A038573 -id:A038573 - cp's OEIS frontend

A256263 Triangle read by rows: T(j,k) = 2*k-1 if k is a power of 2, otherwise, between positions that are powers of 2 we have the initial terms of A016969, with j>=0, 1<=k<=A011782(j) and T(0,1) = 0.

0, 1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5, 11, 17, 15, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 63, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89

Offset: 0

Omar E. Pol, Mar 30 2015

Partial sums give A256264.
First differs from A160552 at a(27).
Appears to be a canonical sequence partially related to the cellular automata of A139250, A147562, A162795, A169707, A255366, A256250. See also A256264 and A256260.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
0;
1;
1,3;
1,3,5,7;
1,3,5,7,5,11,17,15;
1,3,5,7,5,11,17,15,5,11,17,23,29,35,41,31;
1,3,5,7,5,11,17,15,5,11,17,23,29,35,41,31,5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,63;
...
Right border gives A000225.
Apart from the initial 0 the row sums give A000302.
Rows converge to A256258.
.
Illustration of initial terms in the fourth quadrant of the square grid:
---------------------------------------------------------------------------
n   a(n)                 Compact diagram
---------------------------------------------------------------------------
0    0     _
1    1    |_|_ _
2    1      |_| |
3    3      |_ _|_ _ _ _
4    1          |_| | | |
5    3          |_ _| | |
6    5          |_ _ _| |
7    7          |_ _ _ _|_ _ _ _ _ _ _ _
8    1                  |_| | | |_ _  | |
9    3                  |_ _| | |_  | | |
10   5                  |_ _ _| | | | | |
11   7                  |_ _ _ _| | | | |
12   5                  | | |_ _ _| | | |
13  11                  | |_ _ _ _ _| | |
14  17                  |_ _ _ _ _ _ _| |
15  15                  |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
16   1                                  |_| | | |_ _  | |_ _ _ _ _ _  | |
17   3                                  |_ _| | |_  | | |_ _ _ _ _  | | |
18   5                                  |_ _ _| | | | | |_ _ _ _  | | | |
19   7                                  |_ _ _ _| | | | |_ _ _  | | | | |
20   5                                  | | |_ _ _| | | |_ _  | | | | | |
21  11                                  | |_ _ _ _ _| | |_  | | | | | | |
22  17                                  |_ _ _ _ _ _ _| | | | | | | | | |
23  15                                  |_ _ _ _ _ _ _ _| | | | | | | | |
24   5                                  | | | | | | |_ _ _| | | | | | | |
25  11                                  | | | | | |_ _ _ _ _| | | | | | |
26  17                                  | | | | |_ _ _ _ _ _ _| | | | | |
27  23                                  | | | |_ _ _ _ _ _ _ _ _| | | | |
28  29                                  | | |_ _ _ _ _ _ _ _ _ _ _| | | |
29  35                                  | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
30  41                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
31  31                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
A256264(n) gives the total number of cells after n-th stage.

Ivan Neretin, Table of n, a(n) for n = 0..8191
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000225, A000302, A011782, A038573, A006257, A016969, A139251, A160552, A256250, A256258, A256260, A256261, A256264, A256265.

Flatten@Join[{0}, NestList[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 6]] (* Ivan Neretin, Feb 14 2017 *)

Terms a(95) to a(98) fixed by Ivan Neretin, Feb 14 2017

A160721 First differences of A160720.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 12, 28, 4, 12, 12, 28, 12, 28, 28, 60, 4, 12, 12, 28, 12, 28, 28, 60, 12, 28, 28, 60, 28, 60, 60, 124, 4, 12, 12, 28, 12, 28, 28, 60, 12, 28, 28, 60, 28, 60, 60, 124, 12, 28, 28, 60, 28, 60, 60, 124, 28, 60, 60, 124, 60, 124, 124, 252, 4, 12, 12, 28, 12, 28, 28

Offset: 1

Omar E. Pol, May 25 2009, May 29 2009

This sequence is related to the Sierpinski triangle and to Gould's sequence A001316. - Omar E. Pol, Jul 23 2009

When written as a irregular triangle in which row lengths are A011782 it appears that right border gives A173033. - Omar E. Pol, Mar 20 2013

