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 1-8 of 8 results.

A269168 Rule 30 binary tree permutation: a(1) = 1, a(2n) = A269160(a(n)), a(2n+1) = A269164(1+a(n)).

Original entry on oeis.org

1, 7, 2, 25, 9, 14, 3, 111, 33, 63, 11, 50, 18, 13, 4, 401, 143, 231, 41, 193, 79, 53, 15, 222, 66, 126, 22, 51, 17, 28, 5, 1783, 529, 945, 175, 825, 295, 223, 55, 839, 257, 497, 95, 203, 69, 49, 19, 802, 286, 462, 82, 386, 158, 106, 30, 221, 67, 119, 21, 100, 36, 27, 6, 6409, 2295, 3703, 657, 3159, 1201, 849, 233
Offset: 1

Views

Author

Antti Karttunen, Feb 21 2016

Keywords

Comments

This sequence can be represented as a binary tree. Each left hand child is produced as A269160(n), and each right hand child as A269164(1+n), when the parent node contains n:
|
...................1...................
7 2
25......../ \........9 14......../ \........3
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
111 33 63 11 50 18 13 4
401 143 231 41 193 79 53 15 222 66 126 22 51 17 28 5
etc.
Each maximal leftward branch (e.g. 1, 7, 25, ... (= A110240) or 9, 63, 193, ... or 2, 14, 50, ...) gives a trajectory of Rule 30 cellular automaton starting from a particular "seed configuration" which are given in A269164.

Links

Crossrefs

Inverse: A269167.
Cf. A269160, A269164.
Cf. A110240 (the left edge).

Programs

  • Mathematica
    nmax = (* sequence length *) 100; terms (* from A269164 *) = 2000; Clear[a, f]; A269160[n_] := BitXor[n, BitOr[2 n, 4 n]]; f[max_] := f[max] = (s = Sort[Table[A269160[n], {n, 0, max}]]; Complement[Range[Last[s]], s][[1 ;; terms]]); f[terms]; f[max = 2 terms]; While[f[max] != f[max/2], max = 2 max]; A269164[n_Integer] := If[n > Length[f[max]], 0, f[max][[n]]]; a[1] = 1; a[n_] := a[n] = If[EvenQ[n], A269160[a[n/2]], A269164[1 + a[(n - 1)/2]]]; A269168 = Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Feb 23 2016 *)

Formula

a(1) = 1, after which, a(2n) = A269160(a(n)), a(2n+1) = A269164(1+a(n)).

A269167 Permutation of natural numbers: a(1) = 1, a(A269160(n)) = 2*a(n), a(A269164(n+1)) = 1+(2*a(n)).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 2, 127, 5, 255, 11, 511, 14, 6, 23, 1023, 29, 13, 47, 2047, 59, 27, 95, 4095, 4, 126, 62, 30, 119, 55, 191, 8191, 9, 253, 125, 61, 239, 111, 383, 16383, 19, 507, 251, 123, 479, 223, 767, 32767, 46, 12, 28, 1022, 22, 510, 39, 254, 1015, 503, 247, 959, 447, 1535, 10, 65535, 93, 25, 57, 2045, 45
Offset: 1

Views

Author

Antti Karttunen, Feb 21 2016

Keywords

Links

Crossrefs

Inverse: A269168.
Cf. A269160, A269162, A269163, A269164, A269169.

Programs

  • Mathematica
    terms = 100; A269160[n_] := BitXor[n, BitOr[2 n, 4 n]]; f[max_] := f[max] = (s = Sort[Table[A269160[n], {n, 0, max}]]; Complement[Range[Last[s]], s][[1 ;; terms]]); f[terms]; f[max = 2 terms]; While[f[max] != f[max/2], max = 2 max]; A269164[n_] := f[max][[n]]; a[1]=1; eq[n_] := a[A269160[n]] == 2*a[n] && a[A269164[n+1]] == 1 + 2*a[n]; A269167 = Array[a, terms-1] /. Solve[Array[eq, terms-1]] // First (* Jean-François Alcover, Feb 23 2016 *)

Formula

a(1) = 1, for n > 1, if A269162(n) > 0 [when n is in A269163], a(n) = 2*a(A269162(n)), otherwise [when n is in A269164], a(n) = 1 + 2*a(A269169(n)-1).

A269169 The least monotonic left inverse for A269164.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 21 2016

Keywords

Comments

a(n) = number of terms of A269164 <= n.

Links

Crossrefs

Cf. A269164, A269167.

