A181133 - OEIS

%I #19 Feb 10 2023 14:47:13

%S 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 32,33,35,36,37,38,39,40,41,42,44,45,46,47,48,49,50,51,52,54,55,56,57,

%U 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 a(n) = n + A003056(n).

%C Obtained starting with a triangle with 1's and a trailing 2, and accumulating a partial sum along rows and columns:

%C 2; # 2

%C 1,2; # 3,5

%C 1,1,2; # 6,7,9

%C 1,1,1,2; # 10,11,12,14

%C 1,1,1,1,2; # 15,16,17,18,20

%C 1,1,1,1,1,2;

%H Robert Israel, <a href="/A181133/b181133.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 2 + Sum_{k=1..n-1} A042974(k). - _R. J. Mathar_, Oct 08 2010

%F 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 A003056:= [seq(n$(n+1),n=1..20)]:

%p A003056+[$1..nops(A003056)]; # _Robert Israel_, Dec 24 2017

%t Array[# + Floor[(Sqrt[1 + 8 #] - 1)/2] &, 74] (* _Michael De Vlieger_, Dec 24 2017 *)

%t Accumulate[Flatten[Table[Join[PadRight[{},n,1],{2}],{n,0,15}]]] (* _Harvey P. Dale_, Aug 14 2022 *)

%o (PARI) a(n) = n + floor((sqrt(1+8*n)-1)/2) \\ _Iain Fox_, Dec 25 2017

%o (Python)

%o from math import isqrt

%o def A181133(n): return n+(isqrt((n<<3)+1)-1>>1) # _Chai Wah Wu_, Feb 10 2023

%Y Cf. A003056, A042974.

%K nonn,easy

%O 1,1

%A Craig Michoski (michoski(AT)google.com), Oct 05 2010

%E Definition re-fitted to something precise, sequence extended beyond a(15), and comment added by _R. J. Mathar_, Oct 24 2010