			From _Omar E. Pol_, Mar 20 2013 (Start):
Triangle begins:
1;
4;
4,12;
4,12,12,28;
4,12,12,28,12,28,28,60;
4,12,12,28,12,28,28,60,12,28,28,60,28,60,60,124;
4,12,12,28,12,28,28,60,12,28,28,60,28,60,60,124,12,28,28,60,28,60,60,124,28,60,60,124,60,124,124,252;
(End)

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000120, A001316, A038573, A139250, A139251, A160720, A160722, A160723.

a(1)=1. Observation: It appears that a(n) = 4*A038573(n-1), n>1. [From Omar E. Pol, Jul 23 2009]. This formula is correct! - N. J. A. Sloane, Jan 23 2016

More terms from R. J. Mathar, Jul 14 2009

A053581 First differences of the Poly-Bernoulli numbers B_n^(k) with k=-2 (A027649).

Original entry on oeis.org

1, 3, 10, 32, 100, 308, 940, 2852, 8620, 25988, 78220, 235172, 706540, 2121668, 6369100, 19115492, 57362860, 172121348, 516429580, 1549419812, 4648521580, 13946089028, 41839315660, 125520044132

Offset: 0

Barry E. Williams, Jan 18 2000

Also the second differences of A001047.

Equals sum of "terms added" to current row of the triangle version of A038573 to get the next row. a(3) = 32 sum of (3, 7, 7, 15) = terms appended to row 2 of the triangle in A038573. - Gary W. Adamson, Jun 04 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Cf. A001047, A027649.
Cf. A001045.
Cf. A038573. - Gary W. Adamson, Jun 04 2009

List([0..30], n-> 4*3^(n-1) +(0^n -3*2^n)/6) # G. C. Greubel, May 16 2019

[4*3^n/3+0^n/6-2^n/2: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

CoefficientList[Series[(1-x)^2/((1-2x)(1-3x)),{x,0,30}],x]  (* Harvey P. Dale, Apr 22 2011 *)

vector(30, n, n--; 4*3^(n-1) +(0^n -3*2^n)/6) \\ G. C. Greubel, May 16 2019

[4*3^(n-1) +(0^n -3*2^n)/6 for n in (0..30)] # G. C. Greubel, May 16 2019

a(n) = 5*a(n-1) - 6*a(n-2) + C(2,2-n), n>1, with a(0)=1, a(1)=3, where C(2, 2-n)=1 for n=2 and =0 for n>2.
From Paul Barry, Jun 26 2003: (Start)
Binomial transform of A000975(n+1).
G.f.: (1-x)^2/((1-2*x)*(1-3*x)).
a(n) = 4*3^n/3 + 0^n/6 - 2^n/2. (End)
a(n) = Sum_{k=0..n+1} binomial(n+1, k) * Sum_{j=0..floor(k/2)} A001045(k-2*j). - Paul Barry, Apr 17 2005
E.g.f.: (1 - 3*exp(2*x) + 8*exp(3*x))/6. - G. C. Greubel, May 16 2019

A340666 A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 1, 0, 1, 8, 3, 4, 3, 0, 1, 16, 3, 16, 5, 3, 0, 1, 32, 3, 64, 9, 6, 7, 0, 1, 64, 3, 256, 17, 12, 7, 1, 0, 1, 128, 3, 1024, 33, 24, 7, 8, 3, 0, 1, 256, 3, 4096, 65, 48, 7, 64, 9, 3, 0, 1, 512, 3, 16384, 129, 96, 7, 512, 33, 10, 7

Offset: 0

Alois P. Heinz, Jan 15 2021

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,       0,        0, ...
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 2,  4,   8,   16,    32,     64,     128,      256, ...
  3, 3,  3,   3,    3,     3,      3,       3,        3, ...
  1, 4, 16,  64,  256,  1024,   4096,   16384,    65536, ...
  3, 5,  9,  17,   33,    65,    129,     257,      513, ...
  3, 6, 12,  24,   48,    96,    192,     384,      768, ...
  7, 7,  7,   7,    7,     7,      7,       7,        7, ...
  1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Columns k=0-2, 4 give: A038573, A001477, A084471, A084473.
Rows n=0..17, 19 give: A000004, A000012, A000079, A010701, A000302, A000051(k+1), A007283, A010727, A001018, A087289, A007582(k+1), A062709(k+2), A164346, A181565(k+1), A005009, A181404(k+3), A001025, A199493, A253208(k+1).
Main diagonal gives A340667.
Cf. A000120, A023416.

