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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 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

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