This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A071232 #45 Mar 11 2025 15:42:55
%S A071232 0,1,36,378,2080,7875,23436,58996,131328,266085,500500,886446,1493856,
%T A071232 2414503,3766140,5697000,8390656,12071241,17009028,23526370,32004000,
%U A071232 42887691,56695276,74024028,95558400,122078125,154466676,193720086,240956128,297423855,364513500
%N A071232 a(n) = (n^6 + n^3)/2.
%C A071232 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 A071232 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 A071232 C. Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60 (2001), 85-96.
%D A071232 T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
%D A071232 T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
%H A071232 Vincenzo Librandi, <a href="/A071232/b071232.txt">Table of n, a(n) for n = 0..2000</a>
%H A071232 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A071232 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 A071232 G.f.: x*(28*x^4 + 155*x^3 + 147*x^2 + 29*x + 1)/(1-x)^7. - _Colin Barker_, Oct 12 2012
%F A071232 From _Robert A. Russell_, Nov 14 2018: (Start)
%F A071232 a(n) = (A001014(n) + A000578(n)) / 2 = (n^6 + n^3) / 2.
%F A071232 a(n) = A001014(n) - A085744(n) = A085744(n) + A000578(n).
%F A071232 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 A071232 G.f.: x*Sum_{k=0..5} A145882(6,k) * x^k / (1-x)^7.
%F A071232 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 A071232 For n>6, a(n) = Sum_{j=1..7} -binomial(j-8,j) * a(n-j). (End)
%F A071232 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
%F A071232 a(n) = A000217(n^3), sum of the integers up to the n'th cube. - _R. J. Mathar_, Mar 11 2025
%t A071232 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 A071232 (Magma) [(n^6 + n^3)/2: n in [0..50]]; // _Vincenzo Librandi_, Jun 14 2011
%o A071232 (PARI) vector(50, n, n--; (n^6 + n^3)/2) \\ _G. C. Greubel_, Nov 15 2018
%o A071232 (Sage) [(n^6 + n^3)/2 for n in range(50)] # _G. C. Greubel_, Nov 15 2018
%o A071232 (GAP) List([0..50], n -> (n^6 + n^3)/2); # _G. C. Greubel_, Nov 15 2018
%Y A071232 Row 6 of A277504.
%Y A071232 Cf. A001014 (oriented), A085744 (chiral), A000578 (achiral).
%K A071232 nonn,easy
%O A071232 0,3
%A A071232 _N. J. A. Sloane_, Jun 11 2002