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A237501
Number of binary strings of length 2n which contain the reversals of each of their two halves.
3
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
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
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