A102370 -id:A102370 - cp's OEIS frontend

A159966 Lodumo_4 of A102370 (sloping binary numbers).

0, 3, 2, 1, 4, 7, 6, 5, 8, 11, 10, 9, 12, 15, 14, 13, 16, 19, 18, 17, 20, 23, 22, 21, 24, 27, 26, 25, 28, 31, 30, 29, 32, 35, 34, 33, 36, 39, 38, 37, 40, 43, 42, 41, 44, 47, 46, 45, 48, 51, 50, 49, 52, 55, 54, 53, 56, 59, 58, 57, 60, 63, 62, 61, 64, 67, 66, 65, 68, 71, 70, 69, 72

Offset: 0

Philippe Deléham, Apr 28 2009

A permutation of the nonnegative integers.
A092486 preceded by a zero. - Philippe Deléham, May 05 2009
Fixed points are the even numbers. - Wesley Ivan Hurt, Oct 16 2015

G. C. Greubel, Table of n, a(n) for n = 0..10000
OEIS wiki, Lodumo transform.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A002162, A005843, A102370, A132429, A158459, A166549.

[n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4) : n in [0..100]]; // Wesley Ivan Hurt, Oct 16 2015

A159966:=n->n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4): seq(A159966(n), n=0..100); # Wesley Ivan Hurt, Oct 16 2015

Table[n - (1 - (-1)^n) (-1)^((2 n + 1 - (-1)^n)/4), {n, 0, 40}] (* or *) CoefficientList[Series[(3 x - 4 x^2 + 3 x^3)/((x - 1)^2 (1 + x^2)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Oct 16 2015 *)
LinearRecurrence[{2,-2,2,-1},{0,3,2,1},80] (* Harvey P. Dale, Jul 02 2022 *)

concat(0, Vec((3*x-4*x^2+3*x^3)/((x-1)^2*(1+x^2)) + O(x^100))) \\ Altug Alkan, Oct 17 2015

a(n) = lod_4 (A102370(n)).
From Wesley Ivan Hurt, Oct 16 2015: (Start)
G.f.: (3*x-4*x^2+3*x^3)/((x-1)^2*(1+x^2)).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4), n>3.
a(n) = n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4).
a(2n) = A005843(n); a(2n+1) = A166549(n).
a(n+1) - a(n) = A132429(n)*(-1)^n. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Nov 28 2023

A159970 Lodumo_5 of A102370 (sloping binary numbers) .

Original entry on oeis.org

0, 3, 1, 5, 4, 10, 15, 9, 8, 6, 14, 13, 18, 23, 28, 2, 11, 19, 7, 16, 20, 21, 26, 25, 24, 12, 30, 31, 29, 34, 39, 33, 17, 35, 38, 22, 36, 27, 32, 41, 40, 43, 46, 45, 50, 55, 60, 44, 48, 51, 49, 53, 37, 58, 63, 42, 56, 54, 61, 68, 66, 71, 76, 65, 59, 47, 70, 64

Offset: 0

Philippe Deléham, Apr 28 2009

Permutation of nonnegative integers .

The Lodumo_m transformation of a sequence s(n) is defined by a(n) = smallest nonnegative integer not yet in the list a(.) such that a(n)=s(n) (mod m). [From R. J. Mathar, Apr 30 2009]

Cf. A102370, A159059.

Corrected by R. J. Mathar, Apr 30 2009

A103543 Consider those values of k for which A102370(k) = k: 0, 4, 8, 16, 20, 24, 32, 36, 40, 48, 52, 56, 64, ... and divide by 4: 0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, ...; sequence gives missing numbers.

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 62, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 123, 126, 127, 131, 135, 139, 143, 147, 151, 155, 159, 163, 167, 171, 175, 179, 183, 187, 190, 191, 195, 199, 203, 207, 211, 215, 219, 223

Offset: 1

N. J. A. Sloane, Mar 23 2005

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

Cf. A102370, A103584.

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s]; Complement[ Range[225], Select[ Range[900], f[ # ] == 0 &]/4] (* Robert G. Wilson v, Mar 23 2005 *)

Numbers of the form 4k+3 together with the terms of A103584.
It is shown in the reference that A102370(k) = k iff n == 0 (mod 4) and n does not belong to any of the arithmetic progressions Q_r := {2^(4r)*j - 4r, j >= 1} for r = 1, 2, 3, ...
In other words, the sequence consists of the numbers of the form j*2^(4k-2) - k for k >=2 and j >= 1.

More terms from Robert G. Wilson v, Mar 23 2005

A103863 Hamming distance between n and A102370(n) (in binary).

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Mar 31 2005

The Hamming distance between two strings of the same length is the number of places where they differ. - Robert G. Wilson v, Apr 12 2005

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, p. 8.

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Saleem Bhatti, Channel coding; Hamming distance.
Alexander Bogomolny, Distance Between Strings.
National Institute of Standards and Technology, Hamming distance.

Cf. A102370, A103542, A104235.

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[ Log[2, n + 1] + 2]]}, While[ k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s]; hammingdistance[n_] := Count[ IntegerDigits[ BitXor[n, f[n] + n], 2], 1]; Table[ hammingdistance[n], {n, 0, 104}] (* Robert G. Wilson v, Apr 12 2005 *)

a(A104235(n)) = 0.

More terms from Robert G. Wilson v, Apr 12 2005

A103202 A102370 sorted.

