A101642 a(n) = Knuth's Fibonacci (or circle) product "3 o n".

8, 13, 21, 29, 34, 42, 47, 55, 63, 68, 76, 84, 89, 97, 102, 110, 118, 123, 131, 136, 144, 152, 157, 165, 173, 178, 186, 191, 199, 207, 212, 220, 228, 233, 241, 246, 254, 262, 267, 275, 280, 288, 296, 301, 309, 317, 322, 330, 335, 343, 351, 356, 364, 369, 377

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

a(n) = A101345(n) + A000201(n+1) + n + 1. - 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

Third row of array in A101330.

Cf. A101345 = Knuth's Fibonacci (or circle) product "2 o n".

Sequence in context: A266212 A063849 A273980 * A269354 A195984 A019535

Adjacent sequences: A101639 A101640 A101641 * A101643 A101644 A101645

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