A367360 -id:A367360 - cp's OEIS frontend

A367578 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) - CT(a(n-2),a(n-1)) if nonnegative and not already in the sequence, else a(n) = a(n-1) + CT(a(n-2),a(n-1)), where CT(a,b) is the Comma transform (cf. A367360) of a and b.

0, 1, 2, 14, 35, 78, 21, 103, 92, 53, 28, 60, 146, 145, 84, 26, 68, 134, 215, 173, 122, 91, 62, 46, 22, 84, 56, 11, 72, 55, 30, 83, 75, 38, 91, 180, 169, 168, 77, 164, 93, 44, 10, 51, 56, 41, 105, 94, 153, 112, 81, 109, 98, 197, 116, 45, 109, 58, 153, 234, 202, 160, 139, 138, 47, 131, 202, 190

Offset: 0

Scott R. Shannon, Nov 25 2023

This is a variation of Recamán's sequence A005132, where the step size is calculated from the Comma transform of the previous two terms, see A367360 and A121805. As the maximum step size is 99, it is likely that many numbers never appear. In the first 10 million terms the smallest numbers that do appear are 0,1,2,8,10,11,14,17,21,22. The first number to appear twice is 84. The terms show a broadly repetitive pattern that repeats every order of magnitude, although slight differences are still present; see the two attached images.

			a(2) = 2 as CT(a(0),a(1)) = CT(0,1) = 1, so a(2) = a(1) + 1 = 2.
a(3) = 14 as CT(a(1),a(2)) = CT(1,2) = 12, so a(3) = a(2) + 12 = 14.
a(7) = 21 as CT(a(5),a(6)) = CT(35,78) = 57, so a(7) = a(6) - 57 = 21, as 21 is nonnegative and not already in the sequence.

Scott R. Shannon, Table of n, a(n) for n = 0..10000.
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.
Scott R. Shannon, Image of the first 10000000 terms.

Cf. A367360, A005132, A121805.

A367646 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) - GCT(a(n-2),a(n-1)) if nonnegative and not already in the sequence, else a(n) = a(n-1) + CT(a(n-2),a(n-1)), where CT(a,b) is the Comma transform (cf. A367360) of a and b, while GCT(a,b) is the largest possible generalized Command transform (cf. A367635) where at least one digit of both a and b can be chosen.

Original entry on oeis.org

0, 1, 2, 14, 35, 78, 21, 103, 92, 53, 28, 60, 146, 132, 71, 44, 30, 73, 66, 102, 41, 17, 6, 82, 150, 129, 117, 26, 98, 29, 111, 20, 8, 16, 97, 166, 95, 164, 113, 72, 109, 88, 186, 105, 166, 115, 54, 109, 68, 164, 83, 131, 100, 89, 81, 179, 62, 158, 137, 56, 131, 70, 87, 79, 156, 65, 131, 80

Offset: 0

Scott R. Shannon, Nov 25 2023

This is a variation of A367578, where one can choose more than one digit from both a(n-2) and a(n-1) to create the largest possible step to a nonnegative number which has not previously appeared to form a(n). If all such numbers have already appeared the smallest possible forward step is chosen, which is just the standard Comma transform of a(n-2) and a(n-1).

It is conjectured that all nonnegative numbers appear in the sequence. After the first 10 million terms the only fixed points are 0, 1, 2, 29, 65, 84, 222, 377, 491, 499, and it is likely no more exist. The first number to appear twice is a(35) = a(44) = 166.

			a(3) = 14 as CT(a(1),a(2)) = CT(1,2) = 12, so a(3) = a(2) + 12 = 14.
a(6) = 21 as GCT(a(4),a(5)) = GCT(35,78) = 57, so a(6) = a(5) - 57 = 21, as 21 is nonnegative and not already in the sequence.
a(13) = 132 as GCT(a(11),a(12)) = GCT(60,146) = 14, so a(13) = a(12) - 14 = 132, as 132 is nonnegative and not already in the sequence. This is the first term to differ from A367578.

