A386384 -id:A386384 - cp's OEIS frontend

A386385 Period-32 block rewriting of A157196 (blocks 11 and 2): for block index i, keep if i mod 32 in {3,4,11,12,15,16,19,20,27,28}, else swap 11<->2.

2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1

Offset: 0

Daniel Hoyt, Aug 17 2025

The runs of '2', and '1,1' are interleaved in the continued fraction expansion of Sum_{k>=0} (-1)^k/(k!)! (A386384).
Parse A157196 into blocks A = 11 and B = 2, and index the blocks i = 0,1,2,.... For each block i, keep it when i mod 32 in {3, 4, 11, 12, 15, 16, 19, 20, 27, 28}; otherwise swap A <-> B (i.e, swap 1,1 <-> 2) at all other i. Finally expand blocks by A -> 1,1 and B -> 2.
We index the blocks starting at i=0. "Keep" residues are {3,4,11,12,15,16,19,20,27,28} (mod 32); at all other residues we swap (11) <-> (2). After that, expand by (11)->1,1 and (2)->2.
Each block is handled independently by its own i mod 32.

			Starting from A157196 parsed as (11)(2)(11)(11)(2)(11)(2)... = ABAABAB...
i=0: (11), residue 0 not in keep set -> swap to (2)  -> output 2.
i=1: (2),  residue 1 not in keep set -> swap to (11) -> output 1, 1.
i=2: (11), residue 2 not in keep set -> swap to (2)  -> output 2.
i=3: (11), residue 3 is in keep set  -> keep (11)    -> output 1, 1.
i=4: (2),  residue 4 is in keep set  -> keep (2)     -> output 2.
i=5: (11), residue 5 not in keep set -> swap to (2)  -> output 2.
i=6: (2),  residue 6 not in keep set -> swap to (11) -> output 1, 1.
Concatenating: 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, ...

Daniel Hoyt, Table of n, a(n) for n = 0..1000
Daniel Hoyt, Justification for the explicit formula and program provided.

Cf. A157196, A386384.

def a386385(n):
    m=n+1
    if m<=0: raise ValueError("n>=0")
    M=0x18199818
    def run(N,mode):
        s=[1,1];h=2;p=0;k=0;c=0;y='B'
        def g(u):
            nonlocal s,h
            while len(s)>(k&31))&1) else ('A' if xb=='B' else 'B')
            c += 2 if y=='A' else 1; k+=1
        return c if mode==0 else y
    lo,hi=0,m
    while lo=m: hi=md
        else: lo=md+1
    return 2 if run(lo,1)=='B' else 1

A387268 Decimal expansion of Sum_{k>=0} (-1)^k/(k!)!.

Original entry on oeis.org

4, 9, 8, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 2, 2, 8, 4, 8, 6, 8, 2, 2, 0, 7, 2, 2, 9, 4, 6, 0, 1, 5, 9, 8, 2, 4, 4, 1, 5, 9, 1, 2, 2, 8, 2, 9, 1, 2, 4, 3, 5, 8, 3, 7, 2, 1, 3, 7, 1, 8, 3, 3, 9, 0, 8, 4, 6, 4, 0, 4, 0, 1, 1, 4, 2, 1, 2, 1, 5, 6, 1, 6, 5, 9, 6, 3, 7, 9

Offset: 0

Daniel Hoyt, Aug 24 2025

This constant has an interesting simple continued fraction representation.

359/720 approximates this constant to 19 significant digits.

			0.498611111111111111127228486822072294...

Daniel Hoyt, Table of n, a(n) for n = 0..999

Cf. A386384 (continued fraction expansion).

Cf. A363842, A336810.

RealDigits[Sum[(-1)^k/(k!)!, {k, 0, 6}], 10, 100][[1]]

PARI
```
suminf(k=0, (-1)^k/(k!)!)
```

cp's OEIS Frontend

A386385 Period-32 block rewriting of A157196 (blocks 11 and 2): for block index i, keep if i mod 32 in {3,4,11,12,15,16,19,20,27,28}, else swap 11<->2.

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