A000523 -id:A000523 - cp's OEIS frontend

A329325 Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n with first digit removed.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 1, 4, 2, 1, 1, 2, 2, 3, 1, 4, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 1, 5, 3, 1, 1, 5, 4, 1, 5, 2, 1

Offset: 1

Gus Wiseman, Nov 11 2019

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).

			Triangle begins:
   1: ()        21: (22)       41: (23)       61: (1112)
   2: (1)       22: (31)       42: (221)      62: (11111)
   3: (1)       23: (4)        43: (5)        63: (11111)
   4: (11)      24: (1111)     44: (311)      64: (111111)
   5: (2)       25: (13)       45: (32)       65: (6)
   6: (11)      26: (121)      46: (41)       66: (51)
   7: (11)      27: (13)       47: (5)        67: (6)
   8: (111)     28: (1111)     48: (11111)    68: (411)
   9: (3)       29: (112)      49: (14)       69: (6)
  10: (21)      30: (1111)     50: (131)      70: (51)
  11: (3)       31: (1111)     51: (14)       71: (6)
  12: (111)     32: (11111)    52: (1211)     72: (3111)
  13: (12)      33: (5)        53: (122)      73: (33)
  14: (111)     34: (41)       54: (131)      74: (51)
  15: (111)     35: (5)        55: (14)       75: (6)
  16: (1111)    36: (311)      56: (11111)    76: (411)
  17: (4)       37: (5)        57: (113)      77: (6)
  18: (31)      38: (41)       58: (1121)     78: (51)
  19: (4)       39: (5)        59: (113)      79: (6)
  20: (211)     40: (2111)     60: (11111)    80: (21111)
For example, the trimmed binary expansion of 41 is (01001), with Lyndon factorization (01)(001), so row 41 is {2,3}.

Row lengths are A211097.
Row sums are A000523.
Keeping the first digit gives A329314.
Positions of singleton rows are A329327.
Binary Lyndon words are counted by A001037 and ranked by A102659.
Numbers whose reversed binary expansion is a Lyndon word are A328596.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A059966, A211097, A257250, A275692, A296372, A328594, A329313, A329315, A329318.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#1]]&]]]];
Table[Length/@lynfac[Rest[IntegerDigits[n,2]]],{n,100}]

A080541 In binary representation: keep the first digit and left-rotate the others.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 10, 12, 14, 9, 11, 13, 15, 16, 18, 20, 22, 24, 26, 28, 30, 17, 19, 21, 23, 25, 27, 29, 31, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80

Offset: 1

Reinhard Zumkeller, Feb 20 2003

Permutation of natural numbers: let r(n,0)=n, r(n,k)=a(r(n,k-1)) for k>0, then r(n,floor(log_2(n))) = n and for n>1: r(n,floor(log_2(n))-1) = A080542(n).

Discarding their most significant bit, binary representations of numbers present in each cycle of this permutation form a distinct equivalence class of binary necklaces, thus there are A000031(n) separate cycles in each range [2^n .. (2^(n+1))-1] (for n >= 0) of this permutation. A256999 gives the largest number present in n's cycle. - Antti Karttunen, May 16 2015

			a(20)=a('10100')='11000'=24; a(24)=a('11000')='10001'=17.

Inverse: A080542.
Cf. A000031, A003986, A053644, A053645, A000523, A007088, A079944, A080413, A080543, A054429, A256999, A257250.
The set of permutations {A059893, A080541, A080542} generates an infinite dihedral group.

f:= proc(n) local d;
   d:= ilog2(n);
   if n >= 3/2*2^d then 2*n+1-2^(d+1) else 2*n - 2^d fi
end proc:
map(f, [$1..100]); # Robert Israel, May 19 2015

A080541[n_] := FromDigits[Join[{First[#]}, RotateLeft[Rest[#]]], 2] & [IntegerDigits[n, 2]];
Array[A080541, 100] (* Paolo Xausa, May 13 2025 *)

def A080541(n): return ((n&(m:=1< 1 else n  # Chai Wah Wu, Jan 22 2023

maxlevel <- 6 # by choice
a <- 1:3
for(m in 1:maxlevel) for(k in 0:(2^(m-1)-1)){
a[2^(m+1)       + 2*k    ] = 2*a[2^m           + k]
a[2^(m+1)       + 2*k + 1] = 2*a[2^m + 2^(m-1) + k]
a[2^(m+1) + 2^m + 2*k    ] = 2*a[2^m           + k] + 1
a[2^(m+1) + 2^m + 2*k + 1] = 2*a[2^m + 2^(m-1) + k] + 1
}
a
# Yosu Yurramendi, Oct 12 2020