A:= (n, k)-> Bits[Join](subs(0=[0$k][], Bits[Split](n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n<2, n,
     `if`(irem(n, 2, 'r')=1, A(r, k)*2+1, A(r, k)*2^k))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := FromDigits[IntegerDigits[n, 2] /. 0 -> Sequence @@ Table[0, {k}], 2];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 02 2021 *)

A000120(A(n,k)) = A000120(n) = log_2(A(n,0)+1).

A023416(A(n,k)) = k * A023416(n) for n >= 1.

A090971 Sierpiński's triangle, read by rows, starting from 1: T(n,k) = (T(n-1,k) + T(n-1,k-1)) mod 2.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Feb 28 2004

Row sums give A038573.

			Triangle begins with:
  1;
  0, 1;
  1, 1, 1;
  0, 0, 0, 1;
  1, 0, 0, 1, 1;
  0, 1, 0, 1, 0, 1;
  1, 1, 1, 1, 1, 1, 1; ...

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A007318, A038573, A047999, A062534, A074909.

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==n, 1, Mod[T[n-1,k] + T[n-1, k-1], 2]]]; Table[T[n, k], {n,1,10}, {k,1,n}] (* G. C. Greubel, Feb 03 2019 *)
Table[Boole[BitAnd[n, k] == k], {n, 1, 14}, {k, 1, n}] // Flatten (* Amiram Eldar, Aug 24 2024 *)

T(n,k)=if(k<0 || k>n, 0, if(n==0, 1, (T(n-1,k)+T(n-1,k-1))%2))

From Philippe Deléham, Feb 29 2004: (Start)
Triangle A047999(n, k) for n,k > 0; A047999: Pascal's triangle mod 2.
a(n) = A062534(n-1) mod 2.
T(n-1, k-1) = A074909(n, n-k) mod 2. (End)
T(n, k) = 1 if bitand(n, k) = k, and 0 otherwise. - Amiram Eldar, Aug 24 2024

A159913 a(n) = 2^(A000120(n) + 1) - 1, where A000120(n) = number of nonzero bits in n.

Original entry on oeis.org

1, 3, 3, 7, 3, 7, 7, 15, 3, 7, 7, 15, 7, 15, 15, 31, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31, 63, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31, 63, 7, 15, 15, 31, 15, 31, 31, 63, 15, 31, 31, 63, 31, 63, 63, 127, 3, 7, 7, 15, 7, 15, 15, 31, 7, 15, 15, 31, 15, 31, 31

Offset: 0

M. F. Hasler, May 03 2009

nonn

Essentially the same sequence as A117973 and A001316. The latter entry has much more information. - N. J. A. Sloane, Jun 05 2009
First differences of A159912; every other term of A038573.
Equals Sierpinski's gasket, A047999; as an infinite lower triangular matrix * [1,2,2,2,...] as a vector. - Gary W. Adamson, Oct 16 2009
a(n) is also the number of cells turned ON at n-th generation in the outward corner version of the Ulam-Warburton cellular automaton of A147562, and a(n) is also the number of Y-toothpicks added at n-th generation in the outward corner version of the Y-toothpick structure of A160120. - David Applegate and Omar E. Pol, Jan 24 2016

			From _Michael De Vlieger_, Jan 25 2016: (Start)
The number n converted to binary, "0" represented by "." for better visibility of 1's, totaling the 1's and calculating the sequence:
n    Binary   Total                         a(n)
0 -> .     ->     0, thus 2^(0+1)-1 =  2-1 =  1
1 -> 1     ->     1,   "  2^(1+1)-1 =  4-1 =  3
2 -> 1.    ->     1,   "  2^(1+1)-1 =  4-1 =  3
3 -> 11    ->     2,   "  2^(2+1)-1 =  8-1 =  7
4 -> 1..   ->     1,   "  2^(1+1)-1 =  4-1 =  3
5 -> 1.1   ->     2,   "  2^(2+1)-1 =  8-1 =  7
6 -> 11.   ->     2,   "  2^(2+1)-1 =  8-1 =  7
7 -> 111   ->     3,   "  2^(3+1)-1 = 16-1 = 15
8 -> 1...  ->     1,   "  2^(1+1)-1 =  4-1 =  3
9 -> 1..1  ->     2,   "  2^(2+1)-1 =  8-1 =  7
10-> 1.1.  ->     2,   "  2^(2+1)-1 =  8-1 =  7
(End)

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

Rows of triangle in A038573 converge to this sequence. - N. J. A. Sloane, Jun 05 2009

Cf. A000120, A038573, A047999, A159912, A117973, A001316, A147582, A160121.

