A071235 - OEIS

%I #26 Sep 08 2022 08:45:06

%S 0,1,2080,266085,8390656,122078125,1088414496,6920702425,34359869440,

%T 141215033961,500000500000,1569215074141,4458051717120,11649044974645,

%U 28346959952416,64873174640625,140737496743936,291311130683665,578415707719200,1106657483056021

%N a(n) = (n^12 + n^6)/2.

%C Number of unoriented rows of length 12 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)=2080, there are 2^12=4096 oriented arrangements of two colors. Of these, 2^6=64 are achiral. That leaves (4096-64)/2=2016 chiral pairs. Adding achiral and chiral, we get 2080. - _Robert A. Russell_, Nov 13 2018

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

%H Vincenzo Librandi, <a href="/A071235/b071235.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

%F a(n) = n^6*(n^2 + 1)*(n^4 - n^2 + 1)/2.

%F From _Robert A. Russell_, Nov 13 2018: (Start)

%F a(n) = (A008456(n) + A001014(n)) / 2 = (n^12 + n^6) / 2.

%F G.f.: (Sum_{j=1..12} S2(12,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..6} S2(6,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.

%F G.f.: x*Sum_{k=0..11} A145882(12,k) * x^k / (1-x)^13.

%F E.g.f.: (Sum_{k=1..12} S2(12,k)*x^k + Sum_{k=1..6} S2(6,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.

%F For n>12, a(n) = Sum_{j=1..13} -binomial(j-14,j) * a(n-j). (End)

%F From _G. C. Greubel_, Nov 15 2018: (Start)

%F G.f.: x*(1 +2067*x +239123*x^2 +5093505*x^3 +33160062*x^4 + 81255642*x^5 +81255642*x^6 +33160062*x^7 +5093505*x^8 +239123*x^9 +2067*x^10 +x^11)/( 1-x)^13.

%F E.g.f.: x*(2 +2078*x +86616*x^2 +611566*x^3 +1379415*x^4 +*1323653*x^5 + 627396*x^6 +159027*x^7 +22275*x^8 +1705*x^9 +66*x^10 +x^11)*exp(x)/2. (End)

%t Table[(n^12 + n^6)/2, {n,0,30}] (* _Robert A. Russell_, Nov 13 2018 *)

%o (Magma) [n^6*(n^2+1)*(n^4-n^2+1)/2: n in [0..40]]; // _Vincenzo Librandi_, Jun 14 2011

%o (PARI) vector(40, n, n--; ) \\ _G. C. Greubel_, Nov 15 2018

%o (Sage) [n^6*(1 + n^6)/2 for n in range(40)] # _G. C. Greubel_, Nov 15 2018

%o (GAP) List([0..40], n -> (n^12 + n^6)/2); # _G. C. Greubel_, Nov 15 2018

%o (Python) for n in range(0,20): print(int((n**12 + n**6)/2), end=', ') # _Stefano Spezia_, Nov 15 2018

%Y Row 12 of A277504.

%Y Cf. A008456 (oriented), A001014 (achiral).

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Jun 12 2002

%E New name from _G. C. Greubel_, Nov 15 2018