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%I #30 Aug 13 2023 17:33:11
%S 1,3,8,10,12,17,19,21,26,28,30,35,37,39,44,46,48,53,55,57,62,64,66,71,
%T 73,75,80,82,84,89,91,93,98,100,102,107,109,111,116,118,120,125,127,
%U 129,134,136,138,143,145,147
%N Numbers congruent to 1, 3, and 8 modulo 9: positions of 1 in A159955.
%C This sequence, together with A350666 and A350668, gives a 3-set partition of the nonnegative integers.
%C This sequence {a(n)}_{n>=0}, gives the indices of the row sequences of array A = A347834, that are modulo 6 periodic with period length 3, namely {A347834(a(n), m) mod 6}_{m>=0} = {repeat(1, 5, 3)}.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F A159955(a(n)) = 1.
%F Trisection: a(3*k) = 1 + 9*k, a(3*k+1) = 3 + 9*k, and a(3*k+3) = 8 + 9*k, for k >= 0.
%F G.f.: (1 + 2*x + 5*x^2 + x^3)/((1 - x)*(1 - x^3)).
%F a(n) = 1 + 3*n - 2*sin(2*n*Pi/3)/sqrt(3). - _Stefano Spezia_, Jan 30 2022
%F a(n) = 1 + 3*n - S(n-1,-1), with S(-1, x) = 0, with the Chebyshev S polynomials from A049310. From the g.f., or from the previous formula (see also Spezia's formula in A350666).
%e Rows of array {A347834(a(n), m)}_{m>=0}, with modulo 6 congruence:
%e n = 0: row 1: {1, 5, 21, 85, 341, 1365, 5461, ...} mod 6 = {repeat(1, 5, 3)},
%e n = 1: row 3: {7, 29, 117, 469, 1877, 7509, ...} mod 6 = {repeat(1, 5, 3)},
%e ...
%t Select[Range[0, 150], MemberQ[{1, 3, 8}, Mod[#, 9]] &] (* _Amiram Eldar_, Jan 29 2022 *)
%Y Cf. A049310, A055264, A159955, A347834, A350666, A350668.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Jan 29 2022
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