A274743 -id:A274743 - cp's OEIS frontend

A274755 Repunits with even indices multiplied by 99, i.e., 99*(11, 1111, 111111, 11111111, ...).

1089, 109989, 10999989, 1099999989, 109999999989, 10999999999989, 1099999999999989, 109999999999999989, 10999999999999999989, 1099999999999999999989, 109999999999999999999989, 10999999999999999999999989, 1099999999999999999999999989

Offset: 1

Rodolfo A. Fiorini, Jul 04 2016

The reciprocals of the terms give a sequence of even growing periods, starting from 22, with delta = 22 (i.e., 22,44,66,88,110,132,...).

			a(3) = 101*109989 - 100*1089 = 10999989.

R. A. Fiorini, How Random is Your Tomographic Noise? A Number Theoretic Transform (NTT) Approach, Fundamenta Informaticae, 135(1-2), 2014, 135-170.
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 (101,-100).

Cf. A002275, A099814, A274743.

Magma
```
[11*(10^(2*n) - 1): n in [1..20]];
```

A274755:= n-> 11*(10^(2*n) - 1) : seq(A274755(n), n=1..20);

Array[99(10^(2 #)- 1)/9&, 15]
LinearRecurrence[{101, -100}, {1089, 109989}, 20] (* Vincenzo Librandi, Jul 07 2016 *)

Vec(1089*x/((1-x)*(1-100*x)) + O(x^99)) \\ Altug Alkan, Jul 06 2016

a(n) = 101*a(n-1) - 100*a(n-2), with a(1)= 1089 and a(2)= 109989.
G.f.: 1089*x/((1 - x)*(1 - 100*x)). - Ilya Gutkovskiy, Jul 04 2016
a(n) = 99*A099814(n). - Michel Marcus, Jul 04 2016
a(n) = 11*(10^(2*n)-1). - Robert Israel, Jul 06 2016
E.g.f.: 11*exp(x)*(exp(99*x) - 1). - Elmo R. Oliveira, Jun 09 2025

A274766 Multiplication of pair of contiguous repunits, i.e., (01, 111, 11111, 1111111, 1111*11111, ...).

Original entry on oeis.org

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

Offset: 0

Rodolfo A. Fiorini, Jul 05 2016

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).

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

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).

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

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

concat(0, Vec(11*x/((1-x)*(1-10*x)*(1-100*x)) + O(x^99))) \\ Altug Alkan, Jul 05 2016

a(n) = (1-11*10^n+10^(1+2*n))/81 \\ Colin Barker, Jul 05 2016

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).

Edited and added formulae by Bruno Berselli, Jul 05 2016

Last term corrected by Colin Barker, Jul 05 2016

cp's OEIS Frontend

A274755 Repunits with even indices multiplied by 99, i.e., 99*(11, 1111, 111111, 11111111, ...).

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A274766 Multiplication of pair of contiguous repunits, i.e., (01, 111, 11111, 1111111, 1111*11111, ...).

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cp's OEIS Frontend

A274755 Repunits with even indices multiplied by 99, i.e., 99*(11, 1111, 111111, 11111111, ...).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A274766 Multiplication of pair of contiguous repunits, i.e., (01, 111, 11111, 1111111, 1111*11111, ...).