(define (A080541 n) (if (< n 2) n (A003986bi (A053644 n) (+ (* 2 (A053645 n)) (A079944off2 n))))) ;; A003986bi gives the bitwise OR of its two arguments. See A003986.
;; Where A079944off2 gives the second most significant bit of n. (Cf. A079944):
(define (A079944off2 n) (A000035 (floor->exact (/ n (A072376 n)))))
;; Antti Karttunen, May 16 2015

From Antti Karttunen, May 16 2015: (Start)
a(1) = 1; for n > 1, a(n) = A053644(n) bitwise_OR (2*A053645(n) + second_most_significant_bit_of(n)). [Here bitwise_OR is a 2-argument function given by array A003986 and second_most_significant_bit_of gives the second most significant bit (0 or 1) of n larger than 1. See A079944.]
Other identities. For all n >= 1:
a(n) = A059893(A080542(A059893(n))).
a(n) = A054429(a(A054429(n))).
(End)
A080542(a(n)) = a(A080542(n)) = n. [A080542 is the inverse permutation.]
From Robert Israel, May 19 2015: (Start)
Let d = floor(log[2](n)).  If n >= 3*2^(d-1) then a(n) = 2*n + 1 - 2^(d+1), otherwise a(n) = 2*n - 2^d.
G.f.: 2*x/(x-1)^2 + Sum_{n>=1} x^(2^n)+(2^n-1)*x^(3*2^(n-1)))/(x-1). (End)

A080542 In binary representation: keep the first digit and rotate right the others.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 12, 9, 13, 10, 14, 11, 15, 16, 24, 17, 25, 18, 26, 19, 27, 20, 28, 21, 29, 22, 30, 23, 31, 32, 48, 33, 49, 34, 50, 35, 51, 36, 52, 37, 53, 38, 54, 39, 55, 40, 56, 41, 57, 42, 58, 43, 59, 44, 60, 45, 61, 46, 62, 47, 63, 64, 96, 65, 97, 66, 98, 67, 99, 68

Offset: 1

Reinhard Zumkeller, Feb 20 2003

Permutation of natural numbers with inverse = A080541: A080541(a(n)) = a(A080541(n)) = n;
let r(n,0)=n, r(n,k)=a(r(n,k-1)) for k>0, then r(n,floor(log_2(n))) = n and for n>1: r(n,floor(log_2(n))-1) = A080541(n).
Discarding their most significant bit, binary representations of numbers present in each cycle of this permutation form a distinct equivalence class of binary necklaces, thus there are A000031(n) separate cycles in each range [2^n .. (2^(n+1))-1] (for n >= 0) of this permutation.  A256999 gives the largest number present in n's cycle. - Antti Karttunen, May 16 2015

			a(20) = a('10100') = '10010' = 18.
a(25) = a('11001') = '11100' = 28.

Inverse: A080541.
Cf. A000031, A000035, A000523, A004526, A007088, A053644, A053645, A054429, A072376, A080414, A080544, A256999, A257250.
The set of permutations {A059893, A080541, A080542} generates an infinite dihedral group.

kfd[n_]:=Module[{a,b},{a,b}=TakeDrop[IntegerDigits[n,2],1];FromDigits[ Join[a,RotateRight[b]],2]]; Array[kfd,80] (* The program uses the TakeDrop function from Mathematica version 10 *) (* Harvey P. Dale, Feb 12 2016 *)

def A080542(n): return (1+(n&1))*(1<>1) if n > 1 else n # Chai Wah Wu, Jan 22 2023

nmax <- 31 # by choice
a <- 1:3
for(n in 1:nmax) for(k in 0:3)
a[4*n + k] = 2*a[2*n + (k == 1 | k == 3)] + (k == 2 | k == 3)
a
# Yosu Yurramendi, Sep 05 2020

(define (A080542 n) (if (< n 2) n (+ (A053644 n) (+ (* (A000035 n) (A072376 n)) (A004526 (A053645 n))))))  ;; Antti Karttunen, May 16 2015

a(n) = 2^log2(n) + floor((n-2^log2(n))/2) + (n mod 2)*2^(log2(n)-1), where log2(n) is the integer part of base-2 logarithm.
From Antti Karttunen, May 16 2015: (Start)
a(1) = 1; for n > 1, a(n) = A053644(n) + (A000035(n)*A072376(n)) + A004526(A053645(n)). [Essentially the same formula but represented with A-numbers.]
Other identities. For all n >= 1:
a(n) = A059893(A080541(A059893(n))).
a(n) = A054429(a(A054429(n))).
(End)

A125985 Signature-permutation of Vaillé's 1997 bijection on 'bridges' (Dyck paths).

