A071232 - OEIS

%I #43 Sep 08 2022 08:45:06

%S 0,1,36,378,2080,7875,23436,58996,131328,266085,500500,886446,1493856,

%T 2414503,3766140,5697000,8390656,12071241,17009028,23526370,32004000,

%U 42887691,56695276,74024028,95558400,122078125,154466676,193720086,240956128,297423855,364513500

%N a(n) = (n^6 + n^3)/2.

%C 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

%C 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

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

%D T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

%D T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

%H Vincenzo Librandi, <a href="/A071232/b071232.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F 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

%F G.f.: x*(28*x^4 + 155*x^3 + 147*x^2 + 29*x + 1)/(1-x)^7. - _Colin Barker_, Oct 12 2012

%F From _Robert A. Russell_, Nov 14 2018: (Start)

%F a(n) = (A001014(n) + A000578(n)) / 2 = (n^6 + n^3) / 2.

%F a(n) = A001014(n) - A085744(n) = A085744(n) + A000578(n).

%F 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.

%F G.f.: x*Sum_{k=0..5} A145882(6,k) * x^k / (1-x)^7.

%F 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.

%F For n>6, a(n) = Sum_{j=1..7} -binomial(j-8,j) * a(n-j). (End)

%F 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

%t 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 *)

%o (Magma) [(n^6 + n^3)/2: n in [0..50]]; // _Vincenzo Librandi_, Jun 14 2011

%o (PARI) vector(50, n, n--; (n^6 + n^3)/2) \\ _G. C. Greubel_, Nov 15 2018

%o (Sage) [(n^6 + n^3)/2 for n in range(50)] # _G. C. Greubel_, Nov 15 2018

%o (GAP) List([0..50], n -> (n^6 + n^3)/2); # _G. C. Greubel_, Nov 15 2018

%Y Row 6 of A277504.

%Y Cf. A001014 (oriented), A085744 (chiral), A000578 (achiral).

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Jun 11 2002