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%I #59 Apr 04 2024 10:03:58
%S 1,2,4,4,8,4,12,8,12,8,20,8,24,12,16,16,32,12,36,16,24,20,44,16,40,24,
%T 36,24,56,16,60,32,40,32,48,24,72,36,48,32,80,24,84,40,48,44,92,32,84,
%U 40,64,48,104,36,80,48,72,56,116,32,120
%N Number of new visible points created at each step in an n X n grid.
%C Equals row sums of triangle A143467. - _Gary W. Adamson_, Aug 17 2008
%C Equals first differences of A018805: (1, 3, 7, 11, 19, 23, 35, ...). - _Gary W. Adamson_, Aug 17 2008
%C a(n) is the number of rationals p/q such that |p| + |q| = n. - _Geoffrey Critzer_, Oct 11 2011
%C a(n) is the number of nonempty lists of positive integers whose continuants are equal to n. For example, for n = 6 these continuants are [6], [5,1], [1,5], and [1,4,1]. - _Jeffrey Shallit_, May 18 2016
%C a(n) is the number of Christoffel words of length n, for n>=2. Here a binary word w is a Christoffel word if its first and last letters are different, say w = axb with a<>b, and x is a palindrome, and w is the concatenation of two palindromes. See the book of Reutenauer. - _Jeffrey Shallit_, Apr 04 2024
%D C. Reutenauer, From Christoffel words to Markoff numbers, Oxford University Press, 2019.
%H Reinhard Zumkeller, <a href="/A140434/b140434.txt">Table of n, a(n) for n = 1..1000</a>
%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)
%F a(n) = 2*phi(n), where phi is Euler's phi function, A000010, for n >= 2.
%F Sum_{k=1..n} a(k)*floor(n/k) = n^2. - _Benoit Cloitre_, Nov 09 2016
%F G.f.: Sum_{k>=1} mu(k) * x^k * (1 + x^k)/(1 - x^k)^2. - _Seiichi Manyama_, May 24 2021
%e G.f. = x + 2*x^2 + 4*x^3 + 4*x^4 + 8*x^5 + 4*x^6 + 12*x^7 + 8*x^8 + 12*x^9 + ...
%t f[n_] := FoldList[Plus, 1, 2 Array[EulerPhi, n, 2]] // Differences // Prepend[#, 1]&
%t a[ n_] := If[ n < 3, Max[0, n], Sum[ MoebiusMu[d] (2 n/d - 1 - Mod[n/d, 2]), {d, Divisors@n}]]; (* _Michael Somos_, Jul 24 2015 *)
%o (Haskell)
%o a140434 n = a140434_list !! (n-1)
%o a140434_list = 1 : zipWith (-) (tail a018805_list) a018805_list
%o -- _Reinhard Zumkeller_, May 04 2014
%o (PARI) {a(n) = if( n<3, max(0, n), sumdiv(n, d, moebius(d) * (2*n/d - 1 - (n/d)%2)))}; /* _Michael Somos_, Jul 24 2015 */
%o (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k*(1+x^k)/(1-x^k)^2)) \\ _Seiichi Manyama_, May 24 2021
%Y Cf. A018805, A100613, A140435. Equals twice A000010 (for n >= 2).
%K nonn
%O 1,2
%A _Gregg Whisler_, Jun 25 2008, Jun 28 2008
%E Mathematica simplified by _Jean-François Alcover_, Jun 06 2013
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