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 5, 6, 4, 22, 21, 18, 20, 17, 13, 12, 19, 15, 16, 10, 11, 14, 9, 64, 63, 59, 62, 58, 50, 49, 61, 55, 57, 46, 48, 54, 45, 36, 35, 32, 34, 31, 60, 56, 41, 52, 40, 47, 53, 43, 44, 27, 26, 33, 29, 30, 51, 38, 39, 42, 24, 25, 28, 37, 23, 196, 195, 190, 194, 189

Offset: 0

Antti Karttunen, Jan 02 2007

nonn

Vaillé shows in 1997 paper that this automorphism transforms a 'derivation' of a Dyck path to its 'compression', i.e., in OEIS terms, A125985(A126310(n)) = A126309(A125985(n)) holds for all n. He also proves that A057515(A125985(n)) = A126307(n) and A057514(A125985(n)) = A072643(n) - A057514(n) + 1 (the latter identity for all n >= 1).

A. Karttunen, Table of n, a(n) for n = 0..2055
J. Vaillé, Une Bijection Explicative de Plusieurs Propriétés Remarquables des Ponts, European J. Combin. 18 (1997), no. 1, 117-124.
Index entries for signature-permutations of Catalan automorphisms

Inverse: A125986. The number of cycles, maximum cycle sizes and LCM's of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A126291, A126292 and A126293. The fixed points are given by A126300/A126301.

(define (A125985 n) (A080300 (rising-list->binexp (A125985-aux2 (A014486 n)))))
(define (A125985-aux2 n) (let loop ((lists (A125985-aux1 n)) (z (list)) (m 1)) (if (null? lists) z (loop (cdr lists) (m-join z (car lists) m) (+ m 1)))))
(define (A125985-aux1 n) (if (zero? n) (list) (let ((begin_from (<< 1 (- (- (A000523 n) (A090996 n)) 1)))) (let loop ((s (A090996 n)) (t 0) (nth_list 1) (p begin_from) (b (if (= 0 (A004198bi n begin_from)) 0 1)) (lists (list (list)))) (cond ((< s 1) (cond ((< p 1) (reverse! lists)) (else (loop (- t (- 1 b)) b (+ 1 nth_list) (>> p 1) (if (= 0 (A004198bi n (>> p 1))) 0 1) (cons (list (+ b 1 nth_list)) lists))))) (else (loop (- s (- 1 b)) (+ t b) nth_list (>> p 1) (if (= 0 (A004198bi n (>> p 1))) 0 1) (cons (cons (+ b nth_list) (car lists)) (cdr lists)))))))))
(define (A125985-aux2 n) (let loop ((lists (A125985-aux1 n)) (z (list)) (m 1)) (if (null? lists) z (loop (cdr lists) (m-join z (car lists) m) (+ m 1)))))
(define (m-join a b m) (let loop ((a a) (b b) (c (list))) (cond ((and (not (pair? a)) (not (pair? b))) (reverse! c)) ((not (pair? a)) (loop a (cdr b) (cons (car b) c))) ((not (pair? b)) (loop (cdr a) b (cons (car a) c))) ((equal? (car a) (car b)) (loop (cdr a) (cdr b) (cons (car a) c))) ((> (car b) m) (loop a (cdr b) (cons (car b) c))) (else (loop (cdr a) b (cons (car a) c))))))
(define (rising-list->binexp rising-list) (let loop ((s 0) (i 0) (h 0) (fs rising-list)) (cond ((null? fs) (+ s (<< (- (<< 1 h) 1) i))) ((> (car fs) h) (loop s (+ i 1) (car fs) (cdr fs))) (else (loop (+ s (<< (- (<< 1 (+ 1 (- h (car fs)))) 1) i)) (+ i 2 (- h (car fs))) (car fs) (cdr fs))))))
(define (>> n i) (if (zero? i) n (>> (floor->exact (/ n 2)) (- i 1))))
(define (<< n i) (if (<= i 0) (>> n (- i)) (<< (+ n n) (- i 1))))

A126441 Tabular arrangement of the natural numbers: the row on which any nonzero term a(n) appears in is A053645(a(n))=A053645(n+1), and the column is A161511(a(n)). Table is presented by columns with 2^{k-1} items in column k, unused positions are filled with 0's.