Scott R. Shannon, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.
Scott R. Shannon, Image of the first 10 million terms.

Cf. A367360, A367635, A367578, A005132, A121805.

A369303 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused number whose string value contains the comma transform (cf. A367360) of the previous two terms.

Original entry on oeis.org

1, 2, 12, 21, 22, 112, 121, 210, 120, 10, 11, 13, 110, 31, 3, 113, 131, 231, 122, 111, 211, 123, 114, 310, 43, 4, 34, 143, 41, 134, 115, 141, 51, 15, 116, 151, 61, 16, 117, 161, 71, 17, 118, 171, 81, 18, 119, 181, 91, 19, 311, 93, 190, 312, 23, 220, 32, 30, 223, 20, 132, 14, 212, 42, 24, 221

Offset: 1

Scott R. Shannon, Jan 19 2024

The sequence is conjectured to be a permutation of the positive integers. The fixed points begin 1, 2, 10, 11, 2863, 3164, 3545, 3947, 6835, 6947, 7052, ... although it is likely there are infinitely more.

			a(3) = 12 as the comma transform of 1 and 2 is 12.
a(6) = 112 as the comma transform of 21 and 22 is 12, but 12 has already appeared so the next lowest unused number to contain '12' is 112.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first one million terms. The green line is a(n) = n.

Cf. A367360, A121805, A367362.

A121805 The "comma sequence": the lexicographically earliest sequence of positive numbers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms is the same as the sequence of successive differences between the terms.

Original entry on oeis.org

1, 12, 35, 94, 135, 186, 248, 331, 344, 387, 461, 475, 530, 535, 590, 595, 651, 667, 744, 791, 809, 908, 997, 1068, 1149, 1240, 1241, 1252, 1273, 1304, 1345, 1396, 1457, 1528, 1609, 1700, 1701, 1712, 1733, 1764, 1805, 1856, 1917, 1988, 2070

Offset: 1

Eric Angelini, Dec 11 2006

An equivalent, but more formal definition, is: a(1) = 1; for n > 1, let x be the least significant digit of a(n-1); then a(n) = a(n-1) + x*10 + y where y is the most significant digit of a(n) and is the smallest such y, if such a y exists. If no such y exists, stop.
The sequence contains exactly 2137453 terms, with a(2137453)=99999945. The next term does not exist. - W. Edwin Clark, Dec 11 2006
It is remarkable that the sequence persists for so long. - N. J. A. Sloane, Dec 15 2006
The similar sequence A139284, which starts at a(1)=2, persists even longer, ending at a(194697747222394) = 9999999999999918. - Giovanni Resta, Nov 30 2019
Conjecture: This sequence is finite, for any initial term. - N. J. A. Sloane, Nov 14 2023
The base 2 analog (suggested by William Cheswick) is 1, 4, 5, 8, 9, 12, 13, ..., (see A042948) with successive differences 3, 1, 3, 1, ... (repeat). - N. J. A. Sloane, Nov 15 2023
Does not satisfy Benford's Law. - Michael S. Branicky, Nov 16 2023
Using the notion of "comma transform" of a sequence, as defined in A367360, this is the lexicographically earliest sequence of positive integers with the property that its first differences and comma transform coincide. - N. J. A. Sloane, Nov 23 2023

			Replace each comma in the original sequence by the pair of digits adjacent to the comma; the result is the sequence of first differences between the terms of the sequence:
Sequence:   1, 12, 35, 94, 135, 186, 248, 331, 344, 387, 461, 475, ...
Differences: 11, 23, 59, 41 , 51 , 62 , 83 , 13 , 43 , 74 , 14 , ...
To illustrate the formula in the comment: a(6) = 186 and a(7) = 248 = 186 + 62.

