cp's OEIS Frontend

A026274 Greatest k such that s(k) = n, where s = A026272.

%I A026274 #78 Jan 19 2023 02:05:58
%S A026274 3,5,8,11,13,16,18,21,24,26,29,32,34,37,39,42,45,47,50,52,55,58,60,63,
%T A026274 66,68,71,73,76,79,81,84,87,89,92,94,97,100,102,105,107,110,113,115,
%U A026274 118,121,123,126,128,131,134,136,139,141,144
%N A026274 Greatest k such that s(k) = n, where s = A026272.
%C A026274 This is the upper s-Wythoff sequence, where s(n)=n+1.
%C A026274 See comments at A026273.
%C A026274 Conjecture: This sequence consists precisely of those numbers without a 1 or 2 in their Zeckendorf representation. [In other words, numbers which are the sum of distinct nonconsecutive Fibonacci numbers greater than 2.] - _Charles R Greathouse IV_, Jan 28 2015
%C A026274 A Beatty sequence with complement A026273. - _Robert G. Wilson v_, Jan 30 2015
%C A026274 A035612(a(n)+1) = 1. - _Reinhard Zumkeller_, Jul 20 2015
%C A026274 From _Michel Dekking_, Mar 12 2018: (Start)
%C A026274 One has r*r*(n-2*r+3) = n*r^2 -2r^3+3*r^2 = (n+1)*r^2 -2, where r = (1+sqrt(5))/2.
%C A026274 So a(n) = floor((n+1)*r^2)-2, and we see that this sequence is simply the Beatty sequence of the square of the golden ratio, shifted spatially and temporally. In other words, if w = A001950 = 2,5,7,10,13,15,18,20,... is the upper Wythoff sequence, then a(n) = w(n+1) - 2.
%C A026274 (End)
%C A026274 From _Michel Dekking_, Apr 05 2020: (Start)
%C A026274 Proof of the conjecture by _Charles R Greathouse IV_.
%C A026274 Let Z(n) = d(L)...d(1)d(0) be the Zeckendorf expansion of n. Well-known is:
%C A026274       d(0) = 1 if and only if n = floor(k*r^2) - 1
%C A026274 for some integer k (see A003622).
%C A026274 Then the same characterization holds for n with d(1)d(0) = 01, since 11 does not appear in a Zeckendorf expansion. But such an n has predecessor n-1 which always has an expansion with d(1)d(0) = 00. Combined with my comment from March 2018, this proves the conjecture (ignoring n = 0). (End)
%C A026274 It appears that these are the integers m for which A007895(m+1) > A007895(m) where A007895(m) is the number of terms in Zeckendorf representation of m. - _Michel Marcus_, Oct 30 2020
%C A026274 This follows directly from Theorem 4 in my paper "Points of increase of the sum of digits function of the base phi expansion". - _Michel Dekking_, Oct 31 2020
%H A026274 Reinhard Zumkeller, <a href="/A026274/b026274.txt">Table of n, a(n) for n = 1..10000</a>
%H A026274 Hung Viet Chu, <a href="https://arxiv.org/abs/2010.15592">Difference in the Number of Summands in the Zeckendorf Partitions of Consecutive Integers</a>, arXiv:2010.15592 [math.NT], 2020.
%H A026274 Michel Dekking, <a href="https://arxiv.org/abs/2003.14125">Points of increase of the sum of digits function of the base phi expansion</a>, arXiv:2003.14125 [math.CO], 2020.
%H A026274 F. Michel Dekking, <a href="https://doi.org/10.1016/j.tcs.2021.01.011">The sum of digits functions of the Zeckendorf and the base phi expansions</a>, Theoretical Computer Science (2021) Vol. 859, 70-79.
%F A026274 a(n) = floor(r*r*(n+2r-3)), where r = (1+sqrt(5))/2 = A001622. [Corrected by _Tom Edgar_, Jan 30 2015]
%F A026274 a(n) = 3*n - floor[(n+1)/(1+phi)], phi = (1+sqrt(5))/2. - Joshua Tobin (tobinrj(AT)tcd.ie), May 31 2008
%F A026274 a(n) = A003622(n+1) - 1 for n>1 (conjectured). - _Michel Marcus_, Oct 30 2020
%F A026274 This conjectured formula follows directly from the formula a(n) = floor((n+1)*r^2)-2 in my Mar 12 2018 comment above. - _Michel Dekking_, Oct 31 2020
%t A026274 r=(1+Sqrt[5])/2;
%t A026274 a[n_]:=Floor[r*r*(n+2r-3)];
%t A026274 Table[a[n],{n,200}]
%t A026274 Table[Floor[GoldenRatio^2 (n+2*GoldenRatio-3)],{n,60}] (* _Harvey P. Dale_, Dec 23 2022 *)
%o A026274 (Haskell)
%o A026274 a026274 n = a026274_list !! (n-1)
%o A026274 a026274_list = map (subtract 1) $ tail $ filter ((== 1) . a035612) [1..]
%o A026274 -- _Reinhard Zumkeller_, Jul 20 2015
%o A026274 (PARI) a(n)=my(w=quadgen(20),phi=(1+w)/2); phi^2*(n+2*phi-3)\1 \\ _Charles R Greathouse IV_, Nov 10 2021
%o A026274 (Python)
%o A026274 from math import isqrt
%o A026274 def A026274(n): return (n+1+isqrt(5*(n+1)**2)>>1)+n-1 # _Chai Wah Wu_, Aug 17 2022
%Y A026274 Cf. A184117, A026273, A001950, A003622.
%K A026274 nonn,easy
%O A026274 1,1
%A A026274 _Clark Kimberling_
%E A026274 Extended by _Clark Kimberling_, Jan 14 2011