Table[2^(DigitCount[n, 2][[1]] + 1) - 1, {n, 0, 78}] (* or *)
Table[2^(Total@ IntegerDigits[n, 2] + 1) - 1, {n, 0, 78}] (* Michael De Vlieger, Jan 25 2016 *)

PARI
```
A159913(n)=2<
				
```

def A159913(n): return (1<Chai Wah Wu, Nov 15 2022

a(n) = 2^A000120(2n+1) - 1 = A038573(2n+1) = 2*A038573(n) + 1 = A159912(n+1) - A159912(n).
a(n) = A160019(n,n). - Philippe Deléham, Nov 15 2011
a(n) = n - Sum_{k=0..n} (-1)^binomial(n, k). - Peter Luschny, Jan 14 2018

A331856 a(n) is the least value obtained by partitioning the binary representation of n into consecutive blocks, and then reversing those blocks.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 5, 7, 3, 7, 7, 15, 1, 3, 5, 7, 5, 11, 11, 15, 3, 7, 11, 15, 7, 15, 15, 31, 1, 3, 5, 7, 9, 11, 11, 15, 5, 11, 21, 23, 13, 23, 23, 31, 3, 7, 13, 15, 11, 23, 27, 31, 7, 15, 23, 31, 15, 31, 31, 63, 1, 3, 5, 7, 9, 11, 11, 15, 9, 19, 21

Offset: 0

Rémy Sigrist, Jan 29 2020

			For n = 6:
- the binary representation of 6 is "110",
- we can split it in 4 ways:
      "110" -> "011" -> 3
      "1" and "10" -> "1" and "01" -> 5
      "11" and "0" -> "11" and "0" -> 6
      "1" and "1" and "0" -> "1" and "1" and "0" -> 6
- we have 3 distinct values, the least being 3,
- hence a(6) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Rémy Sigrist, PARI program for A331856
Index entries for sequences related to binary expansion of n

See A331855 for the number of distinct values, and A331857 for the greatest value.

Cf. A000225, A023416, A038573.

PARI
```
See Links section.
```

a(n)^A023416(n) = A038573(n) (where a^k denotes the k-th iterate of a).

a(n) <= n with equality iff n belongs to A000225.

A038585 Write n in binary, delete 0's.

Original entry on oeis.org

1, 1, 11, 1, 11, 11, 111, 1, 11, 11, 111, 11, 111, 111, 1111, 1, 11, 11, 111, 11, 111, 111, 1111, 11, 111, 111, 1111, 111, 1111, 1111, 11111, 1, 11, 11, 111, 11, 111, 111, 1111, 11, 111, 111, 1111, 111, 1111, 1111, 11111, 11, 111, 111, 1111, 111, 1111, 1111

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to binary expansion of n

Cf. A000120, A002275, A038573.

a038585 0 = 0
a038585 n = (9 * m + 1) * (a038585 n') + m where (n', m) = divMod n 2
-- Reinhard Zumkeller, Oct 10 2012

Table[FromDigits[DeleteCases[IntegerDigits[n,2],?(#==0&)]],{n,60}] (* _Harvey P. Dale, Feb 27 2012 *)
FromDigits/@(IntegerDigits[Range[60],2]/.(0->Nothing)) (* Harvey P. Dale, Apr 09 2017 *)

def A038585(n):
    return int(bin(n)[2:].replace('0','')) # Chai Wah Wu, May 19 2020

a(n) = (9 * n mod 2 + 1) * a(floor(n/2)) + n mod 2. - Reinhard Zumkeller, Oct 10 2012

a(n) = A002275(A000120(n)). - Chai Wah Wu, May 19 2020

More terms from Erich Friedman

A225985 List the positive numbers, remove even digits (including zeros) from each term; sequence = remaining terms.

