A005945 - OEIS

%I M4966 #49 Sep 08 2022 08:44:34

%S 0,1,15,60,154,315,561,910,1380,1989,2755,3696,4830,6175,7749,9570,

%T 11656,14025,16695,19684,23010,26691,30745,35190,40044,45325,51051,

%U 57240,63910,71079,78765,86986,95760,105105,115039,125580,136746

%N Number of n-step mappings with 4 inputs.

%C a(n) is the coefficient of x^4/4! in n-th iteration of exp(x)-1.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A005945/b005945.txt">Table of n, a(n) for n = 0..1000</a>

%H T. Hogg and B. A. Huberman, <a href="https://doi.org/10.1103/PhysRevA.32.2338">Attractors on finite sets: the dissipative dynamics of computing structures</a>, Phys. Review A 32 (1985), 2338-2346.

%H T. Hogg and B. A. Huberman, <a href="/A000258/a000258.pdf">Attractors on finite sets: the dissipative dynamics of computing structures</a>, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)

%H B. A. Huberman, T. H. Hogg, & N. J. A. Sloane, <a href="/A005945/a005945.pdf">Correspondence, 1985</a>

%H Pierpaolo Natalini, Paolo E. Ricci, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Ricci/ricci3.html">Integer Sequences Connected with Extensions of the Bell Polynomials</a>, Journal of Integer Sequences, 2017, Vol. 20, #17.10.2.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: x*(1+11*x+6*x^2)/(1-x)^4. a(n)=n*(3*n-1)*(2*n-1)/2.

%F For n>0, a(n) = n*A000567(n) - A000217(n-1). - _Bruno Berselli_, Apr 25 2010; Feb 01 2011

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 18 2012

%F a(n) = -A094952(-n) for all n in Z. - _Michael Somos_, Jan 23 2014

%e G.f. = x + 15*x^2 + 60*x^3 + 154*x^4 + 315*x^5 + 561*x^6 + 910*x^7 + ...

%t LinearRecurrence[{4,-6,4,-1},{0,1,15,60},50] (* _Vincenzo Librandi_, Jun 18 2012 *)

%t a[ n_] := 3 n^3 - 5/2 n^2 + 1/2 n; (* _Michael Somos_, Jun 10 2015 *)

%o (PARI) {a(n) = 3*n^3 - 5/2*n^2 + 1/2*n}; /* _Michael Somos_, Jan 23 2014 */

%o (Magma) I:=[0, 1, 15, 60]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // _Vincenzo Librandi_ Jun 18 2012

%Y Cf. for recursive method [Ar(m) is the m-th term of a sequence in the OEIS] a(n) = n*Ar(n) - A000217(n-1) or a(n) = (n+1)*Ar(n+1) - A000217(n) or similar: A081436, A005920, A006003 and the terms T(2, n) or T(3, n) in the sequence A125860. [_Bruno Berselli_, Apr 25 2010]

%Y Cf. A094952.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

%E Edited by _Michael Somos_, Oct 29 2002