Rodolfo A. Fiorini - cp's OEIS frontend

User: Rodolfo A. Fiorini

Rodolfo A. Fiorini has authored 3 sequences.

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

A274743 Repunits with odd indices multiplied by 99, i.e., 99*(1, 111, 11111, 1111111, ...).

Original entry on oeis.org

99, 10989, 1099989, 109999989, 10999999989, 1099999999989, 109999999999989, 10999999999999989, 1099999999999999989, 109999999999999999989, 10999999999999999999989, 1099999999999999999999989, 109999999999999999999999989, 10999999999999999999999999989

Offset: 1

Rodolfo A. Fiorini, Jul 04 2016

It is apparent that the reciprocals of the terms in the sequence give an increasing sequence of periodic terms similar to A095372, but with the initial term equal to "01". The leading zero is important (see links). Furthermore, the reciprocals of the terms give a sequence of even growing periods, starting from 2, with delta = 4 (i.e., 2, 6, 10, 14, 18, ...).

Adding "11" to each term gives the binary representation of the n-th iteration of "Rule 14" elementary cellular automaton starting with a single ON (black cell) as in A266299.

			a(2) = 101*10989 - 100*99 = 1099989.

Rodolfo A. Fiorini, How Random is Your Tomographic Noise? A Number Theoretic Transform (NTT) Approach, Fundamenta Informaticae, 135(1-2), 2014, 135-170.
Rodolfo 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, A095372, A100706, A266299, A274755.

[11*(10^(2*n-1)-1) : n in [1..20]]; // Wesley Ivan Hurt, Jul 04 2016

A274743:=n->11*(10^(2*n-1)-1): seq(A274743(n), n=1..20); # Wesley Ivan Hurt, Jul 04 2016

Array[99(10^(2 # - 1) - 1)/9 &, 15] (* Michael De Vlieger, Jul 04 2016 *)
99*Table[FromDigits[PadRight[{},2n+1,1]],{n,0,15}] (* Harvey P. Dale, Jul 22 2019 *)

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

a002275(n) = (10^n-1)/9
a(n) = 99*a002275(2*n-1) \\ Felix Fröhlich, Jul 05 2016

a(n) = 101*a(n-1) - 100*a(n-2) for n>2, with a(0)= 99 and a(1)= 10989.
a(n) = 99*A100706(n-1).
G.f.: 99*x*(1 + 10*x)/((1 - x)*(1 - 100*x)). - Ilya Gutkovskiy, Jul 04 2016
a(n) = 11*(10^(2*n-1)-1). - Wesley Ivan Hurt, Jul 04 2016
E.g.f.: 11*(9 - 10*exp(x) + exp(100*x))/10. - Stefano Spezia, Aug 05 2024

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User: Rodolfo A. Fiorini

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., (0*1, 1*11, 11*111, 111*1111, 1111*11111, ...).

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A274743 Repunits with odd indices multiplied by 99, i.e., 99*(1, 111, 11111, 1111111, ...).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

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