A183293 Complement of A056000.

1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89

Offset

1,2

Links

Table of n, a(n) for n=1..80.

Formula

(See the Mathematica code.)

From Chai Wah Wu, Nov 05 2025: (Start)

a(n) = n+m-4 if 2*n <= (m-3)*(m+4) and a(n) = n+m-3 otherwise, where m = floor(sqrt(8*n+49)/2).

In general, the complement of the sequence b(n) = n*(n+2*k+1)/2 is given by: a(n) = n+m-k if n <= (m-k+1)*(m+k)/2 and a(n) = n+m-k+1 otherwise, where m = floor(sqrt(8*n+(2k-1)^2)/2). (End)

Mathematica

a=1/2; b=9/2;
F[n_]:=a*n^2+b*n;
R[n_]:=(n/a+((b-1)/(2a))^2)^(1/2);
G[n_]:=n-1+Ceiling[R[n]-(b-1)/(2a)];
Table[F[n], {n, 60}]
Table[G[n], {n, 100}]

Prog

(Python)
from math import isqrt
def A183293(n): return n-4+(m:=isqrt((n<<3)+49)>>1)+((n<<1)>(m-3)*(m+4)) # Chai Wah Wu, Nov 05 2025

Crossrefs

Cf. A056000.

Sequence in context: A175970 A286689 A278373 * A184524 A047226 A059537

Adjacent sequences: A183290 A183291 A183292 * A183294 A183295 A183296

Keyword

nonn

Author

Clark Kimberling, Jan 03 2011

Status

approved