Original entry on oeis.org

1, 3, 5, 7, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, 3, 5, 7, 9, 3, 31, 3, 33, 3, 35, 3, 37, 3, 39, 1, 3, 5, 7, 9, 5, 51, 5, 53, 5, 55, 5, 57, 5, 59, 1, 3, 5, 7, 9, 7, 71, 7, 73, 7, 75, 7, 77, 7, 79, 1, 3, 5, 7, 9, 9, 91, 9, 93, 9, 95, 9, 97, 9, 99, 1, 11, 1, 13

Offset: 1

Dave Durgin, May 22 2013

			The natural numbers with at least one odd digit in their decimal representation are: 1, 3, 5, 7, 9, 10, 11, 12, 13, ...
By excluding their even digits, we obtain: 1, 3, 5, 7, 9, 1, 11, 1, 13, ...
Hence: a(1)=1, a(2)=3, a(3)=5, a(4)=7, a(5)=9, a(6)=1, a(7)=11, a(8)=1, a(9)=13, .... [Example corrected by _Paul Tek_, May 24 2013]

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A038573.

Cf. A014261 (duplicates removed), A226091.

a225985 n = a225985_list !! (n-1)
a225985_list = map read $ filter (not . null) $
    map (filter (`elem` "13579") . show) [0..] :: [Integer]
-- Reinhard Zumkeller, May 26 2013

FromDigits[DeleteCases[IntegerDigits[#],?EvenQ]]&/@Range[200]/. (0-> Nothing) (* _Harvey P. Dale, Apr 04 2017 *)

from itertools import count, islice
def agen(): # generator of terms
    for n in count(1):
        removed  = "".join(d if d in "13579" else "" for d in str(n))
        if removed != "": yield int(removed)
print(list(islice(agen(), 80))) # Michael S. Branicky, Aug 05 2022

a(A226091(n)) = A014261(n). - Reinhard Zumkeller, May 26 2013

Definition clarified by N. J. A. Sloane, Aug 06 2022 at the suggestion of Michel Marcus

A115378 a(n) = number of positive integers k < n such that n XOR k = (n+k).

Original entry on oeis.org

0, 1, 0, 3, 1, 1, 0, 7, 3, 3, 1, 3, 1, 1, 0, 15, 7, 7, 3, 7, 3, 3, 1, 7, 3, 3, 1, 3, 1, 1, 0, 31, 15, 15, 7, 15, 7, 7, 3, 15, 7, 7, 3, 7, 3, 3, 1, 15, 7, 7, 3, 7, 3, 3, 1, 7, 3, 3, 1, 3, 1, 1, 0, 63, 31, 31, 15, 31, 15, 15, 7, 31, 15, 15, 7, 15, 7, 7, 3, 31, 15, 15, 7, 15, 7, 7, 3, 15, 7, 7, 3, 7, 3

Offset: 1

Paul D. Hanna, Jan 21 2006

nonn

The number of positive integers k < n such that n XOR k = (n-k) is A038573(n).

Cf. A080791, A080100, A038573.

a(n)=sum(k=1,n,if(bitxor(n,k)==(n+k),1,0))

a(n) = -1 + 2^(number of 0's in binary expansion of n). a(n) = 2^A080791(n) - 1 = A080100(n) - 1.

cp's OEIS Frontend

A256263 Triangle read by rows: T(j,k) = 2*k-1 if k is a power of 2, otherwise, between positions that are powers of 2 we have the initial terms of A016969, with j>=0, 1<=k<=A011782(j) and T(0,1) = 0.

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A160721 First differences of A160720.

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A053581 First differences of the Poly-Bernoulli numbers B_n^(k) with k=-2 (A027649).

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Sage

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A340666 A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

Mathematica

Formula

A090971 Sierpiński's triangle, read by rows, starting from 1: T(n,k) = (T(n-1,k) + T(n-1,k-1)) mod 2.

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Mathematica

PARI

Formula

A159913 a(n) = 2^(A000120(n) + 1) - 1, where A000120(n) = number of nonzero bits in n.

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Mathematica

PARI

Python

Formula

A331856 a(n) is the least value obtained by partitioning the binary representation of n into consecutive blocks, and then reversing those blocks.

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