Eric Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

Michael S. Branicky, Table of n, a(n) for n = 1..20000 (terms 1..1001 from Zak Seidov)
Eric Angelini, The Commas Sequence, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Fibonacci Quarterly 62:3 (2024), 215-232.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
Lorenzo Angelini, Happy birthday Éric!!, Youtube video.
Michael S. Branicky, Graph of a(n)/n over entire sequence
Simon Demers, Table of n, a(n) for n = 1..2137453 (full sequence)
Robert Dougherty-Bliss, The Comma Sequence is WILD, 2024 video.
Robert Dougherty-Bliss and Natalya Ter-Saakov, The Comma Sequence is Finite in Other Bases, arXiv:2408.03434 [math.NT], 2024.
Carlos Rivera, Puzzle 980. The "Commas" sequence, The Prime Puzzles and Problems Connection.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides
Index entries for sequences related to Benford's law

See A366487 and A367349 for first differences.
Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.
Cf. A330128, A330129, A367338 (comma-successor), A367360.
See also A260261, A042948.

digits:=n->ListTools:-Reverse(convert(n,base,10)):
nextK:=proc(K) local i,L; for i from 0 to 9 do L:=K+digits(K)[ -1]*10+i; if i = digits(L)[1] then return L; fi; od; FAIL; end:
A121805:=proc(n) option remember: if n = 1 then return 1; fi; return nextK(A121805(n-1)); end: # W. Edwin Clark

a[1] = 1; a[n_] := a[n] = For[x=Mod[a[n-1], 10]; y=0, y <= 9, y++, an = a[n-1] + 10*x + y; If[y == IntegerDigits[an][[1]], Return[an]]]; Array[a, 45] (* Jean-François Alcover, Nov 25 2014 *)

a=1; for(n=1,1000, print1(a", "); a+=a%10*10; for(k=1, 9, digits(a+k)[1]==k&&(a+=k)&&next(2)); error("blocked at a("n")=",a-a%10*10)) \\ M. F. Hasler, Jul 21 2015

from itertools import islice
def agen(): # generator of terms
    an, y = 1, 1
    while y < 10:
        yield an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
print(list(islice(agen(), 45))) # Michael S. Branicky, Apr 08 2022

A121805 <- data.frame(n=seq(from=1,to=2137453),a=integer(2137453)); A121805$a[1]=1; for (i in seq(from=2,to=2137453)){LSD=A121805$a[i-1] %% 10; k = 1; while (k != as.integer(substring(A121805$a[i-1]+LSD*10+k,1,1))){k = k+1; if(k>9) break} A121805$a[i]=A121805$a[i-1]+LSD*10+k} # Simon Demers, Oct 19 2017

More terms from Zak Seidov, Dec 11 2006
Edited by N. J. A. Sloane, Sep 17 2023
Changed name from "commas sequence" to "comma sequence". - N. J. A. Sloane, Dec 20 2023

A166499 Concatenation of the rightmost digit of the n-th prime and the leftmost digit of the (n+1)th prime.

Original entry on oeis.org

23, 35, 57, 71, 11, 31, 71, 92, 32, 93, 13, 74, 14, 34, 75, 35, 96, 16, 77, 17, 37, 98, 38, 99, 71, 11, 31, 71, 91, 31, 71, 11, 71, 91, 91, 11, 71, 31, 71, 31, 91, 11, 11, 31, 71, 92, 12, 32, 72, 92, 32, 92, 12, 12, 72, 32, 92, 12, 72, 12, 32, 33, 73, 13, 33, 73, 13, 73, 73

Offset: 1

Zak Seidov, Oct 15 2009

This is the comma transform of the primes (see A367360).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A007652, A077648, A367360.

a:= n-> parse(cat(""||(ithprime(n))[-1],""||(ithprime(n+1))[1])):
seq(a(n), n=1..99);  # Alois P. Heinz, Nov 22 2023

With[{nmax=100},Map[10Mod[#[[1]],10]+IntegerDigits[#[[2]]][[1]]&,Partition[Prime[Range[nmax+1]],2,1]]] (* Paolo Xausa, Nov 24 2023 *)

a(n) = 10 * A007652(n) + A077648(n+1). - Alois P. Heinz, Nov 23 2023

A367362 Comma transform of the nonnegative integers.

