A321672 Number of chiral pairs of rows of length 5 using up to n colors.

0, 0, 12, 108, 480, 1500, 3780, 8232, 16128, 29160, 49500, 79860, 123552, 184548, 267540, 378000, 522240, 707472, 941868, 1234620, 1596000, 2037420, 2571492, 3212088, 3974400, 4875000, 5931900, 7164612, 8594208, 10243380, 12136500

Offset

0,3

Links

Table of n, a(n) for n=0..30.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = (n^5 - n^3) / 2.

a(n) = (A000584(n) - A000578(n)) / 2.

a(n) = A000584(n) - A168178(n) = A168178(n) - A000578(n).

G.f.: (Sum_{j=1..5} S2(5,j)*j!*x^j/(1-x)^(j+1) - Sum_{j=1..3} S2(3,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

G.f.: x * Sum_{k=1..4} A145883(5,k) * x^k / (1-x)^6.

E.g.f.: (Sum_{k=1..5} S2(5,k)*x^k - Sum_{k=1..3} S2(3,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

For n>5, a(n) = Sum_{j=1..6} -binomial(j-7,j) * a(n-j).

From Amiram Eldar, Jun 21 2026: (Start)

Sum_{n>=2} 1/a(n) = 5/2 - 2*zeta(3).

Sum_{n>=2} (-1)^n/a(n) = 3*zeta(3)/2 + 4*log(2) - 9/2. (End)

Example

For a(0) = 0 and  a(1) = 0, there are no chiral rows using fewer than two colors.
For a(2) = 12, the chiral pairs are AAAAB-BAAAA, AAABA-ABAAA, AAABB-BBAAA, AABAB-BABAA, AABBA-ABBAA, AABBB-BBBAAA, ABAAB-BAABA, ABABB-BBABA, ABBAB-BABBA, ABBBB-BBBBA, BAABB-BBAAB, and BABBB-BBBAB.

Mathematica

Table[(n^5-n^3)/2, {n, 0, 40}]
(* Alternative: *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 12, 108, 480, 1500}, 40]

Prog

(PARI) a(n)=(n^5-n^3)/2 \\ Charles R Greathouse IV, Oct 21 2022

Crossrefs

Row 5 of A293500.

Cf. A000584 (oriented), A168178 (unoriented), A000578 (achiral), A008277.

Sequence in context: A271559 A154671 A390575 * A241230 A353047 A037972

Adjacent sequences: A321669 A321670 A321671 * A321673 A321674 A321675

Keyword

nonn,easy,changed

Author

Robert A. Russell, Nov 16 2018

Status

approved