A185381 -id:A185381 - cp's OEIS frontend

A350678 Partial sums of A185381.

0, 1, 3, 6, 14, 35, 69, 158, 302, 679, 1666, 3263, 7444, 18390, 36101, 82469, 157494, 353912, 868141, 1700181, 3878490, 7403068, 16630533, 40788350, 79876519, 182210674, 450124970, 883619407, 2018522577, 3854834480, 8662361456, 21248630481, 41613641555, 94929932728, 234513795173

Offset: 0

Michel Marcus, Jan 11 2022

nonn

Martin Griffiths, The Zeckendorf Representation of a Beatty-Related Fibonacci Sum, Fibonacci Quart. 53 (2015), no. 3, 230-236.

Cf. A000045, A000201, A185381.

f[n_] := Fibonacci[Floor[GoldenRatio * n]]; Accumulate @ Array[f, 35, 0] (* Amiram Eldar, Jan 11 2022 *)

B(n) = (n+sqrtint(5*n^2))\2; \\ A000201
f(n) = fibonacci(B(n)); \\ A185381
a(n) = sum(k=1, n, f(k));

from math import isqrt
from sympy import fibonacci
def A350678(n): return sum(fibonacci((i+isqrt(5*i**2))//2) for i in range(n+1)) # Chai Wah Wu, Jan 11 2022

a(n) = Sum_{k=1..n} A185381(k).

A107858 a(n) = b(k), where b(k) = Fibonacci(n-1) and k = floor( n*(1+sqrt(5))/2 ).

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 28, 45, 117, 189, 494, 799, 2091, 3383, 8856, 14329, 37513, 60697, 158906, 257115, 673135, 1089155, 2851444, 4613733, 12078909, 19544085, 51167078, 82790071, 216747219, 350704367, 918155952, 1485607537, 3889371025

Offset: 1

Roger L. Bagula, Jun 12 2005

nonn

Limit_{n -> oo} a(n+1)/a(n) does not exist.

Apparently the same as A107857. - Georg Fischer, Nov 02 2018

Cf. A000045, A107857, A185381.

F[1] = 0; F[2] = 1; F[n__] := F[n] = F[n - 1] + F[n - 2]
Table[F[ Floor[(Sqrt[5] + 1)*n/2]], {n, 1, 50}] (* F[n] are the Fibonacci numbers, A000045, with offset 1 *)

Edited by N. J. A. Sloane, May 06 2012

cp's OEIS Frontend

A350678 Partial sums of A185381.

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