A283733 a(n) = a(n-1) + 1 + floor(n*golden ratio), with a(0) = 1.

1, 3, 7, 12, 19, 28, 38, 50, 63, 78, 95, 113, 133, 155, 178, 203, 229, 257, 287, 318, 351, 385, 421, 459, 498, 539, 582, 626, 672, 719, 768, 819, 871, 925, 981, 1038, 1097, 1157, 1219, 1283, 1348, 1415, 1483, 1553, 1625, 1698, 1773, 1850, 1928, 2008, 2089

Offset: 0

Author

Clark Kimberling, Mar 16 2017

Keywords

• nonn
• easy

Comments

This is column 1 of the rank array, A283734, of the golden ratio.

Links

• Clark Kimberling, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000201, A001622, A026351 (first differences), A054347, A283734.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = a[n - 1] + 1 + Floor[n*GoldenRatio]; Table[a[n], {n, 0, 100}]

PARI

g = (1 + sqrt(5))/2;
a(n) = if(n<1, 1, a(n - 1) + 1 + floor(n*g));
for(n=0, 100, print1(a(n),", ")) \\ Indranil Ghosh, Mar 17 2017

Formula

a(n) = A054347(n) + n + 1. - Alan Michael Gómez Calderón, Dec 03 2024