OFFSET
0,3
COMMENTS
Number of unoriented rows of length 6 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)=36, there are 2^6=64 oriented arrangements of two colors. Of these, 2^3=8 are achiral. That leaves (64-8)/2=28 chiral pairs. Adding achiral and chiral, we get 36. - Robert A. Russell, Nov 14 2018
For n > 0, a(2n+1) is the number of non-isomorphic 8C_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.
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
FORMULA
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(0)=0, a(1)=1, a(2)=36, a(3)=378, a(4)=2080, a(5)=7875, a(6)=23436. - Harvey P. Dale, Nov 06 2011
G.f.: x*(28*x^4 + 155*x^3 + 147*x^2 + 29*x + 1)/(1-x)^7. - Colin Barker, Oct 12 2012
From Robert A. Russell, Nov 14 2018: (Start)
G.f.: (Sum_{j=1..6} S2(6,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..5} A145882(6,k) * x^k / (1-x)^7.
E.g.f.: (Sum_{k=1..6} S2(6,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>6, a(n) = Sum_{j=1..7} -binomial(j-8,j) * a(n-j). (End)
E.g.f.: x*(2 +34*x +91*x^2 +65*x^3 +15*x^4 +x^5)*exp(x)/2. - G. C. Greubel, Nov 15 2018
MATHEMATICA
Table[(n^6+n^3)/2, {n, 0, 40}] (* or *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 1, 36, 378, 2080, 7875, 23436}, 40] (* Harvey P. Dale, Nov 06 2011 *)
PROG
(Magma) [(n^6 + n^3)/2: n in [0..50]]; // Vincenzo Librandi, Jun 14 2011
(PARI) vector(50, n, n--; (n^6 + n^3)/2) \\ G. C. Greubel, Nov 15 2018
(Sage) [(n^6 + n^3)/2 for n in range(50)] # G. C. Greubel, Nov 15 2018
(GAP) List([0..50], n -> (n^6 + n^3)/2); # G. C. Greubel, Nov 15 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 11 2002
STATUS
approved