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A283971
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a(n) = n except a(4*n + 2) = 2*n + 1.
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3
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0, 1, 1, 3, 4, 5, 3, 7, 8, 9, 5, 11, 12, 13, 7, 15, 16, 17, 9, 19, 20, 21, 11, 23, 24, 25, 13, 27, 28, 29, 15, 31, 32, 33, 17, 35, 36, 37, 19, 39, 40, 41, 21, 43, 44, 45, 23, 47, 48, 49, 25, 51, 52, 53, 27, 55, 56, 57, 29, 59, 60, 61, 31, 63, 64, 65, 33, 67
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OFFSET
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0,4
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COMMENTS
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For n > 1, a(n) is also the cycle length generated by the cycle lengths of the digital roots, in base n, of the powers of k, with k > 0.
Example for n=10 (decimal base): for every h >= 0, the digital roots of 2^h generate a periodic cycle {1,2,4,8,7,5} with period 6; 3^h generates {1,3,9,9,9,9,...} so the periodic cycle {9} has period 1; 4^h generates the periodic cycle {1,4,7} with period 3; etc. So, for n=10 (decimal base representation) the sequence generated by the periods of the digital roots of powers of k (with k > 0) is also periodic {1,6,1,3,6,1,3,2,1} with period 9, hence a(10) = 9. (End)
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LINKS
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FORMULA
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a(2*n) = A022998(n), a(1+2*n) = 1 + 2*n.
a(n) = 2*a(n-4) - a(n-8).
G.f.: x*(1 + x + 3*x^2 + 4*x^3 + 3*x^4 + x^5 + x^6) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).
a(n) = -((-1)^n - (-i)^n - i^n - 7)*n/8, where i = sqrt(-1).
(End)
a(n) = A060819(n) * periodic sequence of length 4: repeat [4, 1, 1, 1].
a(n) = a(n-4) + periodic sequence of length 4: repeat [4, 4, 2, 4].
For n > 0, a(n) is multiplicative with a(p^e) = p^e for prime p >= 2 and e >= 0 except a(2^1) = 1.
Dirichlet g.f.: (1 - 1/2^s - 1/2^(2*s-1)) * zeta(s-1).
(End)
a(n) = n*(7 + cos(n*Pi/2) - cos(n*Pi) + cos(3*n*Pi/2))/8. - Wesley Ivan Hurt, Oct 04 2018
Sum_{k=1..n} a(k) ~ (7/16) * n^2. - Amiram Eldar, Nov 28 2022
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MAPLE
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seq(coeff(series(x*(1+x+3*x^2+4*x^3+3*x^4+x^5+x^6)/((1-x)^2*(1+x)^2*(1+x^2)^2), x, n+1), x, n), n=0..80); # Muniru A Asiru, Jul 20 2018
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MATHEMATICA
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Table[If[Mod[n, 4] == 2, (n - 2)/2 + 1, n], {n, 67}] (* or *)
CoefficientList[Series[x (1 + x + 3 x^2 + 4 x^3 + 3 x^4 + x^5 + x^6)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2), {x, 0, 67}], x] (* Michael De Vlieger, Mar 19 2017 *)
LinearRecurrence[{0, 0, 0, 2, 0, 0, 0, -1}, {0, 1, 1, 3, 4, 5, 3, 7}, 70] (* Robert G. Wilson v, Jul 23 2018 *)
Table[Length[FindTransientRepeat[(Length[FindTransientRepeat[Mod[#1^Range[b]-1, b-1]+1, 2][[2]]]&)/@Range[2, 2*b], 2][[2]]], {b, 2, 100}] (* Federico Provvedi, Nov 13 2018 *)
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PROG
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(PARI) concat(0, Vec(x*(1 + x + 3*x^2 + 4*x^3 + 3*x^4 + x^5 + x^6) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2) + O(x^40))) \\ Colin Barker, Mar 19 2017
(GAP) a:=[0, 1, 1, 3, 4, 5, 3, 7];; for n in [9..85] do a[n]:=2*a[n-4]-a[n-8]; od; a; # Muniru A Asiru, Jul 20 2018
(Python)
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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