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 A111063 a(0) = 1; a(n) = (n-1)*a(n-1) + n. 6
 1, 1, 3, 9, 31, 129, 651, 3913, 27399, 219201, 1972819, 19728201, 217010223, 2604122689, 33853594971, 473950329609, 7109254944151, 113748079106433, 1933717344809379, 34806912206568841, 661331331924807999, 13226626638496160001, 277759159408419360043 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Frank Ruskey, Nov 24 2009: (Start) If the initial 1 were deleted, the recurrence relation becomes a(n) = n+1+n*a(n-1) with a(0) = 1. Furthermore, it can then be shown that a(n) is the number of nonempty subsets of binary strings with n 1's and 2 0's that are closed under the operation of replacing the leftmost 01 with 10. Taking the maximal elements under this relation, a(2) = 9 = |{0011},{0101},{1001},{1010},{1100},{0110}, {0110,1001},{0101,0110},{0011,0110}|. We also have the e.g.f. (1+x)/(1-x) e^x and the formula a(n) = -1 + 2*n!*sum_{j=0..n} 1/j!. (End) a(n+1) = sum of n-th row in triangle A245334. - Reinhard Zumkeller, Aug 30 2014 [A-number corrected by N. J. A. Sloane, May 03 2017] Eigensequence of triangle with (1, 2, 3, ...) as the right and left borders and the rest zeros. - Gary W. Adamson, Aug 01 2016 The following remarks apply to the sequence without the initial term a(0) = 1: For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k. It follows that for each k the sequence a(n) taken modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes 1, 3, 9, 1, 9, 1, 3, 9, 1, 9, ... with period 5. Cf. A000522. - Peter Bala, Nov 19 2017 REFERENCES F. Drewes et al., Tight Bounds for Cut-Operations on Deterministic Finite Automata, in Lecture Notes in Computer Science, Volume 9288 2015, Machines, Computations, and Universality, 7th International Conference, MCU 2015, Famagusta, North Cyprus, September 9-11, 2015, Editors: Jerome Durand-Lose, Benedek Nagy, ISBN: 978-3-319-23110-5 (Print) 978-3-319-23111-2 (Online). ["In the On-Line Encyclopedia of Integer Sequences (OEIS) this matches the sequence A111063."] LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..400 FORMULA a(n+1) = Sum_{k=0..2*n} C(n,floor(k/2))(n-floor(k/2))!. - Paul Barry, May 04 2007 a(n) = A030297(n)/n, n>0. a(n) = A007526(n) + A000522(n). - Gary Detlefs, Jun 10 2010 a(n) = 2*floor(e*n!) - 1, n>1. - Gary Detlefs, Jun 10 2010 E.g.f.: exp(x)*(1+x)/(1-x), - N. J. A. Sloane, May 03 2017 a(n) ~ 2*sqrt(2*Pi)*exp(1)*n^n*sqrt(n)/exp(n). - Ilya Gutkovskiy, Aug 02 2016 a(n) = 2*exp(1)*GAMMA(n, 1) - 1 for n>=1. - Peter Luschny, Nov 21 2017 MAPLE a:=proc(n) option remember; if n=0 then RETURN(1); fi; (n-1)*a(n-1)+n; end; # Alternatively: a := n -> `if`(n=0, 1, 2*exp(1)*GAMMA(n, 1) - 1): seq(simplify(a(n)), n=0..22); # Peter Luschny, Nov 21 2017 MATHEMATICA FoldList[#1*#2 + #2 + 1 &, 1, Range] (* Robert G. Wilson v, Jul 07 2012 *) PROG (Haskell) a111063 n = a111063_list !! n a111063_list = 1 : zipWith (+) [1..] (zipWith (*) [0..] a111063_list) -- Reinhard Zumkeller, Aug 30 2014 CROSSREFS Cf. A000522, A007526, A030297, A245334, A240993. Sequence in context: A349970 A090595 A027040 * A245116 A255382 A089475 Adjacent sequences:  A111060 A111061 A111062 * A111064 A111065 A111066 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Oct 08 2005 STATUS approved

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Last modified July 3 05:09 EDT 2022. Contains 355030 sequences. (Running on oeis4.)