Original entry on oeis.org

1, 12, 23, 34, 45, 56, 67, 78, 89, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81, 92, 2, 12, 22, 32, 42, 52, 62, 72, 82, 93, 3, 13, 23, 33, 43, 53, 63, 73, 83, 94, 4, 14, 24, 34, 44, 54, 64, 74, 84, 95, 5, 15, 25, 35, 45, 55, 65, 75, 85, 96, 6, 16, 26, 36, 46, 56, 66, 76, 86, 97, 7, 17, 27, 37, 47, 57, 67, 77, 87, 98, 8, 18

Offset: 0

N. J. A. Sloane, Nov 22 2023

See A367360 for further information.

Michael S. Branicky, Table of n, a(n) for n = 0..10000
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Youtube

Cf. A000030, A010879, A166499, A367360, A367361, A368362.

a:= n-> parse(cat(""||n[-1], ""||(n+1)[1])):
seq(a(n), n=0..99);  # Alois P. Heinz, Nov 22 2023

FromDigits /@ Partition[Rest@ Flatten[{First[#], Last[#]} & /@ IntegerDigits[Range[0, 120]]], 2, 2] (* Michael De Vlieger, Nov 22 2023 *)

from itertools import count, islice, pairwise
def S(): yield from (str(i) for i in count(0))
def agen(): yield from (int(t[-1]+u[0]) for t, u in pairwise(S()))
print(list(islice(agen(), 82))) # Michael S. Branicky, Nov 22 2023

def A367362(n): return (n%10)*10+int(str(n+1)[0]) # Chai Wah Wu, Dec 22 2023

a(n) = 10 * A010879(n) + A000030(n+1). - Alois P. Heinz, Nov 22 2023

A367361 Comma transform of powers of 2.

Original entry on oeis.org

12, 24, 48, 81, 63, 26, 41, 82, 65, 21, 42, 84, 68, 21, 43, 86, 61, 22, 45, 81, 62, 24, 48, 81, 63, 26, 41, 82, 65, 21, 42, 84, 68, 21, 43, 86, 61, 22, 45, 81, 62, 24, 48, 81, 63, 27, 41, 82, 65, 21, 42, 84, 69, 21, 43, 87, 61, 22, 45, 81, 62, 24, 49, 81, 63, 27, 41, 82, 65, 21, 42, 84, 69, 21, 43, 87, 61, 23, 46, 81

Offset: 0

N. J. A. Sloane, Nov 22 2023

See A367360 for further information.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000079, A000689, A008952, A166499, A367360.

FromDigits /@ Partition[Rest@ Flatten[{First[#], Last[#]} & /@ IntegerDigits[2^Range[0, 120]]], 2, 2] (* Michael De Vlieger, Nov 22 2023 *)

from itertools import count, islice, pairwise
def S(): yield from (str(2**i) for i in count(0))
def agen(): yield from (int(t[-1]+u[0]) for t, u in pairwise(S()))
print(list(islice(agen(), 80))) # Michael S. Branicky, Nov 22 2023

def A367361(n): return (60,20,40,80)[n&3]+int(str(1<Chai Wah Wu, Dec 22 2023

a(n) = 10 * A000689(n) + A008952(n+1). - Alois P. Heinz, Nov 22 2023

A368362 Inverse comma transform of 1,2,3,4,5,...,99.

