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 A168178 #36 Sep 08 2022 08:45:48
%S A168178 0,1,20,135,544,1625,3996,8575,16640,29889,50500,81191,125280,186745,
%T A168178 270284,381375,526336,712385,947700,1241479,1604000,2046681,2582140,
%U A168178 3224255,3988224,4890625,5949476,7184295,8616160,10267769,12163500
%N A168178 a(n) = n^3*(n^2 + 1)/2.
%C A168178 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
%C A168178 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
%D A168178 C. Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60(2001), 85-96.
%H A168178 Vincenzo Librandi, <a href="/A168178/b168178.txt">Table of n, a(n) for n = 0..595</a>
%H A168178 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A168178 From _Robert A. Russell_, Nov 14 2018: (Start)
%F A168178 a(n) = (A000584(n) + A000578(n)) / 2 = (n^5 + n^3) / 2.
%F A168178 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.
%F A168178 G.f.: x*Sum_{k=0..4} A145882(5,k) * x^k / (1-x)^6.
%F A168178 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.
%F A168178 For n>5, a(n) = Sum_{j=1..6} -binomial(j-7,j) * a(n-j). (End)
%F A168178 From _G. C. Greubel_, Nov 15 2018: (Start)
%F A168178 G.f.: x*(1 + 14*x + 30*x^2 + 14*x^3 + x^4)/(1-x)^6.
%F A168178 E.g.f.: x*(2 + 18*x + 26*x^2 + 10*x^3 + x^4)*exp(x)/2. (End)
%t A168178 Table[(n^5+n^3)/2,{n,0,80}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 18 2011 *)
%t A168178 LinearRecurrence[{6, -15, 20, -15, 6, -1},{0, 1, 20, 135, 544, 1625},40] (* _Robert A. Russell_, Nov 14 2018 *)
%o A168178 (Magma) [n^3*(n^2+1)/2: n in [0..50]]; // _Vincenzo Librandi_, Apr 25 2011
%o A168178 (PARI) vector(40, n, n--; n^3*(1+n^2)/2) \\ _G. C. Greubel_, Nov 15 2018
%o A168178 (Sage) [n^3*(1 + n^2)/2 for n in range(40)] # _G. C. Greubel_, Nov 15 2018
%o A168178 (GAP) List([0..40], n -> n^3*(1 +n^2)/2); # _G. C. Greubel_, Nov 15 2018
%Y A168178 Cf. A155977.
%Y A168178 Row 5 of A277504.
%Y A168178 Cf. A000584 (oriented), A000578 (achiral).
%K A168178 nonn,easy
%O A168178 0,3
%A A168178 _N. J. A. Sloane_, Dec 11 2009