cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A030101 a(n) is the number produced when n is converted to binary digits, the binary digits are reversed and then converted back into a decimal number.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 13, 3, 11, 7, 15, 1, 17, 9, 25, 5, 21, 13, 29, 3, 19, 11, 27, 7, 23, 15, 31, 1, 33, 17, 49, 9, 41, 25, 57, 5, 37, 21, 53, 13, 45, 29, 61, 3, 35, 19, 51, 11, 43, 27, 59, 7, 39, 23, 55, 15, 47, 31, 63, 1, 65, 33, 97, 17, 81, 49, 113, 9, 73, 41, 105, 25, 89, 57
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

As with decimal reversal, initial zeros are ignored; otherwise, the reverse of 1 would be 1000000... ad infinitum.
Numerators of the binary van der Corput sequence. - Eric Rowland, Feb 12 2008
It seems that in most cases A030101(x) = A000265(x) and that if A030101(x) <> A000265(x), the next time A030101(y) = A000265(x), A030101(x) = A000265(y). Also, it seems that if a pair of values exist at one index, they will exist at any index where one of them exist. It also seems like the greater of the pair always shows up on A000265 first. - Dylan Hamilton, Aug 04 2010
The number of occasions A030101(n) = A000265(n) before n = 2^k is A053599(k) + 1. For n = 0..2^19, the sequences match less than 1% of the time. - Andrew Woods, May 19 2012
For n > 0: a(a(n)) = n if and only if n is odd; a(A006995(n)) = A006995(n). - Juli Mallett, Nov 11 2010, corrected: Reinhard Zumkeller, Oct 21 2011
n is binary palindromic if and only if a(n) = n. - Reinhard Zumkeller, corrected: Jan 17 2012, thanks to Hieronymus Fischer, who pointed this out; Oct 21 2011
Given any n > 1, the set of numbers A030109(i) = (A030101(i) - 1)/2 for indexes i ranging from 2^n to 2^(n + 1) - 1 is a permutation of the set of consecutive integers {0, 1, 2, ..., 2^n - 1}. This is important in the standard FFT algorithms (starting or ending bit-reversal permutation). - Stanislav Sykora, Mar 15 2012
Row n of A030308 gives the binary digits of a(n), prepended with zero at even positions. - Reinhard Zumkeller, Jun 17 2012
The binary van der Corput sequence is the infinite sequence of fractions { A030101(n)/A062383(n), n = 0, 1, 2, 3, ... }, and begins 0, 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, 9/16, 5/16, 13/16, 3/16, 11/16, 7/16, 15/16, 1/32, 17/32, 9/32, 25/32, 5/32, 21/32, 13/32, 29/32, 3/32, 19/32, 11/32, 27/32, 7/32, 23/32, 15/32, 31/32, 1/64, 33/64, 17/64, 49/64, ... - N. J. A. Sloane, Dec 01 2019
Record highs occur at n = A209492(m) (for n>=1) with values a(n) = A224195(m) (for n>=3). - Bill McEachen, Aug 02 2023

Examples

			a(100) = 19 because 100 (base 10) = 1100100 (base 2) and R(1100100 (base 2)) = 10011 (base 2) = 19 (base 10).

References

  • Hlawka E. The theory of uniform distribution. Academic Publishers, Berkhamsted, 1984. See pp. 93, 94 for the van der Corput sequence. - N. J. A. Sloane, Dec 01 2019

Links

Crossrefs

Cf. A030102 - A030109, A036044, A004086, A005408.
Cf. A055944 (reverse and add), A178225, A273258.
Cf. A056539, A057889 (bijective variants), A224195, A209492.

Programs

  • Haskell
    a030101 = f 0 where
   f y 0 = y
   f y x = f (2 * y + b) x'  where (x', b) = divMod x 2
-- Reinhard Zumkeller, Mar 18 2014, Oct 21 2011
  • J
    ([: #. [: |. #:)"0 NB. Stephen Makdisi, May 07 2018
  • Magma
    A030101:=func; // Jason Kimberley, Sep 19 2011
  • Maple
    A030101 := proc(n)
    convert(n,base,2) ;
    ListTools[Reverse](%) ;
    add(op(i,%)*2^(i-1),i=1..nops(%)) ;
end proc: # R. J. Mathar, Mar 10 2015
# second Maple program:
a:= proc(n) local m, r; m:=n; r:=0;
      while m>0 do r:=r*2+irem(m, 2, 'm') od; r
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Nov 17 2015
  • Mathematica
    Table[FromDigits[Reverse[IntegerDigits[i, 2]], 2], {i, 0, 80}]
bitRev[n_] := Switch[Mod[n, 4], 0, bitRev[n/2], 1, 2 bitRev[(n + 1)/2] - bitRev[(n - 1)/4], 2, bitRev[n/2], 3, 3 bitRev[(n - 1)/2] - 2 bitRev[(n - 3)/4]]; bitRev[0] = 0; bitRev[1] = 1; bitRev[3] = 3; Array[bitRev, 80, 0] (* Robert G. Wilson v, Mar 18 2014 *)
  • PARI
    a(n)=if(n<1,0,subst(Polrev(binary(n)),x,2))
  • PARI
    a(n) = fromdigits(Vecrev(binary(n)), 2); \\ Michel Marcus, Nov 10 2017
  • Python
    def a(n): return int(bin(n)[2:][::-1], 2) # Indranil Ghosh, Apr 24 2017
  • Sage
    def A030101(n): return Integer(bin(n).lstrip("0b")[::-1],2) if n!=0 else 0
[A030101(n) for n in (0..78)]  # Peter Luschny, Aug 09 2012
  • Scala
    (0 to 127).map(n => Integer.parseInt(Integer.toString(n, 2).reverse, 2)) // Alonso del Arte, Feb 11 2020