Original entry on oeis.org

0, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88

Offset: 1

Philippe Deléham, Mar 27 2005

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370.

Numbers missing from A102371.

A103585 Consider numbers k such that (A102370(k)-k)/2 = 1; read them mod 4 to get the sequence.

Original entry on oeis.org

1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3

Offset: 1

Benoit Cloitre and Philippe Deléham, Mar 24 2005

Is there a self-contained construction of this two-valued sequence?

Sequence appears to have period 43. - Ralf Stephan, May 18 2007

			The numbers k are 1, 3, 7, 9, 11, 15, 17, 19, ...

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps], J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370, A103587.

A103615 Number of zeros in A103542(n) (binary equivalent of A102370(n)).

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Mar 31 2005

			The sequence has a natural decomposition into blocks (see the paper): 1; 0; 1, 1, 2; 0, 2, 2, 3, 1, 1, 1; 2, 1, 3, 3, 4, 2, 2, 2, 3, 0, 2, 2, 3, 1, 1; 1, 3, ...

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

Cf. A102370, A103542, A089400.

A023416 := proc(n) local digs : digs := convert(n,base,2) : if nops(digs) = 0 then 1: else add(1-j,j=digs) : fi : end: ili := readline("b102370.txt") : while ili <> 0 do na := sscanf(ili,"%d %d") : na := A023416(op(2,na)) ; printf("%d, ",na) ; ili := readline("b102370.txt") : od: # R. J. Mathar, Aug 10 2007

a(n) = A023416(A102370(n)). - R. J. Mathar, Aug 10 2007

More terms from R. J. Mathar, Aug 10 2007

A103813 Partial sums of A102370.

Original entry on oeis.org

0, 3, 9, 14, 18, 33, 43, 52, 60, 71, 85, 98, 126, 149, 167, 184, 200, 219, 241, 262, 282, 313, 339, 364, 388, 415, 445, 506, 550, 589, 623, 656, 688, 723, 761, 798, 834, 881, 923, 964, 1004, 1047, 1093, 1138, 1198, 1253, 1303, 1352, 1400, 1451, 1505, 1558, 1610, 1673

Offset: 0

N. J. A. Sloane and David Applegate, Apr 01 2005

Hardy and Wright, Sect. 18.2, for definition of average order.

David Applegate, Benoit Cloitre, Philippe Deléham, and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham, and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370.

Accumulate[f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s]; Table[ f[n] + n, {n, 0, 53}] ] (* James C. McMahon, Jan 21 2024 *)

The average order of A102370(n) is n + O(log n).

A104378 First differences of A102370.

Original entry on oeis.org

3, 3, -1, -1, 11, -5, -1, -1, 3, 3, -1, 15, -5, -5, -1, -1, 3, 3, -1, -1, 11, -5, -1, -1, 3, 3, 31, -17, -5, -5, -1, -1, 3, 3, -1, -1, 11, -5, -1, -1, 3, 3, -1, 15, -5, -5, -1, -1, 3, 3, -1, -1, 11, -5, -1, -1, 3, 67, -33, -17, -5, -5, -1, -1, 3, 3, -1, -1, 11, -5, -1, -1, 3, 3, -1, 15, -5, -5, -1, -1, 3, 3, -1, -1, 11, -5, -1, -1, 3, 3, 31, -17, -5, -5, -1, -1, 3, 3

Offset: 0

N. J. A. Sloane, Apr 18 2005

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370.

A105023 a(n) = A102370(n) - n. Or, 2*A103185(n).

Original entry on oeis.org

0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4, 2, 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4, 34, 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4, 2, 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 68, 34, 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4, 2, 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4, 34, 16, 10, 4

Offset: 0

N. J. A. Sloane, Apr 03 2005

When written in base 2 as a right justified table, columns have periods 1, 2, 4, 8, ... - Philippe Deléham, Apr 21 2005

			Has a natural decomposition into blocks: 0; 2; 4, 2, 0; 10, 4, 2, 0, 2, 4, 2; 16, 10, 4, 2, 0, 2, 4, 2, 0, 10, 4, 2, 0, 2, 4; 34, 16, 10, 4, ... where the leading term in each block is given by A105024.

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370, A103185, A105024.

s:= proc (n) local t1, l; t1 := 0; for l to n do if `mod`(n+l,2^l) = 0 then t1 := t1+2^l end if end do; t1 end proc;

a(n) = Sum_{ k >= 1 such that n + k == 0 mod 2^k } 2^k.

cp's OEIS Frontend

A159966 Lodumo_4 of A102370 (sloping binary numbers).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A159970 Lodumo_5 of A102370 (sloping binary numbers) .

Views

Author

Keywords

Comments

Crossrefs

Extensions

A103543 Consider those values of k for which A102370(k) = k: 0, 4, 8, 16, 20, 24, 32, 36, 40, 48, 52, 56, 64, ... and divide by 4: 0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, ...; sequence gives missing numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A103863 Hamming distance between n and A102370(n) (in binary).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A103202 A102370 sorted.

Views

Author

Keywords

Links

Crossrefs

A103585 Consider numbers k such that (A102370(k)-k)/2 = 1; read them mod 4 to get the sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A103615 Number of zeros in A103542(n) (binary equivalent of A102370(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A103813 Partial sums of A102370.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

A104378 First differences of A102370.

Views