cp's OEIS Frontend

A016134 Expansion of g.f. 1/((1-2*x)*(1-10*x)).

%I A016134 #42 Mar 29 2025 04:25:49
%S A016134 1,12,124,1248,12496,124992,1249984,12499968,124999936,1249999872,
%T A016134 12499999744,124999999488,1249999998976,12499999997952,
%U A016134 124999999995904,1249999999991808,12499999999983616,124999999999967232,1249999999999934464,12499999999999868928,124999999999999737856
%N A016134 Expansion of g.f. 1/((1-2*x)*(1-10*x)).
%H A016134 Vincenzo Librandi, <a href="/A016134/b016134.txt">Table of n, a(n) for n = 0..900</a>
%H A016134 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-20).
%F A016134 a(n) = 125*10^(n-2) - 2^(n-2) = a(n-1)*10 + 2^n. - _Henry Bottomley_, Jun 06 2000
%F A016134 G.f.: 1/(1-12*x+20*x^2). - _Zerinvary Lajos_, Apr 27 2009 [corrected by _R. J. Mathar_, Mar 14 2011]
%F A016134 From _Michael Somos_, Dec 03 2016: (Start)
%F A016134 Convolution of A000079 and A011557.
%F A016134 a(n) = (5*10^n - 2^n)/4. (End)
%F A016134 From _Elmo R. Oliveira_, Mar 26 2025: (Start)
%F A016134 E.g.f.: exp(2*x)*(5*exp(8*x) - 1)/4.
%F A016134 a(n) = A060458(n+1)/8.
%F A016134 a(n) = 12*a(n-1) - 20*a(n-2). (End)
%e A016134 G.f. = 1 + 12*x + 124*x^2 + 1248*x^3 + 12496*x^4 + 124992*x^5 + ...
%t A016134 f[n_] := Sum[2^(k - 1)*10^(n - k), {k, n}]; Array[f, 18] (* _Robert G. Wilson v_, Dec 03 2016 *)
%t A016134 a[ n_] := (5 * 10^n - 2^n) / 4; (* _Michael Somos_, Dec 03 2016 *)
%t A016134 LinearRecurrence[{12,-20},{1,12},20] (* _Harvey P. Dale_, Jul 28 2020 *)
%o A016134 (Sage) [lucas_number1(n,12,20) for n in range(1, 18)] # _Zerinvary Lajos_, Apr 27 2009
%o A016134 (Sage) [(10^n - 2^n)/8 for n in range(1,19)] # _Zerinvary Lajos_, Jun 05 2009
%o A016134 (Magma) [2^n*(5^(n+1)-1)/4: n in [0..20]]; // _Vincenzo Librandi_, Oct 09 2011
%o A016134 (PARI) Vec(1/((1-2*x)*(1-10*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%Y A016134 Cf. A060458. - _Zerinvary Lajos_, Jun 05 2009
%Y A016134 Cf. A000079, A011557.
%K A016134 nonn,easy
%O A016134 0,2
%A A016134 _N. J. A. Sloane_