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A220018 Number of cyclotomic cosets of 3 mod 10^n.

%I A220018 #36 Jul 03 2023 23:24:40
%S A220018 4,11,29,71,129,203,293,399,521,659,813,983,1169,1371,1589,1823,2073,
%T A220018 2339,2621,2919,3233,3563,3909,4271,4649,5043,5453,5879,6321,6779,
%U A220018 7253,7743,8249,8771,9309,9863,10433,11019,11621,12239,12873,13523,14189
%N A220018 Number of cyclotomic cosets of 3 mod 10^n.
%H A220018 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).
%F A220018 a(n) = A221855(n) - 1 for all n >= 2, a(1) = A221855(1).
%F A220018 Empirical G.f.: x*(8*x^4-13*x^3-8*x^2+x-4) / (x-1)^3. [_Colin Barker_, Feb 03 2013]
%F A220018 Conjecture: a(n) = 8*n^2-14*n-1 for n>2. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>5. [_Colin Barker_, Apr 14 2013]
%e A220018 a(2) = 11 because there are 11 cyclotomic cosets of 3 mod 100:
%e A220018 O(0) = {0},
%e A220018 O(1) = {1, 3, 9, 27, 81, 43, 29, 87, 61, 83, 49, 47, 41, 23, 69, 7, 21, 63, 89, 67},
%e A220018 O(2) = {2, 6, 18, 54, 62, 86, 58, 74, 22, 66, 98, 94, 82, 46, 38, 14, 42, 26, 78, 34},
%e A220018 O(4) = {4, 12, 36, 8, 24, 72, 16, 48, 44, 32, 96, 88, 64, 92, 76, 28, 84, 52, 56, 68},
%e A220018 O(5) = {5, 15, 45, 35},
%e A220018 O(10) = {10, 30, 90, 70},
%e A220018 O(11) = {11, 33, 99, 97, 91, 73, 19, 57, 71, 13, 39, 17, 51, 53, 59, 77, 31, 93, 79, 37},
%e A220018 O(20) = {20, 60, 80, 40},
%e A220018 O(25) = {25, 75},
%e A220018 O(50) = {50},
%e A220018 O(55) = {55, 65, 95, 85}.
%t A220018 a[n_]:=DivisorSum[10^n,EulerPhi[#]/MultiplicativeOrder[3,#]&];Array[a,50] (* _Ray Chandler_, Jul 03 2023, after Jean-François Alcover *)
%o A220018 (PARI) A220018(n)=sumdiv(10^n, d, eulerphi(d)/znorder(Mod(3, d)))
%Y A220018 Cf. A006694.
%K A220018 base,nonn
%O A220018 1,1
%A A220018 _V. Raman_, Jan 27 2013
%E A220018 Edited by _M. F. Hasler_, Jan 28 2013