A274766 - OEIS

%I #48 Jul 16 2017 14:50:29

%S 0,11,1221,123321,12344321,1234554321,123456654321,12345677654321,

%T 1234567887654321,123456789987654321,12345679010987654321,

%U 1234567901220987654321,123456790123320987654321,12345679012344320987654321,1234567901234554320987654321

%N Multiplication of pair of contiguous repunits, i.e., (0*1, 1*11, 11*111, 111*1111, 1111*11111, ...).

%C From the second to the tenth term they look like in A259937, but it is a completely different sequence.

%C The inverse of sequence terms, except the first one, give a sequence of periodic terms with periods as in A002378, the sequence of oblong (or promic, or heteromecic) numbers: a(n) = n*(n+1). Digit string period L of inverse a(n) is given by L = n*(n+1).

%H Colin Barker, <a href="/A274766/b274766.txt">Table of n, a(n) for n = 0..500</a>

%H R. A. Fiorini, <a href="http://dx.doi.org/10.3233/FI-2015-1267">Computerized tomography noise reduction by CICT optimized exponential cyclic sequences (OECS) co-domain</a>, Fundamenta Informaticae, vol.141 (2015), 115-134.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110, 1000).

%F O.g.f.: 11*x/((1 - x)*(1 - 10*x)*(1 - 100*x)).

%F E.g.f.: (1 - 11*exp(9*x) + 10*exp(99*x))*exp(x)/81.

%F a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2, a(0)=0, a(1)=11, a(2)=1221.

%F a(n) = (10^n - 1)*(10^(n+1) - 1)/81 = A002275(n)*A002275(n+1).

%e a(10) = rep(10)*rep(11) = 12345679010987654321, digit string period of 1/a(10) -> L = 10*11 = 110.

%t Table[(10^n - 1) (10^(n + 1) - 1)/81, {n, 0, 20}] (* _Bruno Berselli_, Jul 05 2016 *)

%o (PARI) concat(0, Vec(11*x/((1-x)*(1-10*x)*(1-100*x)) + O(x^99))) \\ _Altug Alkan_, Jul 05 2016

%o (PARI) a(n) = (1-11*10^n+10^(1+2*n))/81 \\ _Colin Barker_, Jul 05 2016

%Y Cf. A002275, A002378, A259937, A274743, A274755.

%K nonn,easy

%O 0,2

%A _Rodolfo A. Fiorini_, Jul 05 2016

%E Edited and added formulae by _Bruno Berselli_, Jul 05 2016

%E Last term corrected by _Colin Barker_, Jul 05 2016