%I #79 Sep 08 2022 08:45:56
%S 7,22,53,106,187,302,457,658,911,1222,1597,2042,2563,3166,3857,4642,
%T 5527,6518,7621,8842,10187,11662,13273,15026,16927,18982,21197,23578,
%U 26131,28862,31777,34882,38183,41686,45397,49322,53467,57838,62441,67282,72367
%N a(n) = n^3 - 4n^2 + 6n - 2.
%C Number of nilpotent elements in the identity difference partial one - one transformation semigroup, denoted by N(IDI_n). For n=3, #N(IDI_n) = 7.
%C a(n+1) is also the Moore lower bound on the order of an (n,7)-cage. - _Jason Kimberley_, Oct 20 2011
%H Vincenzo Librandi, <a href="/A188377/b188377.txt">Table of n, a(n) for n = 3..1000</a>
%H A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1081/AGB-120038637">On the number of nilpotents in the partial symmetric semigroup</a>, Comm. Algebra 32 (2004), 3017-3023.
%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/cages/">Cages of higher valency</a>
%H R. P. Sullivan, <a href="https://doi.org/10.1016/0021-8693(87)90049-4">Semigroups generated by nilpotent transformations</a>, Journal of Algebra 110 (1987), 324-345.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n+1) = (n+1)^3 - 4*(n+1)^2 + 6*(n+1) - 2
%F = (n-1)^3 + 2*(n-1)^2 + 2*(n-1) + 2
%F = 1222 read in base n-1.
%F - _Jason Kimberley_, Oct 20 2011
%F From _Colin Barker_, Apr 06 2012: (Start)
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F G.f.: x^3*(7 - 6*x + 7*x^2 - 2*x^3)/(1-x)^4. (End)
%F E.g.f.: 2 - x - x^2 + exp(x)*(x^3 - x^2 + 3*x - 2). - _Stefano Spezia_, Apr 09 2022
%t Table[n^3 - 4*n^2 + 6*n - 2, {n, 3, 80}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 07 2011 *)
%t LinearRecurrence[{4,-6,4,-1},{7,22,53,106},50] (* _Harvey P. Dale_, May 29 2019 *)
%o (Magma) [n^3 - 4*n^2 + 6*n - 2: n in [3..50]]; // _Vincenzo Librandi_, May 01 2011
%o (Magma) [SequenceToInteger([2^^3,1],n-2):n in [5..50]]; // _Jason Kimberley_, Oct 20 2011
%o (PARI) a(n)=n^3-4*n^2+6*n-2 \\ _Charles R Greathouse IV_, Apr 06 2012
%Y Cf. A188716, A188947.
%Y Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), this sequence (g=7). - _Jason Kimberley_, Oct 30 2011
%K nonn,easy
%O 3,1
%A _Adeniji, Adenike_ & Makanjuola, Samuel (somakanjuola(AT)unilorin.edu.ng) Apr 14 2011
%E Edited by _N. J. A. Sloane_, Apr 23 2011