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A184005 a(n) = n - 1 + ceiling(3*n^2/4); complement of A184004.

%I A184005 #54 Oct 01 2024 04:19:39
%S A184005 1,4,9,15,23,32,43,55,69,84,101,119,139,160,183,207,233,260,289,319,
%T A184005 351,384,419,455,493,532,573,615,659,704,751,799,849,900,953,1007,
%U A184005 1063,1120,1179,1239,1301,1364,1429,1495,1563,1632,1703,1775,1849,1924,2001,2079,2159,2240,2323,2407,2493,2580,2669,2759
%N A184005 a(n) = n - 1 + ceiling(3*n^2/4); complement of A184004.
%H A184005 G. C. Greubel, <a href="/A184005/b184005.txt">Table of n, a(n) for n = 1..5000</a>
%H A184005 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A184005 a(n) = 2*a(n-1) - 2*a(n-3) + 1*a(n-4).
%F A184005 From _Bruno Berselli_, Jan 25 2011: (Start)
%F A184005 G.f.: x*(1 + 2*x + x^2 - x^3)/((1 + x)*(1 - x)^3).
%F A184005 a(n) = (6*n^2 + 8*n - (-1)^n - 7)/8. (End)
%F A184005 a(n) = round((6*n^2 + 8*n - 7)/8). - _Bruno Berselli_, Jan 25 2011
%F A184005 From _Paul Curtz_, Feb 09 2011: (Start)
%F A184005 a(n) - a(n-1)  = A007494(n).
%F A184005 a(n) - a(n-2)  = 3*n - 1 = A016789(n-1).
%F A184005 a(n) - a(n-4)  = 6*n - 8 = A016957(n-2).
%F A184005 a(n) - a(n-8)  = 12*n - 40 = A017617(n-4).
%F A184005 a(n) - a(n-16) = 24*n - 176 = 8*A016789(n-8).
%F A184005 a(n) - a(n-32) = 48*n - 736 = 16*A016789(n-16). (End)
%F A184005 a(n) = n^2 - floor((n-2)^2/4). - _Bruno Berselli_, Jan 17 2017
%F A184005 a(n) = A002061(n+2) - A002620(n+4). - _Anton Zakharov_, May 17 2017
%F A184005 E.g.f.: (1/8)*(8 + (6*x^2 + 14*x -7)*exp(x) - exp(-x)). - _G. C. Greubel_, Jul 22 2017
%t A184005 a=4/3; b=0;
%t A184005 Table[n+Floor[(a*n+b)^(1/2)],{n,80}]
%t A184005 Table[n-1+Ceiling[(n*n-b)/a],{n,60}]
%t A184005 Table[n - 1 + Ceiling[3 n^2/4], {n, 60}] (* or *) CoefficientList[ Series[x (1 + 2 x + x^2 - x^3)/((1 + x) (1 - x)^3), {x, 0, 60}], x] (* or *) Table[Round[(6 n^2 + 8 n - 7)/8], {n, 60}] (* _Michael De Vlieger_, Mar 23 2016 *)
%t A184005 LinearRecurrence[{2,0,-2,1},{1,4,9,15},60] (* _Harvey P. Dale_, Sep 16 2016 *)
%o A184005 (Magma) [(6*n^2 + 8*n - (-1)^n - 7)/8: n in [1..80]]; // _Vincenzo Librandi_, Feb 09 2011
%o A184005 (PARI) my(x='x+O('x^200)); Vec(x*(1+2*x+x^2-x^3)/((1+x)*(1-x)^3)) \\ _Altug Alkan_, Mar 23 2016
%o A184005 (Python)
%o A184005 def A184005(n): return n+((m:=3*n**2)>>2)-(not m&3) # _Chai Wah Wu_, Oct 01 2024
%Y A184005 Cf. A184004.
%K A184005 nonn,easy
%O A184005 1,2
%A A184005 _Clark Kimberling_, Jan 08 2011