15 2021

f[n_] := FromDigits[Flatten[IntegerDigits[n, 2] /. {0 -> {0, 0}}], 2]; Array[f, 60] (* Robert G. Wilson v, Dec 10 2009 *)

a(1)=1, a(2*k+1)=2*a(k)+1, a(2*k)=4*a(k).

A319652 Write n in 4-ary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 3, 1, 5, 6, 7, 2, 6, 10, 11, 3, 7, 11, 15, 1, 5, 6, 7, 5, 21, 22, 23, 6, 22, 26, 27, 7, 23, 27, 31, 2, 6, 10, 11, 6, 22, 26, 27, 10, 26, 42, 43, 11, 27, 43, 47, 3, 7, 11, 15, 7, 23, 27, 31, 11, 27, 43, 47, 15, 31, 47, 63, 1, 5, 6, 7, 5, 21, 22, 23, 6, 22, 26, 27

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..4095

b-ary: A038573 (b=2), A038574 (b=3), this sequence (b=4), A319653 (b=5), A319654 (b=6), A319655 (b=7), A319656 (b=8), A319657 (b=9), A004185 (b=10).

Cf. A165012.

a:= n-> (l-> add(l[-i]*4^(i-1), i=1..nops(l)))(sort(convert(n, base, 4))):
seq(a(n), n=0..75);  # Alois P. Heinz, Aug 07 2024

Table[FromDigits[Sort[IntegerDigits[n, 4]], 4], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 4)), 4); \\ Michel Marcus, Sep 25 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.join.to_i(k)}
end
p A(4, 100)

A319653 Write n in 5-ary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 6, 7, 8, 9, 2, 7, 12, 13, 14, 3, 8, 13, 18, 19, 4, 9, 14, 19, 24, 1, 6, 7, 8, 9, 6, 31, 32, 33, 34, 7, 32, 37, 38, 39, 8, 33, 38, 43, 44, 9, 34, 39, 44, 49, 2, 7, 12, 13, 14, 7, 32, 37, 38, 39, 12, 37, 62, 63, 64, 13, 38, 63, 68, 69, 14, 39, 64, 69, 74, 3

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..3124

b-ary: A038573 (b=2), A038574 (b=3), A319652 (b=4), this sequence (b=5), A319654 (b=6), A319655 (b=7), A319656 (b=8), A319657 (b=9), A004185 (b=10).

Cf. A165032, A319722.

Table[FromDigits[Sort[IntegerDigits[n,5]],5],{n,0,100}] (* Harvey P. Dale, Feb 29 2024 *)

a(n) = fromdigits(vecsort(digits(n, 5)), 5); \\ Michel Marcus, Sep 25 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.join.to_i(k)}
end
p A(5, 100)

A319654 Write n in 6-ary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 7, 8, 9, 10, 11, 2, 8, 14, 15, 16, 17, 3, 9, 15, 21, 22, 23, 4, 10, 16, 22, 28, 29, 5, 11, 17, 23, 29, 35, 1, 7, 8, 9, 10, 11, 7, 43, 44, 45, 46, 47, 8, 44, 50, 51, 52, 53, 9, 45, 51, 57, 58, 59, 10, 46, 52, 58, 64, 65, 11, 47, 53, 59, 65, 71, 2, 8

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..7775

b-ary: A038573 (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), this sequence (b=6), A319655 (b=7), A319656 (b=8), A319657 (b=9), A004185 (b=10).

Cf. A165051, A319723.

Table[FromDigits[Sort[IntegerDigits[n,6]],6],{n,0,80}] (* Harvey P. Dale, Sep 20 2020 *)

a(n) = fromdigits(vecsort(digits(n, 6)), 6); \\ Michel Marcus, Sep 25 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.join.to_i(k)}
end
p A(6, 100)

A319655 Write n in 7-ary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 13, 2, 9, 16, 17, 18, 19, 20, 3, 10, 17, 24, 25, 26, 27, 4, 11, 18, 25, 32, 33, 34, 5, 12, 19, 26, 33, 40, 41, 6, 13, 20, 27, 34, 41, 48, 1, 8, 9, 10, 11, 12, 13, 8, 57, 58, 59, 60, 61, 62, 9, 58, 65, 66, 67, 68, 69, 10, 59, 66, 73

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..2400

b-ary: A038573 (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), this sequence (b=7), A319656 (b=8), A319657 (b=9), A004185 (b=10).

Cf. A165071, A319724.

a:= n-> (l-> add(l[-i]*7^(i-1), i=1..nops(l)))(sort(convert(n, base, 7))):
seq(a(n), n=0..73);  # Alois P. Heinz, Aug 07 2024

Table[FromDigits[Sort[IntegerDigits[n, 7]], 7], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 7)), 7); \\ Michel Marcus, Sep 25 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.join.to_i(k)}
end
p A(7, 100)

A319656 Write n in 8-ary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 2, 10, 18, 19, 20, 21, 22, 23, 3, 11, 19, 27, 28, 29, 30, 31, 4, 12, 20, 28, 36, 37, 38, 39, 5, 13, 21, 29, 37, 45, 46, 47, 6, 14, 22, 30, 38, 46, 54, 55, 7, 15, 23, 31, 39, 47, 55, 63, 1, 9, 10, 11, 12, 13, 14

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..4095

b-ary: A038573 (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), A319655 (b=7), this sequence (b=8), A319657 (b=9), A004185 (b=10).

