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A220020 Number of cyclotomic cosets of 9 mod 10^n.

%I A220020 #26 Jul 12 2025 13:42:17
%S A220020 6,20,56,140,256,404,584,796,1040,1316,1624,1964,2336,2740,3176,3644,
%T A220020 4144,4676,5240,5836,6464,7124,7816,8540,9296,10084,10904,11756,12640,
%U A220020 13556,14504,15484,16496,17540,18616,19724,20864,22036,23240,24476,25744,27044,28376,29740,31136,32564,34024,35516,37040,38596
%N A220020 Number of cyclotomic cosets of 9 mod 10^n.
%H A220020 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).
%F A220020 Conjecture: a(n) = 4*(4*n^2-7*n-1) for n>2. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>5. G.f.: 2*x*(8*x^4-13*x^3-7*x^2-x-3) / (x-1)^3. - _Colin Barker_, Apr 13 2013
%e A220020 a(2) = 20 because there are 20 cyclotomic cosets of 9 mod 100:
%e A220020 {1, 9, 81, 29, 61, 49, 41, 69, 21, 89}
%e A220020 {3, 27, 43, 87, 83, 47, 23, 7, 63, 67}
%e A220020 {11, 99, 91, 19, 71, 39, 51, 59, 31, 79}
%e A220020 {13, 17, 53, 77, 93, 37, 33, 97, 73, 57}
%e A220020 {2, 18, 62, 58, 22, 98, 82, 38, 42, 78}
%e A220020 {4, 36, 24, 16, 44, 96, 64, 76, 84, 56}
%e A220020 {6, 54, 86, 74, 66, 94, 46, 14, 26, 34}
%e A220020 {8, 72, 48, 32, 88, 92, 28, 52, 68, 12}
%e A220020 {10, 90}
%e A220020 {30, 70}
%e A220020 {20, 80}
%e A220020 {40, 60}
%e A220020 {50}
%e A220020 {5, 45}
%e A220020 {15, 35}
%e A220020 {55, 95}
%e A220020 {65, 85}
%e A220020 {25}
%e A220020 {75}
%e A220020 {0}
%t A220020 a[n_] := DivisorSum[10^n, EulerPhi[#]/MultiplicativeOrder[9, #]&]; Array[a, 50] (* _Jean-François Alcover_, Dec 10 2015, adapted from PARI *)
%t A220020 LinearRecurrence[{3,-3,1},{6,20,56,140,256},50] (* _Harvey P. Dale_, Jul 12 2025 *)
%o A220020 (PARI) for(n=1,50,print1(sumdiv(10^n, d, eulerphi(d)/znorder(Mod(9, d)))", "))
%Y A220020 Cf. A006694, A220468.
%K A220020 base,nonn
%O A220020 1,1
%A A220020 _V. Raman_, Jan 27 2013