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

Author

Rodolfo A. Fiorini, Jul 05 2016

Keywords

• nonn
• easy

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).

Examples

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

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).

Crossrefs

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

Programs

Mathematica

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

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

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).

Extensions

Edited and added formulae by Bruno Berselli, Jul 05 2016

Last term corrected by Colin Barker, Jul 05 2016