This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A016186 #34 Nov 14 2024 08:23:53
%S A016186 1,18,244,2952,33616,368928,3951424,41611392,432891136,4463129088,
%T A016186 45705032704,465640261632,4725122093056,47800976744448,
%U A016186 482407813955584,4859262511644672,48874100093157376,490992800745259008,4927942405962072064,49423539247696576512,495388313981572612096,4963106511852580896768
%N A016186 Expansion of 1/((1-8*x)*(1-10*x)).
%C A016186 a(n) is the number of strings of n+1 decimal digits having an odd number of 0's. For 2 digits these are for example the 18 strings 01, 02, 03, ..., 09, 10, 20, 30, ..., 90. - _Geoffrey Critzer_, Jan 24 2011
%H A016186 G. C. Greubel, <a href="/A016186/b016186.txt">Table of n, a(n) for n = 0..990</a>
%H A016186 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-80).
%F A016186 From _R. J. Mathar_, Sep 18 2008: (Start)
%F A016186 a(n) = 5*10^n - 4*8^n = A081203(n+1).
%F A016186 Binomial transform of A081035. (End)
%F A016186 From _Geoffrey Critzer_, Jan 24 2011: (Start)
%F A016186 a(n) = 8*a(n-1) + 10^(n-1).
%F A016186 E.g.f.: exp(9*x)*sinh(x) (with offset 1). (End)
%F A016186 A060531(n) = a(n) - 9*a(n-1). - R. J. Mathar_, Jan 27 2011
%F A016186 From _Vincenzo Librandi_, Feb 09 2011: (Start)
%F A016186 a(n) = 10*a(n-1) + 8^n, a(0)=1.
%F A016186 a(n) = 18*a(n-1) - 80*a(n-2), a(0)=1, a(1)=18. (End)
%F A016186 E.g.f.: exp(9*x)*( cosh(x) + 9*sinh(x) ). - _G. C. Greubel_, Nov 14 2024
%t A016186 Rest@With[{m=30}, CoefficientList[Series[Exp[9 x] Sinh[x], {x,0,m}], x]*Range[0, m]!]
%t A016186 Table[2^n*(5^(n+1)-4^(n+1)), {n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 09 2011 *)
%t A016186 LinearRecurrence[{18,-80},{1,18},30] (* _Harvey P. Dale_, Aug 26 2019 *)
%o A016186 (PARI) Vec(1/((1-8*x)*(1-10*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 24 2012
%o A016186 (Magma) [2^n*(5^(n+1)-4^(n+1)): n in [0..40]]; // _G. C. Greubel_, Nov 14 2024
%o A016186 (SageMath)
%o A016186 A016186=BinaryRecurrenceSequence(18,-80,1,18)
%o A016186 print([A016186(n) for n in range(41)]) # _G. C. Greubel_, Nov 14 2024
%Y A016186 Cf. A016140, A060531, A081035, A081203.
%K A016186 nonn,easy
%O A016186 0,2
%A A016186 _N. J. A. Sloane_
%E A016186 More terms added by _G. C. Greubel_, Nov 14 2024