A214417 -id:A214417 - cp's OEIS frontend

A214414 Fixed points of permutations A105027 and A214417.

0, 1, 5, 7, 18, 22, 26, 30, 1031, 1039, 1047, 1055, 1063, 1071, 1079, 1087, 1095, 1103, 1111, 1119, 1127, 1135, 1143, 1151, 1159, 1167, 1175, 1183, 1191, 1199, 1207, 1215, 1223, 1231, 1239, 1247, 1255, 1263, 1271, 1279, 1287, 1295, 1303, 1311, 1319, 1327

Offset: 1

Reinhard Zumkeller, Jul 21 2012

nonn

A105027(a(n)) = A214417(a(n)) = a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A105271, A214433, A104235.

a214414 n = a214414_list !! (n-1)
a214414_list = [x | x <- [0..], a105027 x == x]

A105027 Write numbers in binary under each other; to get the next block of 2^k (k >= 0) terms of the sequence, start at 2^k, read diagonals in upward direction and convert to decimal.

Original entry on oeis.org

0, 1, 3, 2, 6, 5, 4, 7, 15, 10, 9, 8, 11, 14, 13, 12, 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30, 29, 61, 44, 39, 34, 33, 32, 35, 38, 37, 36, 47, 42, 41, 40, 43, 46, 45, 60, 55, 50, 49, 48, 51, 54, 53, 52, 63, 58, 57, 56, 59, 62, 126, 93, 76, 71, 66, 65, 64, 67, 70

Offset: 0

N. J. A. Sloane, Apr 03 2005

This is a permutation of the nonnegative integers.
Structure: blocks of size 2^k - 1 taken from A102370, interspersed with terms of A102371. - Philippe Deléham, Nov 17 2007
a(A062289(n)) = A102370(n) for n > 0; a(A000225(n)) = A102371(n); a(A214433(n)) = A105025(a(n)). - Reinhard Zumkeller, Jul 21 2012

			        0
        1
       10
       11
   -> 100  Starting here, the upward diagonals
      101  read 110, 101, 100, 111, giving the block 6, 5, 4, 7.
      110
      111
     1000
     1001
     1010
     1011
      ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
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.
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to binary expansion of n

Cf. A102370, A105025, A105026, A105028.
Cf. A070939, A000079.
Cf. A214414 (fixed points), A214417 (inverse).

import Data.Bits ((.|.), (.&.))
a105027 n = foldl (.|.) 0 $ zipWith (.&.)
                  a000079_list $ enumFromTo (n + 1 - a070939 n) n
-- Reinhard Zumkeller, Jul 21 2012

block[k_] := Module[{t}, t = Table[PadLeft[IntegerDigits[n, 2], k+1], {n, 2^(k-1), 2^(k+1)-1}]; Table[FromDigits[Table[t[[n-m+1, m]], {m, 1, k+1}], 2], {n,2^(k-1)+1, 2^(k-1)+2^k}]]; block[0] = {0, 1}; Table[block[k], {k, 0, 6}] // Flatten (* Jean-François Alcover, Jun 30 2015 *)

apply( {A105027(n,L=exponent(n+!n))=sum(k=0,L,bitand(n+k-L,2^k))}, [0..55]) \\ M. F. Hasler, Apr 18 2022

a(2^n - 1) = A102371(n) for n > 0. - Philippe Deléham, May 10 2005

More terms from John W. Layman, Apr 07 2005

A105271 Fixed points of the permutation of the nonnegative integers defined by A105025 (i.e., n such that A105025(n) = n).

Original entry on oeis.org

0, 1, 4, 6, 17, 21, 25, 29, 1024, 1032, 1040, 1048, 1056, 1064, 1072, 1080, 1088, 1096, 1104, 1112, 1120, 1128, 1136, 1144, 1152, 1160, 1168, 1176, 1184, 1192, 1200, 1208, 1216, 1224, 1232, 1240, 1248, 1256, 1264, 1272, 1280, 1288, 1296, 1304, 1312, 1320

Offset: 1

Emeric Deutsch, Apr 16 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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. A105025.

Cf. A214417, A214414, A214433, A104235.

a105271 n = a105271_list !! (n-1)
a105271_list = [x | x <- [0..], a105025 x == x]
-- Reinhard Zumkeller, Jul 21 2012

More terms from John W. Layman, Jun 03 2005

Offset corrected by Reinhard Zumkeller, Jul 21 2012

A214416 Inverse permutation to A105025.

Original entry on oeis.org

0, 1, 3, 2, 4, 7, 6, 5, 15, 10, 9, 8, 11, 14, 13, 12, 22, 17, 16, 31, 18, 21, 20, 19, 30, 25, 24, 23, 26, 29, 28, 27, 37, 32, 63, 46, 33, 36, 35, 34, 45, 40, 39, 38, 41, 44, 43, 42, 53, 48, 47, 62, 49, 52, 51, 50, 61, 56, 55, 54, 57, 60, 59, 58, 68, 127, 94

Offset: 0

Reinhard Zumkeller, Jul 21 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A214417, A105271 (fixed points).

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a214416 = fromJust . (`elemIndex` a105025_list)

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A214414 Fixed points of permutations A105027 and A214417.

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A105027 Write numbers in binary under each other; to get the next block of 2^k (k >= 0) terms of the sequence, start at 2^k, read diagonals in upward direction and convert to decimal.

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A105271 Fixed points of the permutation of the nonnegative integers defined by A105025 (i.e., n such that A105025(n) = n).

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A214416 Inverse permutation to A105025.

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