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 A177251 #36 Apr 29 2024 09:30:47
%S A177251 1,1,2,5,22,114,697,4923,39612,357899,3588836,39556420,475392841,
%T A177251 6187284605,86701097310,1301467245329,20835850494474,354382860600678,
%U A177251 6381494425302865,121290065781743383,2426510081356069016,50969474697328055063,1121571023472780698152
%N A177251 Number of permutations of [n] having no adjacent 3-cycles, i.e., no cycles of the form (i, i+1, i+2).
%H A177251 Seiichi Manyama, <a href="/A177251/b177251.txt">Table of n, a(n) for n = 0..449</a>
%H A177251 R. A. Brualdi and Emeric Deutsch, <a href="http://arxiv.org/abs/1005.0781">Adjacent q-cycles in permutations</a>, arXiv:1005.0781 [math.CO], 2010.
%F A177251 a(n) = Sum_{j=0..floor(n/3)} (-1)^j*(n-2*j)!/j!.
%F A177251 a(n) = A177250(n,0).
%F A177251 a(n) - n*a(n-1) = 2*a(n-3) + 3*(-1)^(n/3) if 3 | n, otherwise a(n) - n*a(n-1) = 2*a(n-3).
%F A177251 lim_{n -> oo} a(n)/n! = 1.
%F A177251 The o.g.f. g(z) satisfies z^2*(1+z^3)*g'(z) - (1+z^3)(1-z-2z^3)g(z) + 1 - 2z^3 = 0; g(0)=1.
%F A177251 G.f.: hypergeometric2F0([1,1], [], x/(1+x^3))/(1+x^3). - _Mark van Hoeij_, Nov 08 2011
%F A177251 D-finite with recurrence a(n) = n*a(n-1) + a(n-3) + (n-3)*a(n-4) + 2*a(n-6). - _R. J. Mathar_, Jul 26 2022
%F A177251 G.f.: Sum_{k>=0} k! * x^k / (1+x^3)^(k+1). - _Seiichi Manyama_, Feb 20 2024
%e A177251 a(4)=22 because the only permutations of {1,2,3,4} having adjacent 3-cycles are (123)(4) and (1)(234).
%p A177251 a := proc (n) options operator, arrow: sum((-1)^j*factorial(n-2*j)/factorial(j), j = 0 .. floor((1/3)*n)) end proc: seq(a(n), n = 0 .. 22);
%t A177251 a[n_] := Sum[(-1)^j*(n - 2*j)!/j!, {j, 0, n/3}];
%t A177251 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Nov 17 2017 *)
%o A177251 (Magma)
%o A177251 A177251:= func< n | (&+[(-1)^j*Factorial(n-2*j)/Factorial(j): j in [0..Floor(n/3)]]) >;
%o A177251 [A177251(n): n in [0..30]]; // _G. C. Greubel_, Apr 28 2024
%o A177251 (SageMath)
%o A177251 def A177251(n): return sum((-1)^j*factorial(n-2*j)/factorial(j) for j in range(1+n//3))
%o A177251 [A177251(n) for n in range(31)] # _G. C. Greubel_, Apr 28 2024
%Y A177251 Cf. A000166, A008290, A177248, A177249, A177250, A177252, A177253.
%Y A177251 Cf. A370324, A370569.
%K A177251 nonn
%O A177251 0,3
%A A177251 _Emeric Deutsch_, May 07 2010
%E A177251 Crossreferences corrected by _Emeric Deutsch_, May 09 2010