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%I #51 Jan 03 2024 12:16:28
%S 1,2,3,6,9,14,19,26,33,42,51,62,73,86,99,114,129,146,163,182,201,222,
%T 243,266,289,314,339,366,393,422,451,482,513,546,579,614,649,686,723,
%U 762,801,842,883,926,969,1014,1059,1106,1153,1202,1251,1302,1353,1406
%N a(n) = ceiling(n^2/2) + 1.
%C a(n+1) gives index of the first occurrence of n in A100795. - _Amarnath Murthy_, Dec 05 2004
%C First term in each group in A074148. - _Amarnath Murthy_, Aug 28 2002
%C From _Christian Barrientos_, Jan 01 2021: (Start)
%C 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.
%C For n = 3, there are 6 square polyominoes with a bounding box of size 2 X 3:
%C _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%C |_|_|_| |_|_|_| |_|_|_| |_|_|_| |_|_|_| |_|_|_
%C |_|_|_| |_|_| |_| |_| |_| |_| |_|_|
%C (End)
%H Harry J. Smith, <a href="/A061925/b061925.txt">Table of n, a(n) for n = 0..1000</a>
%H Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, <a href="https://arxiv.org/abs/2312.05145">Cyclic permutations avoiding patterns in both one-line and cycle forms</a>, arXiv:2312.05145 [math.CO], 2023. See p. 2.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F a(n) = a(n-1) + 2*floor((n-1)/2) + 1 = A061926(3, k) = 2*A002620(n+1) - (n-1) = A000982(n) + 1.
%F a(2*n) = a(2*n-1) + 2*n - 1 = 2*n^2 + 1 = A058331(n).
%F a(2*n+1) = a(2*n) + 2*n + 1 = 2*(n^2 + n + 1) = A051890(n+1).
%F a(n) = floor((n^2+3)/2). - _Gary Detlefs_, Feb 13 2010
%F From _R. J. Mathar_, Feb 19 2010: (Start)
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
%F G.f.: (1-x^2+2*x^3)/((1+x) * (1-x)^3). (End)
%F a(n) = (2*n^2 - (-1)^n + 5)/4. - _Bruno Berselli_, Sep 29 2011
%F a(n) = A007590(n+1) - n + 1. - _Wesley Ivan Hurt_, Jul 15 2013
%F a(n) + a(n+1) = A027688(n). a(n+1) - a(n) = A109613(n). - _R. J. Mathar_, Jul 20 2013
%F E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x))/2. - _Stefano Spezia_, May 07 2021
%p seq(floor((n^2+3)/2),n=0..25); # _Gary Detlefs_, Feb 13 2010
%t Table[Ceiling[n^2/2]+1,{n,0,60}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 02 2011 *)
%t LinearRecurrence[{2,0,-2,1},{1,2,3,6},60] (* _Harvey P. Dale_, Jan 03 2024 *)
%o (PARI) { for (n=0, 1000, write("b061925.txt", n, " ", ceil(n^2/2) + 1) ) } \\ _Harry J. Smith_, Jul 29 2009
%Y Cf. A100795, A074147-A074149.
%K nonn,easy
%O 0,2
%A _Henry Bottomley_, May 17 2001
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 09 2007
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