OFFSET
1,3
COMMENTS
From Travis Scott, Feb 15 2023: (Start)
Let b(i) give the i-th bit of a string indexed {0...j-1}, let s > 1 divide j, define (s-)seam as a decimation b(i == r (mod j/s)), and let n count carbon atoms. Then the stable deoxy-n-itol isomers of Nemba and Fah enumerate as bit strings of length 2n and Hamming weight h = n+1, up to reversal, with b(0) = b(2n-1) = 1, and such that a 2-seam colored 00 destabilizes a string. [Ex. 111100 fails because its last bit is 0 and 101101 fails because it decimates 1 (mod 3) to 00.]
The seam 0 (mod n) may not be 01 and the seam n-1 (mod n) may not be 10; otherwise, 2-seams permute freely as compositions of n+1 into n parts using one kind of 2 (11) and two kinds of 1 (01 10). The number of stable strings follows by inex [exclude the two restrictions disjointly and include them jointly: A001787(n) - 2*A001787(n-1) + A001787(max(n-2, 0))]: combining terms gives A045623(n-1).
Now if n is even, h is odd and no palindromes may be colored, while if n is odd, palindromy induces a monochromatic seam (n-1)/2 (mod n) which may only be colored 11. Stable strings invariant to reversal are thus palindromes of length 2n == 2 (mod 4) with min(n, 2) impermutable bit pairs [their endpoints must also be colored 11; the pairs coincide when n=1] so that, in this case, the inex of curated wreath products exemplified at Table I of NF95 reduces to (A045623(n-1) + A011782((n-1)/2)(n mod 2))/2. (End)
LINKS
K. Balasubramanian, Enumeration of stable stereo and position isomers of polysubstituted alcohols, Ann. N.Y. Acad. Sci. 319.1 (1979), pp. 33-36.
R. M. Nemba and M. Fah, On the application of sieve formula to the enumeration of stable stereo and position isomers of deoxyaldytols [sic], Tetrahedron 51.13 (1995), pp. 3831-3840. See Table II.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,8).
FORMULA
a(n) = 2^max(n-4, -2)(n+2) + 2^((n-5)/2)(n mod 2).
a(2^n - 2) = 2^(2^n + n-6).
From Chai Wah Wu, Jun 04 2024: (Start)
a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 8*a(n-4) for n > 5.
G.f.: x*(x - 1)*(3*x^3 - x^2 - 2*x + 1)/((2*x - 1)^2*(2*x^2 - 1)). (End)
EXAMPLE
From Travis Scott, Feb 15 2023: (Start)
The a(5)=15 stable strings counting deoxypentitols, grouped by position-isomer equivalence [see B79, Fig. 4, cit. NF95 ante Table II]:
(A) 1000011111, 1000101111, 1001011101, 1001101101, 1010011011, 1010101011, 1100010111, 1100100111;
(B) 1001001111, 1011001011, 1100001111, 1101000111;
(C) 1010001111, 1011001101, 1110000111. (End)
MATHEMATICA
Table[(n+2)2^Max[n-4, -2]+Mod[n, 2]2^((n-5)/2), {n, 32}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Feb 09 2019
EXTENSIONS
a(1)-a(2) prepended, more terms added by Travis Scott, Feb 15 2023
STATUS
approved