A274766 Multiplication of pair of contiguous repunits, i.e., (0*1, 1*11, 11*111, 111*1111, 1111*11111, ...).

0, 11, 1221, 123321, 12344321, 1234554321, 123456654321, 12345677654321, 1234567887654321, 123456789987654321, 12345679010987654321, 1234567901220987654321, 123456790123320987654321, 12345679012344320987654321, 1234567901234554320987654321

Offset

0,2

Comments

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

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).

Links

Colin Barker, Table of n, a(n) for n = 0..500

R. A. Fiorini, Computerized tomography noise reduction by CICT optimized exponential cyclic sequences (OECS) co-domain, Fundamenta Informaticae, vol.141 (2015), 115-134.

Index entries for linear recurrences with constant coefficients, signature (111,-1110, 1000).

Formula

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

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

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.

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

Example

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

Mathematica

Table[(10^n - 1) (10^(n + 1) - 1)/81, {n, 0, 20}] (* Bruno Berselli, Jul 05 2016 *)

Prog

(PARI) concat(0, Vec(11*x/((1-x)*(1-10*x)*(1-100*x)) + O(x^99))) \\ Altug Alkan, Jul 05 2016
(PARI) a(n) = (1-11*10^n+10^(1+2*n))/81 \\ Colin Barker, Jul 05 2016

Crossrefs

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

Sequence in context: A180581 A233012 A019524 * A261138 A259937 A078269

Adjacent sequences: A274763 A274764 A274765 * A274767 A274768 A274769

Keyword

nonn,easy

Author

Rodolfo A. Fiorini, Jul 05 2016

Extensions

Edited and added formulae by Bruno Berselli, Jul 05 2016

Last term corrected by Colin Barker, Jul 05 2016

Status

approved