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A181133 a(n) = n + A003056(n).

%I A181133 #19 Feb 10 2023 14:47:13
%S A181133 2,3,5,6,7,9,10,11,12,14,15,16,17,18,20,21,22,23,24,25,27,28,29,30,31,
%T A181133 32,33,35,36,37,38,39,40,41,42,44,45,46,47,48,49,50,51,52,54,55,56,57,
%U A181133 58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74,75,77,78,79,80,81,82,83,84,85
%N A181133 a(n) = n + A003056(n).
%C A181133 Obtained starting with a triangle with 1's and a trailing 2, and accumulating a partial sum along rows and columns:
%C A181133 2; # 2
%C A181133 1,2; # 3,5
%C A181133 1,1,2; # 6,7,9
%C A181133 1,1,1,2; # 10,11,12,14
%C A181133 1,1,1,1,2; # 15,16,17,18,20
%C A181133 1,1,1,1,1,2;
%H A181133 Robert Israel, <a href="/A181133/b181133.txt">Table of n, a(n) for n = 1..10000</a>
%F A181133 a(n) = 2 + Sum_{k=1..n-1} A042974(k). - _R. J. Mathar_, Oct 08 2010
%F A181133 G.f.: (2*x-1)/(1-x)^2 + Theta_2(0,sqrt(x))/(x^(1/8)*(2-2*x)) where Theta_2 is a Jacobi theta function. - _Robert Israel_, Dec 24 2017
%p A181133 A003056:= [seq(n$(n+1),n=1..20)]:
%p A181133 A003056+[$1..nops(A003056)]; # _Robert Israel_, Dec 24 2017
%t A181133 Array[# + Floor[(Sqrt[1 + 8 #] - 1)/2] &, 74] (* _Michael De Vlieger_, Dec 24 2017 *)
%t A181133 Accumulate[Flatten[Table[Join[PadRight[{},n,1],{2}],{n,0,15}]]] (* _Harvey P. Dale_, Aug 14 2022 *)
%o A181133 (PARI) a(n) = n + floor((sqrt(1+8*n)-1)/2) \\ _Iain Fox_, Dec 25 2017
%o A181133 (Python)
%o A181133 from math import isqrt
%o A181133 def A181133(n): return n+(isqrt((n<<3)+1)-1>>1) # _Chai Wah Wu_, Feb 10 2023
%Y A181133 Cf. A003056, A042974.
%K A181133 nonn,easy
%O A181133 1,1
%A A181133 Craig Michoski (michoski(AT)google.com), Oct 05 2010
%E A181133 Definition re-fitted to something precise, sequence extended beyond a(15), and comment added by _R. J. Mathar_, Oct 24 2010