OFFSET
1,2
COMMENTS
a(n) = x(n) + y(n) where x(n)/y(n) is the continued fraction [1,2,3,4,...,n].
Using a(n) = x(n) - y(n) instead of a(n) = x(n) + y(n) would give A058307.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..449
Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020.
Zhicong Lin and Sherry H. F. Yan, Vincular patterns in inversion sequences, Applied Mathematics and Computation (2020), Vol. 364, 124672.
Shuzhen Lv and Philip B. Zhang, Joint equidistributions of mesh patterns 123 and 321 with symmetric and antipodal shadings, arXiv:2501.00357 [math.CO], 2024. See p. 17.
FORMULA
From Paul D. Hanna, Oct 31 2006: (Start)
a(n+1) = Sum_{k=0..n} k!*C(floor((n+k)/2),k)*C(floor((n+k+1)/2),k).
a(n+1) = Sum_{k=0..n} k!*A124428(n+k,k). (End)
MATHEMATICA
a[1]= 1; a[2]= 2; a[n_]:= a[n] = (n-1)*a[n-1]+a[n-2]; Table[a[n], {n, 20}] (* Robert G. Wilson v, Feb 14 2005 *)
RecurrenceTable[{a[1]==1, a[2]==2, a[n+1]==n*a[n]+a[n-1]}, a, {n, 20}] (* Harvey P. Dale, Sep 04 2018 *)
PROG
(PARI) a(n)=sum(k=0, n, k!*binomial((n+k)\2, k)*binomial((n+k+1)\2, k)) \\ Paul D. Hanna, Oct 31 2006
(Magma) I:=[1, 2]; [n le 2 select I[n] else (n-1)*Self(n-1) +Self(n-2): n in [1..30]]; // G. C. Greubel, Feb 23 2019
(Sage) [sum(factorial(k)*binomial(floor((n+k-1)/2), k)*binomial(floor((n+k)/2), k) for k in (0..n)) for n in (1..30)] # G. C. Greubel, Feb 23 2019
(GAP) a:=[1, 2];; for n in [3..30] do a[n]:=(n-1)*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Feb 23 2019
CROSSREFS
KEYWORD
base,easy,nonn
AUTHOR
Eric Angelini, Feb 12 2005
EXTENSIONS
Edited and extended by Robert G. Wilson v, Feb 14 2005
STATUS
approved