Original entry on oeis.org

1, 2, 3, 4, 5, 0, 7, 8, 9, 6, 11, 0, 0, 0, 15, 16, 17, 10, 19, 0, 13, 0, 23, 0, 0, 0, 0, 0, 0, 0, 31, 32, 33, 18, 35, 12, 21, 14, 39, 0, 0, 0, 27, 0, 0, 0, 47, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, 64, 65, 34, 67, 20, 37, 22, 71, 0, 25, 0, 43, 0, 29, 0, 79, 0, 0, 0, 0, 0, 0, 0, 55, 0, 0

Offset: 0

Alford Arnold, Jan 19 2007

Note: 1 might be a more natural starting offset for this sequence, although the identities concerning A053645 and A161511 would have to be changed. - Antti Karttunen, Oct 12 2009.
This can be regarded as an arrangement of the partitions, indexed by position in A125106. The partitions in a given row all have the same remaining partition when the largest part is removed; specifically, the partition indexed by the row number in A125106 (with row 0 having the empty partition remaining).
The first value on row n is A004760(n+1). The second value on each row is A004760(n+1) plus A062383(n); subsequent values increase by ever enlarging powers of two. Or equivalently, each subsequent value on the row after the first nonzero value is given by A004754(previous value on the same row).
A055941(r) tells how many terms the row r (>= 0) has been shifted rightward from its "natural position", i.e. with how many zeros that row has been prepended.
The number of (nonzero) entries in column k is A000041(k).

			The largest power of 2 <= 6 is 4, 6 - 4 = 2, so 6 is in row 2. By A125106, 6 corresponds to the partition [2^2], total 4, so 6 goes in column 4. Thus T(2,4) = 6.
The table begins:
1.2.4..8.16.32.64.128.256.512.1024
..3.5..9.17.33.65.129.257.513.1025
.......6.10.18.34..66.130.258..514
....7.11.19.35.67.131.259.515.1027
............12.20..36..68.132..260
.........13.21.37..69.133.261..517
............14.22..38..70.134..262
......15.23.39.71.135.263.519.1031
...................24..40..72..136
...............25..41..73.137..265
...................26..42..74..138
............27.43..75.139.267..523
.......................28..44...76
...............29..45..77.141..269
...................30..46..78..142
.........31.47.79.143.271.527.1039
...........................48...80
.......................49..81..145
...........................50...82
...................51..83.147..275

A. Karttunen, Table of n, a(n) for n = 0..65534 (first 16 columns)

Cf. A125106, A053645, A000041, A004760, A062383, A000079 (column lengths).
A053645(a(A166274(n))) = A053645(1+A166274(n)) for all n>=1.
Positions of zeros: A166275, this sequence without zeros: A161924. A161920(n) gives the position of the first nonzero term on the row n-1.

columns = 7; row[n_] := n-2^Floor[Log2[n]]; col[0] = 0; col[n_] := If[EvenQ[n], col[n/2] + DigitCount[n/2, 2, 1], col[(n-1)/2]+1]; Clear[T]; T[, ] = 0; Do[T[row[k], col[k]] = k, {k, 1, 2^columns}]; Table[T[n-1, k], {k, 1, columns}, {n, 1, 2^(k-1)}] // Flatten (* Jean-François Alcover, Sep 09 2017 *)

Edited by Franklin T. Adams-Watters, Jan 23 2007

Further edited and Scheme-code added by Antti Karttunen, Oct 12 2009

A244108 Number T(n,k) of permutations of {1,2,...,n} that result in a binary search tree of height k; triangle T(n,k), k>=0, k<=n<=2^k-1, read by columns.

Original entry on oeis.org

1, 1, 2, 2, 4, 16, 40, 80, 80, 8, 64, 400, 2240, 11360, 55040, 253440, 1056000, 3801600, 10982400, 21964800, 21964800, 16, 208, 2048, 18816, 168768, 1508032, 13501312, 121362560, 1099169280, 10049994240, 92644597760, 857213660160, 7907423180800, 72155129446400

Offset: 0

Alois P. Heinz, Dec 21 2015

Empty external nodes are counted in determining the height of a search tree.

