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A101642 a(n) = Knuth's Fibonacci (or circle) product "3 o n". 5
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 (list; graph; refs; listen; history; text; internal format)
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
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
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

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Last modified June 18 11:45 EDT 2024. Contains 373481 sequences. (Running on oeis4.)