0,3

Also number of n X 2 binary matrices under row and column permutations and column complementations (if offset is 0).

Also Molien series for certain 4-D representation of dihedral group of order 8.

With offset 4, number of bracelets (turnover necklaces) of n-bead of 2 colors with 4 red beads. - Washington Bomfim, Aug 27 2008

From Vladimir Shevelev, Apr 23 2011: (Start)

Also number of non-equivalent necklaces of 4 beads each of them painted by one of n colors.

The sequence solves the so-called Reis problem about convex k-gons in case k=4 (see our comment to A032279). (End)

Number of 2 X 2 matrices with nonnegative integer values totaling n under row and column permutations. - Gabriel Burns, Nov 08 2016

From Petros Hadjicostas, Jan 12 2019: (Start)

By "necklace", Vladimir Shevelev (above) means "turnover necklace", i.e., a bracelet. Zagaglia Salvi (1999) also uses this terminology: she calls a bracelet "necklace" and a necklace "cycle".

According to Cyvin et al. (1997), the sequence (a(n): n >= 0) consists of "the total numbers of isomers of polycyclic conjugated hydrocarbons with q + 1 rings and q internal carbons in one ring (class Q_q)", where q = 4 and n is the hydrogen content (i.e., we count certain isomers of C_{n+2*q} H_n with q = 4 and n >= 0). (End)

a(n) is the number of rooted binary perfect phylogenies with sample size n+4 and a 4-leaf symmetric shape. - Noah A Rosenberg, Nov 15 2025

N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000

Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon. Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 1.

S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.

S. N. Ethier and S. E. Hodge, Identity-by-descent analysis of sibship configurations, Amer. J. Medical Genetics, 22 (1985), 263-272.

H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.

W. D. Hoskins and Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.

W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)

M. Klemm, Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4, Arch. Math. (Basel), 53 (1989), 201-207.

P. Lisonek, Quasi-polynomials: A case study in experimental combinatorics, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]

Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.

Vladimir Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.

Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma) (Cf. Section 5), arXiv:1104.4051 [math.CO], 2011.

Chloe E. Shiff and Noah A. Rosenberg, Enumeration of rooted binary perfect phylogenies, Discr. Appl. Math. 380 (2026), 538-561.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

G.f.: (1+x^3)/((1-x)*(1-x^2)^2*(1-x^4)).

G.f.: (1/8)*(1/(1-x)^4+3/(1-x^2)^2+2/(1-x)^2/(1-x^2)+2/(1-x^4)). - Vladeta Jovovic, Aug 05 2000

Euler transform of length 6 sequence [ 1, 2, 1, 1, 0, -1 ]. - Michael Somos, Feb 01 2007

a(2n+1) = A006918(2n+2)/2;

a(2n) = (A006918(2n+1) + A008619(n))/2.

a(n) = -a(-6 - n) for all n in Z. - Michael Somos, Feb 05 2011

From Vladimir Shevelev, Apr 22 2011: (Start)

if n == 0 (mod 4), then a(n) = n*(n^2-3*n+8)/48;

if n == 1, 3 (mod 4), then a(n) = (n^2-1)*(n-3)/48;

if n == 2 (mod 4), then a(n) = (n-2)*(n^2-n+6)/48. (End)

a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-4) - 2*a(n-5) + 2*a(n-7) - a(n-8) with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 8, a(5) = 10, a(6) = 16, a(7) = 20. - Harvey P. Dale, Oct 24 2012

a(n) = ((n+3)*(2*n^2+12*n+19+9*(-1)^n) + 6*(-1)^((2*n-1+(-1)^n)/4)*(1+(-1)^n))/96. - Luce ETIENNE, Mar 16 2015

a(n) = |A128498(n)| + |A128498(n-3)|. - R. J. Mathar, Jun 11 2019

a(n) = A389571(n+4) - A002623(n). - Noah A Rosenberg, Nov 15 2025

G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 10*x^5 + 16*x^6 + 20*x^7 + 29*x^8 + ...

There are 8 4 X 2 matrices up to row and column permutations and column complementations:

[1 1] [1 0] [1 0] [0 1] [0 1] [0 1] [0 1] [0 0]

[1 1] [1 1] [1 0] [1 0] [1 0] [1 0] [0 1] [0 1]

[1 1] [1 1] [1 1] [1 1] [1 0] [1 0] [1 0] [1 0]

[1 1] [1 1] [1 1] [1 1] [1 1] [1 0] [1 0] [1 1].

There are 8 2 X 2 matrices of nonnegative integers totaling 4 up to row and column permutations:

[4 0] [3 1] [2 2] [2 1] [2 1] [3 0] [2 0] [1 1]

[0 0] [0 0] [0 0] [0 1] [1 0] [1 0] [2 0] [1 1].

A005232:=-(-1-z-2*z**3+2*z**2+z**7-2*z**6+2*z**4)/(z**2+1)/(1+z)**2/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence apart from an initial 1

k = 4; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)

CoefficientList[ Series[(1 - x + x^2)/((1 - x)^2(1 - x^2)(1 - x^4)), {x, 0, 51}], x] (* Robert G. Wilson v, Mar 29 2006 *)

LinearRecurrence[{2, 0, -2, 2, -2, 0, 2, -1}, {1, 1, 3, 4, 8, 10, 16, 20}, 60] (* Harvey P. Dale, Oct 24 2012 *)

k=4 (* Number of red beads in bracelet problem *); CoefficientList[Series[(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[(k+2)/2])/2, {x, 0, 50}], x] (* Herbert Kociemba, Nov 04 2016 *)

(PARI) {a(n) = (n^3 + 9*n^2 + (32-9*(n%2))*n + [48, 15, 36, 15][n%4+1]) / 48}; \\ Michael Somos, Feb 01 2007

(PARI) {a(n) = my(s=1); if( n<-5, n = -6-n; s=-1); if( n<0, 0, s * polcoeff( (1 - x + x^2) / ((1 - x)^2 * (1 - x^2) * (1 - x^4)) + x * O(x^n), n))}; \\ Michael Somos, Feb 01 2007

(PARI) a(n) = round((n^3 +9*n^2 +(32-9*(n%2))*n)/48 +0.6) \\ Washington Bomfim, Jul 17 2008

(PARI) a(n) = ceil((n+1)*(2*n^2+16*n+39+9*(-1)^n)/96) \\ Tani Akinari, Aug 23 2013

(Python) a=lambda n: sum((k//2+1)*((n-k)//2+1) for k in range((n-1)//2+1))+(n+1)%2*(((n//4+1)*(n//4+2))//2) # Gabriel Burns, Nov 08 2016

Row n=2 of A343875.

Column k=4 of A052307.

Cf. A006381, A006382, A008805.

Sequence in context: A308844 A131355 A092534 * A165272 A310010 A294085

Adjacent sequences: A005229 A005230 A005231 * A005233 A005234 A005235

nonn,easy,nice,changed

Sequence extended by Christian G. Bower

approved