Programs

  • Mathematica
    terms = 100; Clear[f]; f[max_] := f[max] = (s = Sort[Table[BitXor[n, BitOr[ 2 n, 4 n]], {n, 0, max}]]; Complement[Range[Last[s]], s][[1 ;; terms]]); f[terms]; f[max = 2 terms]; While[Print[max]; f[max] != f[max/2], max = 2 max]; A269164 = f[max]; a[n_] := Count[A269164, k_ /; k <= n]; Table[ a[n], {n, 1, Length[A269164]}] (* Jean-François Alcover, Feb 23 2016 *)

Formula

Other identities. For all n >= 1:
a(A269164(n)) = n.

A269160 Formula for Wolfram's Rule 30 cellular automaton: a(n) = n XOR (2n OR 4n).

Original entry on oeis.org

0, 7, 14, 13, 28, 27, 26, 25, 56, 63, 54, 53, 52, 51, 50, 49, 112, 119, 126, 125, 108, 107, 106, 105, 104, 111, 102, 101, 100, 99, 98, 97, 224, 231, 238, 237, 252, 251, 250, 249, 216, 223, 214, 213, 212, 211, 210, 209, 208, 215, 222, 221, 204, 203, 202, 201, 200, 207, 198, 197, 196, 195, 194, 193, 448, 455, 462
Offset: 0

Views

Author

Antti Karttunen, Feb 20 2016

Keywords

Comments

Take n, write it in binary, see what Rule 30 would do to that state, convert it to decimal: that is a(n). For example, we can see in A110240 that 7 = 111_2 becomes 25 = 11001_2 under Rule 30, which is shown here by a(7) = 25. - N. J. A. Sloane, Nov 25 2016
The sequence is injective: no value occurs more than once.
Fibbinary numbers (A003714) give all integers n>=0 for which a(n) = A048727(n) and for which a(n) = A269161(n).

Links

Crossrefs

Cf. A003714, A003986, A003987, A057889, A070939.
Cf. A110240 (iterates starting from 1).
Cf. A269162 (left inverse).
Cf. A269163 (same sequence sorted into ascending order).
Cf. A269164 (values missing from this sequence).
Cf. also A048727, A269161.

Programs

Formula

a(n) = n XOR (2n OR 4n) = A003987(n, A003986(2*n, 4*n)).
Other identities. For all n >= 0:
a(2*n) = 2*a(n).
a(n) = A057889(A269161(A057889(n))). [Rule 30 is the mirror image of rule 86.]
A269162(a(n)) = n.
For all n >= 1:
A070939(a(n)) - A070939(n) = 2. [The binary length of a(n) is two bits longer than that of n for all nonzero values.]
G.f.: (3*x + 2*x^2 +x^3)/(1 - x^4) + Sum_{k>=1}(2^(k + 1)*x^(2^(k - 1))/((1 + x^(2^(k + 1)))*(1 - x))). - Miles Wilson, Jan 24 2025

A269162 a(0) = 0, for n > 0, a(n) = the least (necessarily also unique) k such that A269160(k) = n, or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 6, 5, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 14, 13, 12, 11, 10, 0, 8, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 30, 29, 28, 27, 26, 0, 24, 23, 22, 21, 20, 0, 0, 25
Offset: 0

Views

Author

Antti Karttunen, Feb 20 2016

Keywords

Comments

If n > 0 and a(n) > 0 then a(n) is the unique finite predecessor of the configuration encoded in the binary representation of n (A007088) when Wolfram's Rule 30 cellular automaton is applied.

Links

Crossrefs

Cf. A110240, A269160, A269163, A269164 (indices of zeros), A269165, A269166.

Programs

  • Mathematica
    (* empirical *) a[n_] := Module[{k}, For[k = Floor[n/7], k <= Ceiling[n/3], k++, If[BitXor[k, BitOr[2k, 4k]] == n, Return[k]]]; 0]; Table[a[n], {n, 0, 16387}] (* Jean-François Alcover, Feb 23 2016 *)
  • Scheme
    (define (A269162 n) (let loop ((p 0)) (cond ((= n (A269160 p)) p) ((> p n) 0) (else (loop (+ 1 p)))))) ;; Very slow implementation.
(define (A269162 n) (if (zero? n) n (let ((nwid-2 (- (A000523 n) 2))) (let loop ((p (if (< n 4) 0 (A000079 nwid-2)))) (let ((k (A269160 p))) (cond ((= n k) p) ((> (A000523 p) nwid-2) 0) (else (loop (+ 1 p))))))))) ;; Somewhat optimized.

