A074992 a(n) = (10^(2*n) + 10^n + 1)/3.

1, 37, 3367, 333667, 33336667, 3333366667, 333333666667, 33333336666667, 3333333366666667, 333333333666666667, 33333333336666666667, 3333333333366666666667, 333333333333666666666667, 33333333333336666666666667, 3333333333333366666666666667, 333333333333333666666666666667

Offset: 0

Author

Amarnath Murthy, Aug 31 2002

Keywords

• nonn
• easy

Comments

Apart from the initial 1, common difference of the arithmetic progression pertaining to the sequence A074991.

This is also a root sequence pertaining to the patterned perfect square sequence 1369, 11336689,111333666889,... i.e., k ones, k threes and k sixes followed by (k-1) 8's and a 9. (37^2 = 1369).

This is a self-complementing sequence: each term has even number of digits (the first one has to be read 01, the leading zero is important). If you add the first half to the second half of any term, you get the sequence A011557, powers of 10. Furthermore, the reciprocals of the sequence terms, except the first one, give a sequence of periodic terms with period sequence as in A008585, a(n) = 3*n, and value given by A086574, a(n)=k where R(k+3)=3. - Rodolfo A. Fiorini, Jul 14 2016

Links

• Rodolfo A. Fiorini, Computerized tomography noise reduction by CICT optimized exponential cyclic sequences (OECS) co-domain, Fundamenta Informaticae, 141(2015), pp. 115-134.
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A008585, A011557, A066138, A074991, A086574, A274766.

Programs

Maple

A074992 := proc(n)
    (10^(2*n)+10^n+1)/3 ;
end proc:
seq(A074992(n),n=0..15) ; # R. J. Mathar, May 06 2017

Mathematica

{01}~Join~Table[FromDigits@ Flatten@ Map[IntegerDigits, {#, 10^n - #}] &@ Floor[10^n/3], {n, 12}] (* Michael De Vlieger, Jul 22 2016 *)

PARI

a(n) = (10^(2*n) + 10^n + 1)/3; \\ Michel Marcus, Sep 14 2013

PARI

Vec(-x*(1000*x^2-740*x+37)/((x-1)*(10*x-1)*(100*x-1)) + O(x^100)) \\ Colin Barker, Sep 23 2013

PARI

a(n)=my(x=10^n); (x^2+x+1)/3 \\ Charles R Greathouse IV, Jul 22 2016

Formula

a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3), for n > 2, a(0)=1, a(1)=37, a(2)=3367.

G.f.: (1 - 74*x + 370*x^2)/((1-x)*(1-10*x)*(1-100*x)). - Colin Barker, Sep 23 2013 and Robert Israel, Jul 22 2016

From Elmo R. Oliveira, Sep 12 2024: (Start)

E.g.f.: exp(x)*(exp(99*x) + exp(9*x) + 1)/3.

a(n) = A066138(n)/3. (End)

Extensions

Entry revised (new definition, new offset, new initial term, etc.) by N. J. A. Sloane, Jul 27 2016 (Some of the old programs may need slight modifications.)