OFFSET
1,1
COMMENTS
Let phi be the golden ratio. Using Fred Lunnon's formula in A101330 for Knuth's circle product, and the fact that phi^{-2} = 2-phi, plus [-x] = -[x]-1 for non-integer x, one obtains the formula below, expressing this sequence in terms of the lower Wythoff sequence. It follows in particular that the sequence of first differences 5,8,8,5,8,5,8,8,5,8,... of this sequence is the Fibonacci word A003849 on the alphabet {8,5}, shifted by 1. - Michel Dekking, Dec 23 2019
Also numbers with suffix string 0000, when written in Zeckendorf representation. - A.H.M. Smeets, Mar 20 2024
LINKS
A.H.M. Smeets, Table of n, a(n) for n = 1..20000
W. F. Lunnon, Proof of formula
FORMULA
a(n) = 3*A000201(n+1) + 2n - 3. - Michel Dekking, Dec 23 2019
MATHEMATICA
zeck[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[ fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr]]; kfp[n_, m_] := Block[{y = Reverse[ IntegerDigits[ zeck[ n]]], z = Reverse[ IntegerDigits[ zeck[ m]]]}, Sum[ y[[i]]*z[[j]]*Fibonacci[i + j + 2], {i, Length[z1]}, {j, Length[z2]}]]; (* Robert G. Wilson v, Feb 04 2005 *)
Table[ kfp[3, n], {n, 50}] (* Robert G. Wilson v, Feb 04 2005 *)
Array[3*Floor[(# + 1)*GoldenRatio] + 2*# - 3 &, 100] (* Paolo Xausa, Mar 23 2024 *)
PROG
(Python)
from math import isqrt
def A101642(n): return 3*(n+1+isqrt(5*(n+1)**2)>>1)+(n<<1)-3 # Chai Wah Wu, Aug 29 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 26 2005
EXTENSIONS
More terms from David Applegate, Jan 26 2005
More terms from Robert G. Wilson v, Feb 04 2005
STATUS
approved