OFFSET
0,2
COMMENTS
a(n+1) gives index of the first occurrence of n in A100795. - Amarnath Murthy, Dec 05 2004
First term in each group in A074148. - Amarnath Murthy, Aug 28 2002
From Christian Barrientos, Jan 01 2021: (Start)
For n >= 3, a(n) is the number of square polyominoes with at least 2n - 2 cells whose bounding box has size 2 X n.
For n = 3, there are 6 square polyominoes with a bounding box of size 2 X 3:
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_|_|_| |_|_|_| |_|_|_| |_|_|_| |_|_|_| |_|_|_
|_|_|_| |_|_| |_| |_| |_| |_| |_|_|
(End)
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..1000
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
a(2*n) = a(2*n-1) + 2*n - 1 = 2*n^2 + 1 = A058331(n).
a(2*n+1) = a(2*n) + 2*n + 1 = 2*(n^2 + n + 1) = A051890(n+1).
a(n) = floor((n^2+3)/2). - Gary Detlefs, Feb 13 2010
From R. J. Mathar, Feb 19 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1-x^2+2*x^3)/((1+x) * (1-x)^3). (End)
a(n) = (2*n^2 - (-1)^n + 5)/4. - Bruno Berselli, Sep 29 2011
a(n) = A007590(n+1) - n + 1. - Wesley Ivan Hurt, Jul 15 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x))/2. - Stefano Spezia, May 07 2021
MAPLE
seq(floor((n^2+3)/2), n=0..25); # Gary Detlefs, Feb 13 2010
MATHEMATICA
Table[Ceiling[n^2/2]+1, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 2, 3, 6}, 60] (* Harvey P. Dale, Jan 03 2024 *)
PROG
(PARI) { for (n=0, 1000, write("b061925.txt", n, " ", ceil(n^2/2) + 1) ) } \\ Harry J. Smith, Jul 29 2009
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, May 17 2001
EXTENSIONS
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 09 2007
STATUS
approved