A048729 -id:A048729 - cp's OEIS frontend

A048725 a(n) = Xmult(n,5) or rule90(n,1).

0, 5, 10, 15, 20, 17, 30, 27, 40, 45, 34, 39, 60, 57, 54, 51, 80, 85, 90, 95, 68, 65, 78, 75, 120, 125, 114, 119, 108, 105, 102, 99, 160, 165, 170, 175, 180, 177, 190, 187, 136, 141, 130, 135, 156, 153, 150, 147, 240, 245, 250, 255, 228, 225, 238, 235, 216, 221, 210

Offset: 0

Antti Karttunen, Apr 26 1999

The orbit of 1 under iteration of this function is the Sierpinski gasket A038183. It is called "rule 90" because the 8 bits of 90 = 01011010 in binary give bit k of the result as function of the value in {0,...,7} made out of bits k,k+1,k+2 of the input (i.e., floor(input / 2^k) mod 8). - M. F. Hasler, Oct 09 2017

			   n (in binary) | 4n [binary] | n XOR 4n [binary] | [decimal] = a(n)
          0      |        0    |           0       |        0
          1      |      100    |         101       |        5
         10      |     1000    |        1010       |       10
         11      |     1100    |        1111       |       15
        100      |    10000    |       10100       |       20
        101      |    10100    |       10001       |       17
   etc.

Alois P. Heinz, Table of n, a(n) for n = 0..16383

Cf. A048720, A048705, A048710, A048724, A048727, A048729.
Cf. A038183.
Cf. A353167 (terms sorted).

a:= n-> Bits[Xor](n*4, n):
seq(a(n), n=0..120);  # Alois P. Heinz, Aug 24 2019

Table[ BitXor[4n, n], {n, 0, 60}] (* Robert G. Wilson v, Jul 06 2006 *)

a(n)=bitxor(n,4*n) \\ Charles R Greathouse IV, Oct 03 2016

def A048725(n): return n^ n<<2 # Chai Wah Wu, Jun 29 2022

a(n) = n XOR n*2 XOR (n XOR n*2)*2 = A048724(A048724(n)). - Reinhard Zumkeller, Nov 12 2004

a(n) = n XOR (4n). - M. F. Hasler, Oct 09 2017

A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 17, 18, 19, 24, 25, 32, 33, 34, 35, 36, 38, 48, 49, 50, 51, 64, 65, 66, 67, 68, 70, 72, 73, 76, 96, 97, 98, 99, 100, 102, 128, 129, 130, 131, 132, 134, 136, 137, 140, 144, 145, 146, 147, 152, 153, 192, 193, 194, 195, 196, 198, 200, 201

Offset: 1

Antti Karttunen, Mar 30 1999

If bit i is 1, then bits i+-2 must be 0. All terms satisfy A048725(n) = 5*n.
It appears that n is in the sequence if and only if C(5n,n) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003
Yes, as remarked in A048715, "This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p." - Jason Kimberley, Dec 21 2011
A116361(a(n)) <= 2. - Reinhard Zumkeller, Feb 04 2006

Superset of A048715 and A048719. Union of A004742 and A003726.

Cf. A048729, A003714, A115845, A115847, A116360.

Reap[Do[If[OddQ[Binomial[5n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Second program: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, MatchQ[bb, {0}|{1}|{1, 1}|{_, 0, , 1, __}|{_ 1, , 0, __}] && !MatchQ[bb, {_, 1, , 1, __}]];
Select[Range[0, 201], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

is(n)=!bitand(n,n>>2) \\ Charles R Greathouse IV, Oct 03 2016

list(lim)=my(v=List(),n,t); while(n<=lim, t=bitand(n,n>>2); if(t, n+=1<Charles R Greathouse IV, Oct 22 2021

A061858 Differences between the ordinary multiplication table A004247 and the carryless multiplication table for GF(2)[X] polynomials A048720, i.e., the effect of the carry bits in binary multiplication.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 0, 0, 0, 0, 12, 0, 8, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 24, 0, 0, 0

Offset: 0

Antti Karttunen, May 11 2001

			From _Peter Munn_, Jan 28 2021: (Start)
The top left 12 X 12 corner of the table:
      |  0   1   2   3   4   5   6   7   8   9  10  11
------+------------------------------------------------
   0  |  0   0   0   0   0   0   0   0   0   0   0   0
   1  |  0   0   0   0   0   0   0   0   0   0   0   0
   2  |  0   0   0   0   0   0   0   0   0   0   0   0
   3  |  0   0   0   4   0   0   8  12   0   0   0   4
   4  |  0   0   0   0   0   0   0   0   0   0   0   0
   5  |  0   0   0   0   0   8   0   8   0   0  16  16
   6  |  0   0   0   8   0   0  16  24   0   0   0   8
   7  |  0   0   0  12   0   8  24  28   0   0  16  28
   8  |  0   0   0   0   0   0   0   0   0   0   0   0
   9  |  0   0   0   0   0   0   0   0   0  16   0  16
  10  |  0   0   0   0   0  16   0  16   0   0  32  32
  11  |  0   0   0   4   0  16   8  28   0  16  32  52
(End)

"Zoomed in" variant: A061859.
Rows/columns 3, 5 and 7 are given by A048728, A048729, A048730.
Main diagonal divided by 4: A213673.
Numbers that generate no carries when multiplied in binary by 11_2: A003714, by 101_2: A048716, by 1001_2: A115845, by 10001_2: A115847, by 100001_2: A114086.
Other sequences related to the presence/absence of a carry in binary multiplication: A116361, A235034, A235040, A236378, A266195, A289726.

a(n) = A004247(n) - A048720(n).

cp's OEIS Frontend

A048725 a(n) = Xmult(n,5) or rule90(n,1).

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A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).

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A061858 Differences between the ordinary multiplication table A004247 and the carryless multiplication table for GF(2)[X] polynomials A048720, i.e., the effect of the carry bits in binary multiplication.

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cp's OEIS Frontend

A048725 a(n) = Xmult(n,5) or rule90(n,1).

Views

Author

Keywords

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Examples

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Programs

Maple

Mathematica

PARI

Python

Formula

A048716 Numbers n such that binary expansion matches ((0)*00(1?)1)*(0*).

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Author

Keywords

Comments

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Programs

Mathematica

PARI

PARI

A061858 Differences between the ordinary multiplication table A004247 and the carryless multiplication table for GF(2)[X] polynomials A048720, i.e., the effect of the carry bits in binary multiplication.

Views

Author

Keywords

Examples

Crossrefs

Formula

A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).