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A274231 Ternary representation with index set {0, 1, 5}. 0
0, 1, 3, 4, 5, 8, 9, 10, 12, 13, 14, 15, 16, 17, 20, 24, 25, 27, 28, 29, 30, 31, 32, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 60, 61, 65, 72, 73, 75, 76, 77, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 101, 105, 106, 108, 109, 110, 111, 112, 113 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A number m is in the sequence if m = b_r * 3^r + b_(r-1) * 3^(r-1) + ... + b_0, where b_i is in {0, 1, 5}.

The maximal sets of consecutive numbers in this sequences can be associated with the Fibonacci numbers (A000045) and Pell numbers (A000129).

REFERENCES

Thomas Koshy, Pell and Pell-Lucas Numbers with Applications, Springer New York Heidelberge Dordrecht London, Cambridge, 2014, p.193

LINKS

Table of n, a(n) for n=1..77.

Wipawee Tangjai, Density and spacing properties of some families of non-standard ternary representations, Doctoral Dissertation, University of Illinois at Urbana-Champaign, 2014.

FORMULA

G.f.:  (1/(1-x))(1-x^2+x^3)(1-x^(2*3)+x^(3^2))...(1-x^(2*3^k)+x^(3^(k+1)) ...

EXAMPLE

a(1) = 0;

a(2) = 3*a(1) + 1 = 1;

a(3) = 3*a(2) = 3;

a(4) = 3*a(2) + 1;

a(5) = 3*a(1) + 5 = 5;

a(6) = 3*a(2) + 5.

MATHEMATICA

Select[Union[Table[FromDigits[IntegerDigits[k, 3] /. 2 -> 5, 3], {k, 0, 3^5 - 1}]], # < 3^5 &] (* Giovanni Resta, Jun 24 2016 *)

PROG

(R) #This program generates numbers from a(1) to a(135)

#it can be increased by changing number of k

m3<-function(x, k){

  for(j in 1:k){

  A=array(3*x[1])

  B=array(3*x[1]+1)

  C=array(3*x[1]+5)

  for(i in 2:length(x)){

    A=c(A, 3*x[i])

    B=c(B, 3*x[i]+1)

    C=c(C, 3*x[i]+5)

    result=sort(union(x, union(A, union(B, C))), decreasing = FALSE)

   }

  x=result

  }

  return(result)

  }

S=array(0)

U=m3(S, 3) #row r-1

V=m3(S, 4) #row r

up=ceiling((V[length(V)]-5)/3) # find the max element in r that less than in r-1

Y1=V[V<up]#take elt in r less than up

Y2=setdiff(Y1, U) #elt in r less than up not in r-1

Y3=m3(Y2, 1) # elt apply the rec to elt in Y2

Re=sort(union(V, Y3))

Re2=paste(Re, sep=", ", collapse=", ")

write.table(Re2, file="A015Sequence.txt")

CROSSREFS

Cf. A000045 (Fibonacci numbers), A000129 (Pell numbers), A001333 (Pell-Lucas numbers). Superset of A005836.

Sequence in context: A188003 A120519 A100614 * A173999 A127427 A286994

Adjacent sequences:  A274228 A274229 A274230 * A274232 A274233 A274234

KEYWORD

nonn

AUTHOR

Wipawee Tangjai, Jun 15 2016

STATUS

approved

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Last modified June 20 09:27 EDT 2019. Contains 324234 sequences. (Running on oeis4.)