A004275 - OEIS

%I #54 Jun 17 2024 15:49:36

%S 0,1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,

%T 48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,

%U 94,96,98,100,102,104,106,108,110,112,114,116,118,120,122,124

%N 1 together with nonnegative even numbers.

%C A091090(a(n)) = 1. - _Reinhard Zumkeller_, Mar 13 2011

%C Base-4 analog of A031149: floor(n^2/4) is a square. - _M. F. Hasler_, Jan 15 2012

%C From _Eric M. Schmidt_, Jul 17 2017: (Start)

%C Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) and e(i) != e(k). [Martinez and Savage, 2.2]

%C Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) >= e(j) and e(i) != e(k). [Martinez and Savage, 2.2]

%C (End)

%H Vincenzo Librandi, <a href="/A004275/b004275.txt">Table of n, a(n) for n = 0..10000</a>

%H Megan A. Martinez and Carla D. Savage, <a href="https://arxiv.org/abs/1609.08106">Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations</a>, arXiv:1609.08106 [math.CO], 2016.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F G.f.: x*(1+x^2)/(1-x)^2. - _Paul Barry_, Feb 28 2003

%F a(n) = floor((2*n^2)/(1 + n)). - _Enrique Pérez Herrero_, Apr 05 2010

%F a(n) = 2n - 2 + floor(2/(n+1)) = max(n, 2n-2) = 2n - 1 + sgn(1-n). Also, a(0)=0, a(1)=1, a(n) = 2n-2 for n > 1. - _Wesley Ivan Hurt_, Nov 05 2013

%F E.g.f.: 2 + 2*exp(x)*(x - 1) + x. - _Stefano Spezia_, Jun 16 2024

%p A004275:= n-> 2*n - 2 + floor(2/(n+1)); seq(A004275(k), k=0..100); # _Wesley Ivan Hurt_, Nov 05 2013

%t A004275[n_]:=Floor[(2 n^2)/(1 + n)]; (* _Enrique Pérez Herrero_, Apr 05 2010 *)

%t Insert[Range[0,110,2],1,2] (* _Harvey P. Dale_, Feb 27 2015 *)

%o (Magma) [Floor((2*n^2)/(1 + n)): n in [0..60] ]; // _Vincenzo Librandi_, Aug 19 2011

%o (Haskell)

%o a004275 n = 2 * n - 1 + signum (1 - n)

%o a004275_list = 0 : 1 : [2, 4 ..]  -- _Reinhard Zumkeller_, Dec 18 2013

%Y Cf. A004277.

%Y Range of A007457.

%K easy,nonn

%O 0,3

%A _N. J. A. Sloane_