%I #22 Sep 08 2022 08:45:33

%S 1,11,101,911,8201,73811,664301,5978711,53808401,484275611,4358480501,

%T 39226324511,353036920601,3177332285411,28595990568701,

%U 257363915118311,2316275236064801,20846477124583211,187618294121248901

%N Expansion of (1+x)/(1-10*x+9*x^2).

%C Orbit starting at 1 of A138893: a(n)=A138893^(n)(1). Partial sums of A003952.

%C Sum of n-th row of triangle of powers of 9: 1; 1 9 1; 1 9 81 9 1; 1 9 81 729 81 9 1; ... - _Philippe Deléham_, Feb 22 2014

%H Vincenzo Librandi, <a href="/A138894/b138894.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-9).

%F G.f.: (1+x)/((1-x)*(1-9x)).

%F a(n) = (5/4)*9^n - 1/4.

%F a(n) = A002452(n) + A002452(n+1).

%F Bisection of A135522/3. a(n+1)=9*a(n)+2. - _Paul Curtz_, Apr 22 2008

%F a(n) = Sum_{k=0..n} A112468(n,k)*10^k. - _Philippe Deléham_, Feb 22 2014

%e a(0) = 1;

%e a(1) = 1 + 9 + 1 = 11;

%e a(2) = 1 + 9 + 81 + 9 + 1 = 101;

%e a(3) = 1 + 9 + 81 + 729 + 81 + 9 + 1 = 911; etc. - _Philippe Deléham_, Feb 22 2014

%t Table[(5*9^n - 1)/4, {n, 0, 18}] (* _L. Edson Jeffery_, Feb 13 2015 *)

%o (Magma) [(5/4)*9^n-1/4: n in [0..20]]; // _Vincenzo Librandi_, Aug 09 2011

%o (PARI) Vec((1+x)/(1-10*x+9*x^2) + O(x^30)) \\ _Michel Marcus_, Feb 13 2015

%Y Cf. A096053 ((3*9^n-1)/2), a(n+1)=9a(n)-4 in A135423.

%Y Cf. A002452, A003952, A135522, A138893.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Apr 02 2008