			Triangle T(n,k) begins:
: 1;
:    1;
:       2;
:       2,  4;
:          16,      8;
:          40,     64,      16;
:          80,    400,     208,      32;
:          80,   2240,    2048,     608,     64;
:               11360,   18816,    8352,   1664,   128;
:               55040,  168768,  104448,  30016,  4352,   256;
:              253440, 1508032, 1277568, 479040, 99200, 11008, 512;

Alois P. Heinz, Columns k = 0..9, flattened
Wikipedia, Binary search tree
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Row sums give A000142.
Column sums give A227822.
Main diagonal gives A011782, lower diagonal gives A076616.
T(n,A000523(n)+1) = A076615(n).
T(2^n-1,n) = A056972(n).
T(2n,n) = A265846(n).
Cf. A195581 (the same read by rows), A195582, A195583, A316944, A317012.

b:= proc(n, k) option remember; `if`(n<2, `if`(k b(n, k)-b(n, k-1):
seq(seq(T(n, k), n=k..2^k-1), k=0..5);

b[n_, k_] := b[n, k] = If[n<2, If[kJean-François Alcover, Feb 19 2017, translated from Maple *)

Sum_{k=0..n} k * T(n,k) = A316944(n).

Sum_{k=n..2^n-1} k * T(k,n) = A317012(n).

A030103 Base 4 reversal of n (written in base 10).

Original entry on oeis.org

0, 1, 2, 3, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15, 1, 17, 33, 49, 5, 21, 37, 53, 9, 25, 41, 57, 13, 29, 45, 61, 2, 18, 34, 50, 6, 22, 38, 54, 10, 26, 42, 58, 14, 30, 46, 62, 3, 19, 35, 51, 7, 23, 39, 55, 11, 27, 43, 59, 15, 31, 47, 63, 1, 65, 129, 193, 17, 81, 145, 209, 33, 97, 161

Offset: 0

David W. Wilson

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A004086, A030101 - A030108, A048703, A055948, A035524.

import Data.List (unfoldr)
a030103 n = foldl (\v d -> 4*v + d) 0 $ unfoldr dig n where
    dig x = if x == 0 then Nothing else Just $ swap $ divMod x 4
-- Reinhard Zumkeller, Oct 10 2011

IntegerReverse[Range[0, 100], 4] (* Paolo Xausa, Aug 07 2024 *)

a(n,b=4)=subst(Polrev(base(n,b)),x,b) /* where */
base(n,b)={my(a=[n%b]);while(0M. F. Hasler, Nov 04 2011
(MIT/GNU Scheme)
(define (A030103 n) (if (zero? n) n (let ((uplim (+ (A000523 n) (- 1 (modulo (A000523 n) 2))))) (add (lambda (i) (* (bit_i n (+ i (expt -1 i))) (expt 2 (- uplim i)))) 0 uplim))))
(define (bit_i n i) (modulo (floor->exact (/ n (expt 2 i))) 2))
;; The functional add implements sum_{i=lowlim..uplim} intfun(i):
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
;; Antti Karttunen, Oct 30 2013

A152487 Triangle read by rows, 0<=k<=n: T(n,k) = Levenshtein distance of n and k in binary representation.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 2, 0, 2, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 2, 0, 3, 2, 2, 1, 2, 1, 1, 0, 3, 3, 2, 3, 1, 2, 2, 3, 0, 3, 3, 2, 2, 1, 1, 2, 2, 1, 0, 3, 3, 2, 2, 1, 1, 1, 2, 1, 2, 0, 3, 3, 2, 2, 2, 1, 2, 1, 2, 1, 1, 0, 3, 3, 2, 2, 1, 2, 1, 2, 1, 2, 2, 3, 0, 3, 3, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 1, 0

Offset: 0

Reinhard Zumkeller, Dec 06 2008

T(n,k) gives number of editing steps (replace, delete and insert) to transform n to k in binary representations;
row sums give A152488; central terms give A057427;
T(n,k) <= Hamming-distance(n,k) for n and k with A070939(n)=A070939(k);
T(n,0) = A000523(n+1);
T(n,1) = A000523(n) for n>0;
T(n,3) = A106348(n-2) for n>2;
T(n,n-1) = A091090(n-1) for n>0;
T(n,n) = A000004(n);
T(A000290(n),n) = A091092(n).
T(n,k) >= A322285(n,k) - Pontus von Brömssen, Dec 02 2018

