A101642 - OEIS

%I #43 Mar 25 2024 06:34:24

%S 8,13,21,29,34,42,47,55,63,68,76,84,89,97,102,110,118,123,131,136,144,

%T 152,157,165,173,178,186,191,199,207,212,220,228,233,241,246,254,262,

%U 267,275,280,288,296,301,309,317,322,330,335,343,351,356,364,369,377

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

%C 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

%C Also numbers with suffix string 0000, when written in Zeckendorf representation. - _A.H.M. Smeets_, Mar 20 2024

%H A.H.M. Smeets, <a href="/A101642/b101642.txt">Table of n, a(n) for n = 1..20000</a>

%H W. F. Lunnon, <a href="/A101330/a101330.txt">Proof of formula</a>

%F a(n) = 3*A000201(n+1) + 2n - 3. - _Michel Dekking_, Dec 23 2019

%F a(n) = A101345(n) + A000201(n+1) + n + 1. - _Michel Dekking_, Dec 23 2019

%t 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 *)

%t Table[ kfp[3, n], {n, 50}] (* _Robert G. Wilson v_, Feb 04 2005 *)

%t Array[3*Floor[(# + 1)*GoldenRatio] + 2*# - 3 &, 100] (* _Paolo Xausa_, Mar 23 2024 *)

%o (Python)

%o from math import isqrt

%o def A101642(n): return 3*(n+1+isqrt(5*(n+1)**2)>>1)+(n<<1)-3 # _Chai Wah Wu_, Aug 29 2022

%Y Third row of array in A101330.

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

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jan 26 2005

%E More terms from _David Applegate_, Jan 26 2005

%E More terms from _Robert G. Wilson v_, Feb 04 2005