A168178 a(n) = n^3*(n^2 + 1)/2.

0, 1, 20, 135, 544, 1625, 3996, 8575, 16640, 29889, 50500, 81191, 125280, 186745, 270284, 381375, 526336, 712385, 947700, 1241479, 1604000, 2046681, 2582140, 3224255, 3988224, 4890625, 5949476, 7184295, 8616160, 10267769, 12163500

Offset: 0

Author

N. J. A. Sloane, Dec 11 2009

Keywords

• nonn
• easy

Comments

Number of unoriented rows of length 5 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=20, there are 8 achiral (AAAAA, AABAA, ABABA, ABBBA, BAAAB, BABAB, BBABB, BBBBB) and 12 chiral pairs (AAAAB-BAAAA, AAABA-ABAAA, AAABB-BBAAA, AABAB-BABAA, AABBA-ABBAA, AABBB-BBBAA, ABAAB-BAABA, ABABB-BBABA, ABBAB-BABBA, ABBBB-BBBBA, BAABB-BBAAB, BABBB-BBBAB). - Robert A. Russell, Nov 14 2018

For n > 0, a(2n+1) is the number of non-isomorphic kC_m-snakes, where m = 2n+1 or m = 2n (for n>=2). A kC_n-snake is a connected graph in which the k>=2 blocks are isomorphic to the cycle C_n and the block-cutpoint graph is a path. - Christian Barrientos, May 16 2019

References

• C. Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60(2001), 85-96.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..595
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A155977.

Row 5 of A277504.

Cf. A000584 (oriented), A000578 (achiral).

Programs

GAP

List([0..40], n -> n^3*(1 +n^2)/2); # G. C. Greubel, Nov 15 2018

Magma

[n^3*(n^2+1)/2: n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

Mathematica

Table[(n^5+n^3)/2,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1},{0, 1, 20, 135, 544, 1625},40] (* Robert A. Russell, Nov 14 2018 *)

PARI

vector(40, n, n--; n^3*(1+n^2)/2) \\ G. C. Greubel, Nov 15 2018

Sage

[n^3*(1 + n^2)/2 for n in range(40)] # G. C. Greubel, Nov 15 2018

Formula

From Robert A. Russell, Nov 14 2018: (Start)

a(n) = (A000584(n) + A000578(n)) / 2 = (n^5 + n^3) / 2.

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=0..4} A145882(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). (End)

From G. C. Greubel, Nov 15 2018: (Start)

G.f.: x*(1 + 14*x + 30*x^2 + 14*x^3 + x^4)/(1-x)^6.

E.g.f.: x*(2 + 18*x + 26*x^2 + 10*x^3 + x^4)*exp(x)/2. (End)