Formula

a(n) = 0, a(2n) = a(n), a(2n+1) = a(n) + 2^(floor(log_2(n)) + 1). For n > 0, a(n) = 2*A030109(n) - 1. - Ralf Stephan, Sep 15 2003
a(n) = b(n, 0) with b(n, r) = r if n = 0, otherwise b(floor(n/2), 2*r + n mod 2). - Reinhard Zumkeller, Mar 03 2010
a(1) = 1, a(3) = 3, a(2n) = a(n), a(4n+1) = 2a(2n+1) - a(n), a(4n+3) = 3a(2n+1) - 2a(n) (as in the Project Euler problem). To prove this, expand the recurrence into binary strings and reversals. - David Applegate, Mar 16 2014, following a posting to the Sequence Fans Mailing List by Martin Møller Skarbiniks Pedersen.
Conjecture: a(n) = 2*w(n) - 2*w(A053645(n)) - 1 for n > 0, where w = A264596. - Velin Yanev, Sep 12 2017

Extensions

Edits (including correction of an erroneous date pointed out by J. M. Bergot) by Jon E. Schoenfield, Mar 16 2014
Name clarified by Antti Karttunen, Nov 09 2017

A226454 The number of squares added in the n-th step of a Pythagoras tree of the 30-60-90 triangle.

Original entry on oeis.org

1, 2, 4, 8, 16, 30, 54, 98, 180, 332, 612, 1120, 2046, 3736
Offset: 1

Views

Author

Kival Ngaokrajang, Jun 07 2013

Keywords

Comments

Growth of the Pythagoras tree based on the triangle with internal angles of 30, 60 and 90 degrees.
The generating rule is expansion in sequential order on each stage; the smaller squares (opposite to the 30 deg angle) come first. The generating order labeled by "stage-number" starts as 1-1; 2-1, 2-2; 3-1, 3-2, 3-3, 3-4;...and so on. Overlap is prohibited, i.e., if any part of a new element in the next generating order cuts into any previous (existing, lower order) one, that new elements will be not be inserted/added: lower generating orders have precedence over higher generating orders.
The non-overlap rule limits the growth of the sequence to a(n+1) <= 2*a(n).
For Pythagoras tree based on isosceles right triangle with the same rule, the sequence will be A053599(n-1) + 1.

Links

Crossrefs

Cf. A053599.

A227298 The number of squares added in the n-th step of a Pythagoras tree of the 30-60-90 triangle, using the rule larger squares come first.

Original entry on oeis.org

1, 2, 4, 8, 16, 30, 56, 102, 186, 340, 624, 1148
Offset: 1

Views

Author

Kival Ngaokrajang, Jul 05 2013

Keywords

Comments

Growth of the Pythagoras tree based on the triangle with internal angles of 30, 60 and 90 degrees.
The generating rule is expansion in sequential order on each stage; the larger squares (opposite to the 60 deg angle) come first. The generating order labeled by "stage-number" starts as 1-1; 2-1, 2-2; 3-1, 3-2, 3-3, 3-4;...and so on. Overlap is prohibited, i.e., if any part of a new element in the next generating order cuts into any previous (existing, lower order) one, that new elements will be not be inserted/added: lower generating orders have precedence over higher generating orders.
The non-overlap rule limits the growth of the sequence to a(n+1) <= 2*a(n).
For Pythagoras tree based on isosceles right triangle with the same rule, the sequence will be A053599(n-1) + 1.

Links

Crossrefs

Cf. A053599, A226454.

A152202 Triangle read by rows, A000012 * A152201.

Original entry on oeis.org

1, 3, 6, 1, 10, 3, 15, 7, 1, 21, 13, 3, 28, 22, 8, 1, 36, 34, 16, 3, 45, 50, 30, 9, 1, 55, 70, 50, 19, 3, 66, 95, 80, 39, 10, 1, 78, 125, 120, 69, 22, 3, 91, 161, 175, 119, 49, 11, 1
Offset: 0

Views

Author

Gary W. Adamson, Nov 29 2008

Keywords

Comments

Row sums = A053599: (1, 3, 7, 13, 23, 37, 59,...).

Examples

			First few rows of the triangle =
1;
3;
6, 1;
10, 3;
15, 7, 1;
21, 13, 3;
28, 22, 8, 1;
36, 34, 16, 3;
45, 50, 30, 9, 1;
55, 70, 50, 19, 3;
66, 95, 80, 39, 10, 1;
78, 125, 120, 69, 22, 3;
91, 161, 175, 119, 49, 11, 1;
...

Crossrefs

A152201, A053599

Formula

Triangle read by rows, A000012 * A152201 = partial sums of A152201 column terms.
Showing 1-4 of 4 results.

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