			The triangle T(n, k) begins:
  n\k  0  1  2  3  4  5  6  7  8  9 10 11 12 13 ...
   0:  0
   1:  1  0
   2:  1  1  0
   3:  2  1  1  0
   4:  2  2  1  2  0
   5:  2  2  1  1  1  0
   6:  2  2  1  1  1  2  0
   7:  3  2  2  1  2  1  1  0
   8:  3  3  2  3  1  2  2  3  0
   9:  3  3  2  2  1  1  2  2  1  0
  10:  3  3  2  2  1  1  1  2  1  2  0
  11:  3  3  2  2  2  1  2  1  2  1  1  0
  12:  3  3  2  2  1  2  1  2  1  2  2  3  0
  13:  3  3  2  2  2  1  1  1  2  1  2  2  1  0
  ...
The distance between the binary representations of 46 and 25 is 4 (via the edits "101110" - "10111" - "10011" - "11011" - "11001"), so T(46,25) = 4. - _Pontus von Brömssen_, Dec 02 2018

Alois P. Heinz, Rows n = 0..200, flattened
Michael Gilleland, Levenshtein Distance
Wikipedia, Levenshtein Distance
Index entries for sequences related to binary expansion of n

Cf. A000004, A000290, A000523, A057427, A070939, A091090, A091092, A106348, A152488, A322285.

T(n,k) = f(n,k) with f(x,y) = if x>y then f(y,x) else if x<=1 then Log2(y)-0^y+(1-x)*0^(y+1-2^(y+1)) else Min{f([x/2],[y/2]) + (x mod 2) XOR (y mod 2), f([x/2],y)+1, f(x,[y/2])+1}, where Log2=A000523.

A212445 a(n) = floor( n + log(n) ).

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 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, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Mohammad K. Azarian, May 17 2012

Complement of A045650. - Michel Marcus, Jun 30 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000523, A062153, A102572, A212446, A212447, A212448, A212449, A212450, A212451, A212452, A212453, A212454.

[Floor(n+Log(n)): n in [1..80]]; // Vincenzo Librandi, Feb 15 2013

Table[Floor[n + Log[n]], {n, 100}] (* T. D. Noe, May 21 2012 *)

a(n)=n+log(n)\1 \\ Charles R Greathouse IV, Aug 07 2012

A256340 Numbers which have only digits 7 and 8 in base 10.

Original entry on oeis.org

7, 8, 77, 78, 87, 88, 777, 778, 787, 788, 877, 878, 887, 888, 7777, 7778, 7787, 7788, 7877, 7878, 7887, 7888, 8777, 8778, 8787, 8788, 8877, 8878, 8887, 8888, 77777, 77778, 77787, 77788, 77877, 77878, 77887, 77888, 78777, 78778, 78787, 78788, 78877, 78878

Offset: 1

M. F. Hasler, Mar 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..8190
Index entries for 10-automatic sequences.

Cf. A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256341 (digits 8 & 9).

[n: n in [1..35000] | Set(IntegerToSequence(n, 10)) subset {7, 8}];

[n: n in [1..100000] | Set(Intseq(n)) subset {7,8}]; // Vincenzo Librandi, Aug 19 2016

Flatten[Table[FromDigits[#,10]&/@Tuples[{7,8},n],{n,5}]]

A256340(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\9*7

def a(n): return int(bin(n+1)[3:].replace('0', '7').replace('1', '8'))
print([a(n) for n in range(1, 45)]) # Michael S. Branicky, Jul 08 2021

a(n) = A007931(n) + A002280(A000523(n+1)) = A256292(n) + A256077(n) etc.

cp's OEIS Frontend

A329325 Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n with first digit removed.

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A080541 In binary representation: keep the first digit and left-rotate the others.

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A080542 In binary representation: keep the first digit and rotate right the others.

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A125985 Signature-permutation of Vaillé's 1997 bijection on 'bridges' (Dyck paths).

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A126441 Tabular arrangement of the natural numbers: the row on which any nonzero term a(n) appears in is A053645(a(n))=A053645(n+1), and the column is A161511(a(n)). Table is presented by columns with 2^{k-1} items in column k, unused positions are filled with 0's.

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A244108 Number T(n,k) of permutations of {1,2,...,n} that result in a binary search tree of height k; triangle T(n,k), k>=0, k<=n<=2^k-1, read by columns.

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A030103 Base 4 reversal of n (written in base 10).

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