A239083 The sequence S = a(1), a(2), ... is defined by a(1)=1, if d,e,f are consecutive digits then we do not have d < e < f, and S is always extended with the smallest integer not yet present in S.

1, 2, 10, 3, 11, 4, 12, 13, 14, 15, 5, 6, 16, 17, 7, 8, 18, 19, 9, 20, 21, 22, 23, 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, 58, 59, 60, 61, 62, 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, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 200, 201, 121, 122, 130, 202

Offset: 1

Author

Michel Marcus and N. J. A. Sloane, Mar 10 2014

Keywords

• nonn
• base
• look

Comments

More than the usual number of terms are given in order to show that the pattern breaks after 120.

Computed by Lars Blomberg.

This is the first (Sa) of a family of 25 similar sequences. For others see

A239083-A239086, A239136-A239139, A239087-A239090, A239215-A239218, A239235.

The sequence So (see link) has d > e = f in the definition. It does not have its own entry in the OEIS because it begins with the numbers 1 through 99. Using x-y to indicate the numbers from x through y, the sequence So begins like this:

1-99,101-109,120,110-112,121,201,113,122-130,114,131,202,132-140,115,141,

203,142-150,116,151,204,152-160,117,161,205,162-170,118,171,206, 172-180,

119,181,207,182-191, 208,192-199,209, 210,212-219,230, 220-223,231, 224,232,

301, 225,233-240,226,241,227,242, ...

Likewise, the sequence Sw is omitted for a similar reason. It has d = e > f in the definition, and begins 1-89,99,999,9999,99999,999999,9999999,..., continuing with strings of 9's.

Again, the sequences Sx and Sy are omitted because they are too close to A130571.

Sx (which has d = e >= f) begins

1-11,20,12-19,21,22,30,23-29,31-33,40,34-39,41-44,50,45-49,51-55,60,56-59,

61-66, 70,67-69,71-77,80,78,79,81-88,90,89,100,91-98,101,120,102-109,

112-119,121,122,300, 123-133,400,134-144,500,145-155,600,156-166,700,

167-177,800,178-188,900,189-198,200-202, ...

and Sy (d = e = f) begins

1-11,20,12-19,21,22,30,23-29,31-33,40,34-39,41-44,50,45-49,51-55,60,56-59,

61-66, 70,67-69,71-77,80,78,79,81-88,90,89,91-110,112-221,223-332,334-443,

445-554,556-665, 667-776,778-887,889-899,1001,900-989,1002,990-998,1003-1010,...

The sequences Sd, Si, Sl, Sq are omitted because they do not have enough terms to warrant their own entries.

References

• Eric Angelini, Posting to Sequence Fans Mailing List, Sep 28 2013

Links

• Gleb Ivanov, Table of n, a(n) for n = 1..10000
• Eric Angelini, Less than <, Equal to =, Greater than > (see sequence Sa)
• Eric Angelini, Less than <, Equal to =, Greater than > [Cached copy, with permission of the author] (see sequence Sa)

Crossrefs

The sequences in this family are given in A239083-A239086, A239136-A239139, A239087-A239090, A239215-A239218, A239235.

Programs

Mathematica

a[1]=1;a[n_]:=a[n]=Block[{k=1},While[MemberQ[s=Array[a,n-1],k]||Or@@(#<#2<#3&@@@Partition[Flatten[IntegerDigits/@Join[s[[-2;;]],{k}]],3,1]),k++];k];Array[a,126] (* Giorgos Kalogeropoulos, May 13 2022 *)

Python

is_ok = lambda s: not any(s[i-2] < s[i-1] < s[i] for i in range(2, len(s)))
terms, appears, digits = [1],{1},'1'
for i in range(100):
    t = 1
    while not(t not in appears and is_ok(digits + str(t))):
        t += 1
    terms.append(t); appears.add(t); digits = digits + str(t)
    digits = digits[-2:]
print(terms) # Gleb Ivanov, Dec 04 2021