OFFSET
0,2
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).
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)
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amarnath Murthy, Aug 31 2002
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.)
STATUS
approved