A183137 -id:A183137 - cp's OEIS frontend

A183136 a(n) = [1/r]+[2/r]+...+[n/r], where r = golden ratio = (1+sqrt(5))/2 and []=floor.

0, 1, 2, 4, 7, 10, 14, 18, 23, 29, 35, 42, 50, 58, 67, 76, 86, 97, 108, 120, 132, 145, 159, 173, 188, 204, 220, 237, 254, 272, 291, 310, 330, 351, 372, 394, 416, 439, 463, 487, 512, 537, 563, 590, 617, 645, 674, 703, 733, 763, 794, 826, 858

Offset: 1

Clark Kimberling, Dec 26 2010

nonn

a(n) + A183137(n) = A000217(n) (the triangular numbers).

			The terms [k/r] are given by A060143 (and A005206): 0,1,1,2,3,3,4,4,5,6,6,7,8,8,...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Hector Zenil, N. Kiani, and J. Tegner, Low Algorithmic Complexity Entropy-deceiving Graphs, arXiv preprint arXiv:1608.05972 [cs.IT], 2016-2017.

Cf. A183137, A000217, A005206, A060143, A060144.

Accumulate[Floor[Range[100]/GoldenRatio]] (* Paolo Xausa, Jun 20 2025 *)

default(realprecision,100); r=(1+sqrt(5))/2; for(n=1,99, print1(sum(k=1,n,floor(k/r)), ", "))

from math import isqrt
def A183136(n): return sum(isqrt(5*i**2)-i>>1 for i in range(1,n+1)) # Chai Wah Wu, Jun 20 2025

a(n) = [1/r]+[2/r]+...+[n/r], where r = golden ratio = (1+sqrt(5))/2 and []=floor.

A258376 Number of edges connecting the subgraph on {1, ..., n} with the complement in the minimal graph on positive natural numbers where degree(n) equals n.

Original entry on oeis.org

1, 1, 2, 4, 5, 7, 8, 10, 13, 15, 18, 22, 25, 29, 32, 36, 41, 45, 50, 54, 59, 65, 70, 76, 83, 89, 96, 102, 109, 117, 124, 132, 141, 149, 158, 166, 175, 185, 194, 204, 213, 223, 234, 244, 255, 267, 278, 290, 301, 313, 326, 338, 351, 363, 376, 390, 403, 417, 432

Offset: 1

John Furey, May 28 2015

nonn

A graph can be constructed using each of the numbers n as vertices wherein the degree of each vertex is itself, i.e. the number n corresponds to the unique vertex of degree n. The minimal such simple graph is defined here to be when each number is maximally connected to smaller numbers. In that case, provably each number is connected to the next A005206(n) (Hofstadter G-sequence) greater numbers, e.g. 5 is connected to the next three greater numbers 6, 7, and 8, and 5 is also connected to the two smaller numbers 3 and 4. During bottom-up construction of the full graph, the order of the finite subgraph upon addition of each vertex n is obviously n, and the size of this subgraph is provably A183137(n). This subgraph has a(n) connections to the rest of the full graph.

			Following along bottom-up construction, the natural number 1 only connects to 2, so a(1) = 1. The subgraph comprising 1 and 2 only connects to 3, so a(2) = 1. 3 also connects to 4 and 5, so a(3) = 2. The three (Hofstadter G) larger connections of 4 and the one remaining larger connection of 3 yield a(4) = 4.

John Furey, Table of n, a(n) for n = 1..1000

Cf. A005206, A183137.

a(n) = Sum_{i=1..n} max(0,A005206(i)-n+i). - Alois P. Heinz, Jun 01 2015

cp's OEIS Frontend

A183136 a(n) = [1/r]+[2/r]+...+[n/r], where r = golden ratio = (1+sqrt(5))/2 and []=floor.

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