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 A026274 Greatest k such that s(k) = n, where s = A026272. 10
 3, 5, 8, 11, 13, 16, 18, 21, 24, 26, 29, 32, 34, 37, 39, 42, 45, 47, 50, 52, 55, 58, 60, 63, 66, 68, 71, 73, 76, 79, 81, 84, 87, 89, 92, 94, 97, 100, 102, 105, 107, 110, 113, 115, 118, 121, 123, 126, 128, 131, 134, 136, 139, 141, 144 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS This is the upper s-Wythoff sequence, where s(n)=n+1. See comments at A026273. 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 Fibonacci numbers greater than 2.] - Charles R Greathouse IV, Jan 28 2015 A Beatty sequence with complement A026273. - Robert G. Wilson v, Jan 30 2015 A035612(a(n)+1) = 1. - Reinhard Zumkeller, Jul 20 2015 From Michel Dekking, Mar 12 2018: (Start) 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. 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. (End) From Michel Dekking, Apr 05 2020: (Start) Proof of the conjecture by Charles R Greathouse IV. Let Z(n) = d(L)...d(1)d(0) be the Zeckendorf expansion of n. Well-known is:       d(0) = 1 if and only if n = floor(k*r^2) - 1 for some integer k (see A003622). 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) LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 Michel Dekking, Points of increase of the sum of digits function of the base phi expansion, arXiv:2003.14125 [math.CO], 2020. FORMULA a(n) = floor(r*r*(n+2r-3)), where r = (1+sqrt(5))/2 = A001622. [Corrected by Tom Edgar, Jan 30 2015] a(n) = 3*n - floor[(n+1)/(1+phi)], phi = (1+sqrt(5))/2. - Joshua Tobin (tobinrj(AT)tcd.ie), May 31 2008 MATHEMATICA r=(1+Sqrt)/2; a[n_]:=Floor[r*r*(n+2r-3)]; Table[a[n], {n, 200}] PROG (Haskell) a026274 n = a026274_list !! (n-1) a026274_list = map (subtract 1) \$ tail \$ filter ((== 1) . a035612) [1..] -- Reinhard Zumkeller, Jul 20 2015 CROSSREFS Cf. A184117, A026273, A001950, A003622. Sequence in context: A213732 A247909 A184659 * A137910 A022850 A008576 Adjacent sequences:  A026271 A026272 A026273 * A026275 A026276 A026277 KEYWORD nonn AUTHOR EXTENSIONS Extended by Clark Kimberling, Jan 14 2011 STATUS approved

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Last modified June 7 05:20 EDT 2020. Contains 334837 sequences. (Running on oeis4.)