Original entry on oeis.org

10, 10, 20, 30, 40, 50, 60, 70, 80, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81, 92, 2, 12, 22, 32, 42, 52, 62, 72, 82, 93, 3, 13, 23, 33, 43, 53, 63, 73, 83, 94, 4, 14, 24, 34, 44, 54, 64, 74, 84, 95, 5, 15, 25, 35, 45, 55, 65, 75, 85, 96, 6, 16, 26, 36, 46, 56, 66, 76, 86, 97, 7, 17, 27, 37, 47, 57, 67, 77, 87, 98, 8, 18, 28, 38, 48, 58, 68, 78, 88, 99, 9, 19, 29, 39, 49, 59, 69, 79, 89

Offset: 1

N. J. A. Sloane, Jan 03 2024

See A367360 for further information.

Cf. A367360, A367362

A367556 Comma transform of the Fibonacci numbers.

Original entry on oeis.org

1, 11, 12, 23, 35, 58, 81, 32, 13, 45, 58, 91, 42, 33, 76, 9, 71, 72, 44, 16, 51, 61, 12, 74, 87, 51, 31, 83, 15, 98, 1, 92, 93, 85, 79, 51, 22, 73, 96, 61, 51, 12, 64, 77, 31, 1, 32, 34, 67, 91, 52, 43, 95, 38, 21, 52, 73, 25, 99, 11, 2, 14, 16, 21, 31, 52, 84

Offset: 0

Alois P. Heinz, Nov 22 2023

See A367360 for further information.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000045, A003893, A008963, A138844, A367360.

F:= combinat[fibonacci]:
a:= n-> parse(cat(""||(F(n))[-1], ""||(F(n+1))[1])):
seq(a(n), n=0..92);

With[{nmax=100},Map[10Mod[#[[1]],10]+IntegerDigits[#[[2]]][[1]]&,Partition[Fibonacci[Range[0,nmax+1]],2,1]]] (* Paolo Xausa, Nov 24 2023 *)

from sympy import fibonacci
from itertools import islice, pairwise, count
def S(): yield from (fibonacci(i) for i in count(0))
def C(g): # generator of comma transform of sequence passed as a generator
    yield from (10*(t%10) + int(str(u)[0]) for t, u in pairwise(g))
def agen(): return C(S())
print(list(islice(agen(), 67))) # Michael S. Branicky, Jan 05 2024

a(n) = 10 * A003893(n) + A008963(n+1).

A367610 Comma transform of A367362.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 21, 22, 23, 24, 25, 26, 27, 28, 29, 33, 31, 32, 33, 34, 35, 36, 37, 38, 39, 44, 41, 42, 43, 44, 45, 46, 47, 48, 49, 55, 51, 52, 53, 54, 55, 56, 57, 58, 59, 66, 61, 62, 63, 64, 65, 66, 67, 68, 69, 77, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

N. J. A. Sloane, Dec 11 2023

This is the second-order comma transform of the nonnegative integers.

See A367360 for further information.

N. J. A. Sloane, Table of n, a(n) for n = 1..9999

Cf. A367360, A367362.

from itertools import count, islice, pairwise
def S(): yield from (str(i) for i in count(0))
def C(): yield from (str(int(t[-1]+u[0])) for t, u in pairwise(S()))
def a(): yield from (int(t[-1]+u[0]) for t, u in pairwise(C()))
print(list(islice(a(), 80))) # Michael S. Branicky, Dec 11 2023

cp's OEIS Frontend

A367578 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) - CT(a(n-2),a(n-1)) if nonnegative and not already in the sequence, else a(n) = a(n-1) + CT(a(n-2),a(n-1)), where CT(a,b) is the Comma transform (cf. A367360) of a and b.

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A369303 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused number whose string value contains the comma transform (cf. A367360) of the previous two terms.

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A121805 The "comma sequence": the lexicographically earliest sequence of positive numbers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms is the same as the sequence of successive differences between the terms.

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A166499 Concatenation of the rightmost digit of the n-th prime and the leftmost digit of the (n+1)th prime.

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A367362 Comma transform of the nonnegative integers.

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A367361 Comma transform of powers of 2.

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A368362 Inverse comma transform of 1,2,3,4,5,...,99.

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