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A221855 Number of cyclotomic cosets of 13 mod 10^n.

%I A221855 #23 Jul 03 2023 23:18:06
%S A221855 4,12,30,72,130,204,294,400,522,660,814,984,1170,1372,1590,1824,2074,
%T A221855 2340,2622,2920,3234,3564,3910,4272,4650,5044,5454,5880,6322,6780,
%U A221855 7254,7744,8250,8772,9310,9864,10434,11020,11622,12240,12874,13524,14190,14872,15570,16284,17014,17760,18522,19300
%N A221855 Number of cyclotomic cosets of 13 mod 10^n.
%H A221855 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).
%F A221855 a(n) = A220018(n) for n = 1.
%F A221855 a(n) = A220018(n) + 1 for all n >= 2.
%F A221855 Conjecture: a(n) = 2*n*(4*n-7) for n>2. a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>5. G.f.: 2*x*(4*x^4-7*x^3-3*x^2-2) / (x-1)^3. - _Colin Barker_, Apr 14 2013
%e A221855 a(2) = 12 because there are 12 cyclotomic cosets of 13 mod 100:
%e A221855 {1, 13, 69, 97, 61, 93, 9, 17, 21, 73, 49, 37, 81, 53, 89, 57, 41, 33, 29, 77}
%e A221855 {3, 39, 7, 91, 83, 79, 27, 51, 63, 19, 47, 11, 43, 59, 67, 71, 23, 99, 87, 31}
%e A221855 {2, 26, 38, 94, 22, 86, 18, 34, 42, 46, 98, 74, 62, 6, 78, 14, 82, 66, 58, 54}
%e A221855 {4, 52, 76, 88, 44, 72, 36, 68, 84, 92, 96, 48, 24, 12, 56, 28, 64, 32, 16, 8}
%e A221855 {5, 65, 45, 85}
%e A221855 {15, 95, 35, 55}
%e A221855 {25}
%e A221855 {75}
%e A221855 {10, 30, 90, 70}
%e A221855 {20, 60, 80, 40}
%e A221855 {50}
%e A221855 {0}
%t A221855 a[n_]:=DivisorSum[10^n,EulerPhi[#]/MultiplicativeOrder[13,#]&]; Array[a,50] (* _Ray Chandler_, Jul 03 2023, after Jean-François Alcover *)
%o A221855 (PARI) for(n=1,50,print1(sumdiv(10^n, d, eulerphi(d)/znorder(Mod(13, d)))", "))
%Y A221855 Cf. A006694, A220468.
%K A221855 base,nonn
%O A221855 1,1
%A A221855 _V. Raman_, Jan 28 2013