Cf. A165090.

a:= n-> (l-> add(l[-i]*8^(i-1), i=1..nops(l)))(sort(convert(n, base, 8))):
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 07 2024

Table[FromDigits[Sort[IntegerDigits[n, 8]], 8], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 8)), 8); \\ Michel Marcus, Sep 25 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.join.to_i(k)}
end
p A(8, 100)

A319657 Write n in 9-ary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 10, 11, 12, 13, 14, 15, 16, 17, 2, 11, 20, 21, 22, 23, 24, 25, 26, 3, 12, 21, 30, 31, 32, 33, 34, 35, 4, 13, 22, 31, 40, 41, 42, 43, 44, 5, 14, 23, 32, 41, 50, 51, 52, 53, 6, 15, 24, 33, 42, 51, 60, 61, 62, 7, 16, 25, 34, 43, 52, 61, 70, 71

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..6560

b-ary: A038573 (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), A319655 (b=7), A319656 (b=8), this sequence (b=9), A004185 (b=10).

Cf. A165110.

a:= n-> (l-> add(l[-i]*9^(i-1), i=1..nops(l)))(sort(convert(n, base, 9))):
seq(a(n), n=0..71);  # Alois P. Heinz, Aug 07 2024

Table[FromDigits[Sort[IntegerDigits[n, 9]], 9], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 9)), 9); \\ Michel Marcus, Sep 25 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.join.to_i(k)}
end
p A(9, 100)

A106151 In binary representation of n: delete one zero in each contiguous block of zeros.

Original entry on oeis.org

1, 1, 3, 2, 3, 3, 7, 4, 5, 3, 7, 6, 7, 7, 15, 8, 9, 5, 11, 6, 7, 7, 15, 12, 13, 7, 15, 14, 15, 15, 31, 16, 17, 9, 19, 10, 11, 11, 23, 12, 13, 7, 15, 14, 15, 15, 31, 24, 25, 13, 27, 14, 15, 15, 31, 28, 29, 15, 31, 30, 31, 31, 63, 32, 33, 17, 35, 18, 19, 19, 39, 20, 21, 11, 23, 22, 23

Offset: 1

Reinhard Zumkeller, May 07 2005

Equivalently, change bits 10 -> 0. - Michael S. Branicky, Nov 12 2021

			n=144 = '10010000' -> '101000' = 40 = a(144);
n=145 = '10010001' -> '101001' = 41 = a(145);
n=146 = '10010010' -> '10101'  = 21 = a(146).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A007088, A030308, A038573, A090078, A175047, A048679, A348710.

import Data.List (group)
a106151 = foldr (\b v -> 2 * v + b) 0 . concatMap
   (\bs'@(b:bs) -> if b == 0 then bs else bs') . group . a030308_row
-- Reinhard Zumkeller, Jul 05 2013

A106151(n) = if(n<=1, n, if(n%2, 1+(2*A106151((n-1)/2)), A106151(n>>valuation(n, 2))<<(valuation(n, 2)-1))); \\ Antti Karttunen, May 13 2018

A106151(n) = { my(s=0, i=0); while(n, if(2!=(n%4), s += (n%2)<>= 1); (s); }; \\ Antti Karttunen, Jul 01 2024

def a(n): return int(bin(n).replace("b", "").replace("10", "1"), 2)
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Nov 12 2021

a(n) <= n; a(n) = n iff n = 2^k-1: a(A000225(n))=A000225(n);
A000120(a(n)) = A000120(n);
A023416(a(n)) = A023416(n) - A087116(n).
a(n) = b(n, 0), where b(n, r) = if n = 1 then 1 else b(floor(n/2), 1 - n mod 2)*(1 + floor((1 + r + n mod 2)/2)) + n mod 2.
For n <= 1, a(n) = n, and for n > 1, if n is odd, then a(n) = 1+2*a((n-1)/2), otherwise, when n is even, a(n) = (2^(A007814(n)-1)) * a(A000265(n)). - Antti Karttunen, May 13 2018

cp's OEIS Frontend

A048967 Number of even entries in row n of Pascal's triangle (A007318).

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A038574 Write n in ternary, sort digits into increasing order.

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A084471 Change 0 to 00 in binary representation of n.

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A319652 Write n in 4-ary, sort digits into increasing order.

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A319653 Write n in 5-ary, sort digits into increasing order.

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A319654 Write n in 6-ary, sort digits into increasing order.

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A319655 Write n in 7-ary, sort digits into increasing order.

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A319656 Write n in 8-ary, sort digits into increasing order.