A237501 Number of binary strings of length 2n which contain the reversals of each of their two halves.

4, 12, 42, 106, 300, 654, 1664, 3300, 7940, 15018, 34948, 64396, 147130, 267404, 604722, 1090998, 2453492, 4409000, 9886266, 17729222, 39693612, 71108358, 159076784, 284820632, 636918540, 1140064414, 2548902598, 4561828606, 10198077780, 18250461694, 40797250536, 73008145904, 163198882506, 292045189424, 652815291522

Offset

1,1

Links

Max Alekseyev, Table of n, a(n) for n = 1..45

Michael S. Branicky, Python program

Example

The two halves of 01111011 are 0111 and 1011. Their reversals are 1110 and 1101, and both are substrings of 01111011. Since there are 105 other strings of length 2*4 with this property, a(4) = 106.

Mathematica

sQ[L_, {s__}] := MatchQ[L, {___, s, ___}]; a[n_] := Length@ Select[ Tuples[{0, 1}, 2*n], sQ[#, Reverse[Take[#, n]]] && sQ[#, Reverse[Take[#, -n ]]] &]; Array[a, 8]

Prog

(Python) # see link for faster version
from itertools import product as prod
def ok(s): return s[:len(s)//2][::-1] in s and s[len(s)//2:][::-1] in s
def a(n): return 2*sum(ok("0"+"".join(p)) for p in prod("01", repeat=2*n-1))
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, Feb 06 2021

Crossrefs

Cf. A237500, A237502, A241208, A241210, A241211.

Sequence in context: A242470 A242471 A239800 * A375547 A300124 A343517

Adjacent sequences: A237498 A237499 A237500 * A237502 A237503 A237504

Keyword

nonn

Author

Giovanni Resta, Feb 08 2014

Extensions

a(17)-a(20) from Michael S. Branicky, Feb 06 2021

Terms a(21) onward from Max Alekseyev, Jul 24 2024

Status

approved