A274231 Ternary representation with index set {0, 1, 5}.

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

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.

Wipawee Tangjai, A Non-standard Ternary Representation of Integers, Thai J. Math (2020) Special Issue: Annual Meeting in Mathematics 2019, 269-283.

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