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

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

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

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

2; # 2

1,2; # 3,5

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

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

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

1,1,1,1,1,2;

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003056, A042974.

Programs

Maple

A003056:= [seq(n$(n+1),n=1..20)]:
A003056+[$1..nops(A003056)]; # Robert Israel, Dec 24 2017

Mathematica

Array[# + Floor[(Sqrt[1 + 8 #] - 1)/2] &, 74] (* Michael De Vlieger, Dec 24 2017 *)
Accumulate[Flatten[Table[Join[PadRight[{},n,1],{2}],{n,0,15}]]] (* Harvey P. Dale, Aug 14 2022 *)

PARI

a(n) = n + floor((sqrt(1+8*n)-1)/2) \\ Iain Fox, Dec 25 2017

Python

from math import isqrt
def A181133(n): return n+(isqrt((n<<3)+1)-1>>1) # Chai Wah Wu, Feb 10 2023

Formula

a(n) = 2 + Sum_{k=1..n-1} A042974(k). - R. J. Mathar, Oct 08 2010

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

Extensions

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