Formula

Other identities. For all n >= 0:
a(A269160(n)) = n. [This sequence works as a left inverse of A269160.]
a(A110240(n+1)) = A110240(n).

A269163 Numbers which have a finite predecessor in Wolfram's Rule 30 cellular automaton; range of A269160 sorted into ascending order.

Original entry on oeis.org

0, 7, 13, 14, 25, 26, 27, 28, 49, 50, 51, 52, 53, 54, 56, 63, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 111, 112, 119, 125, 126, 193, 194, 195, 196, 197, 198, 200, 201, 202, 203, 204, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 221, 222, 223, 224, 231, 237, 238, 249, 250, 251, 252, 385, 386, 387, 388
Offset: 0

Views

Author

Antti Karttunen, Feb 20 2016

Keywords

Comments

Numbers which have a finite predecessor in Wolfram's Rule 30 cellular automaton. The configuration of white and black cells is encoded in the binary representation (A007088) of each number.
The indexing starts from zero, because a(0) = 0 is a special case in this sequence. (Zero is the only number which is its own predecessor).

Links

Crossrefs

Complement: A269164.
Cf. A007088, A269162.

Programs

  • Mathematica
    terms = 100; Clear[f]; f[max_] := f[max] = Sort[Table[BitXor[n, BitOr[2n, 4n]], {n, 0, max}]][[1 ;; terms]]; f[terms]; f[max = 2 terms]; While[ Print[max]; f[max] != f[max/2], max = 2 max]; A269163 = f[max] (* Jean-François Alcover, Feb 23 2016 *)

A269166 If A269162(n) = 0, then a(n) = 0, otherwise a(n) = 1 + a(A269162(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0
Offset: 0

Views

Author

Antti Karttunen, Feb 21 2016

Keywords

Comments

a(n) gives the generational distance to the earliest finite ancestor when the binary expansion of n is interpreted as a pattern in Wolfram's Rule-30 cellular automaton or 0 if that pattern has no finite predecessors.
A110240 gives the record positions (after zero) and particularly, for n > 0, A110240(n) gives the first occurrence of n in this sequence.
See also comments in A269165.

Links

Crossrefs

Cf. A110240, A269160, A269162.
Cf. A269164 (the indices of zeros after the initial zero).
Cf. A269165 (the earliest finite ancestor for n).
Cf. also A268389.

Programs

  • Scheme
    ;; This implementation is based on given recurrence and utilitizes the memoization-macro definec:
(definec (A269166 n) (let ((p (A269162 n))) (if (zero? p) 0 (+ 1 (A269166 p)))))
;; This one computes the same with tail-recursive iteration:
(define (A269166 n) (let loop ((n n) (p (A269162 n)) (s 0)) (if (zero? p) s (loop p (A269162 p) (+ 1 s)))))

Formula

If A269162(n) = 0, then a(n) = 0, otherwise a(n) = 1 + a(A269162(n)).
Other identities. For all n >= 0:
a(A110240(n)) = n. [Works as a left inverse of sequence A110240.]

A269165 If A269162(n) = 0, then a(n) = n, otherwise a(n) = a(A269162(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 3, 2, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 1, 6, 5, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 15, 2, 3, 12, 11, 10, 55, 8, 57, 58, 59, 60, 61, 62, 9, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81
Offset: 0

Views

Author

Antti Karttunen, Feb 21 2016

Keywords

Comments

a(n) is the earliest finite ancestor pattern n in Rule-30 or n itself if n has no finite predecessors.
Starting from k = a(n) with any n and iterating map k -> A269160(k) exactly A269166(n) times yields n back.
Apart from zero no terms of A269163 occur so all terms after zero are in A269164. Each term of A269164 occurs an infinitely many times.

Links

Crossrefs

Cf. A269160, A269163, A269164, A269166 (for a distance in A269162-steps to the ancestor pattern).
Cf. A110240 (indices of ones in this sequence).
Cf. also A268669.

Programs

  • Scheme
    ;; This implementation is based on given recurrence and utilitizes the memoization-macro definec:
(definec (A269165 n) (let ((p (A269162 n))) (if (zero? p) n (A269165 p))))
;; This one computes the same with tail-recursive iteration:
(define (A269165 n) (let loop ((n n) (p (A269162 n))) (if (zero? p) n (loop p (A269162 p)))))

Formula

If A269162(n) = 0, then a(n) = n, otherwise a(n) = a(A269162(n)).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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