A153773 -id:A153773 - cp's OEIS frontend

A153775 Sequence S such that 1 is in S and if x is in S, then 3x-1 and 3x are in S.

1, 2, 3, 5, 6, 8, 9, 14, 15, 17, 18, 23, 24, 26, 27, 41, 42, 44, 45, 50, 51, 53, 54, 68, 69, 71, 72, 77, 78, 80, 81, 122, 123, 125, 126, 131, 132, 134, 135, 149, 150, 152, 153, 158, 159, 161, 162, 203, 204, 206, 207, 212, 213, 215, 216, 230, 231, 233, 234, 239, 240, 242

Offset: 1

Clark Kimberling, Jan 02 2009

nonn

Subsequences include A007051, A000244, A153773, A153774.
First generation: 1
2nd generation: 2, 3
3rd generation: 5, 6, 8, 9
4th generation: 14, 15, 17, 18, 23, 24, 26, 27
Does every generation contain a prime?
From Peter Munn, Feb 10 2022: (Start)
Consider a Sierpinski arrowhead curve formed of edges indexed consecutively from 0 at its axis of symmetry and aligned with an infinite Sierpinski gasket so that each edge is contained in the boundary of either the plane sector occupied by the gasket or a triangular region of the gasket's complement. The numbers {4*a(n)-1 : n >= 1} (that is, 3, 7, 11, 19, 23, 31, 35, 55, 59, ...) index the edges that are contained in the boundaries of certain triangular regions: each is the first region encountered of each successively larger size that does not lie across the axis of symmetry.
Let S be the set of terms. Define c: N -> P(R) so that c(m) is the scaled and translated Cantor ternary set spanning [m-0.5, m], and let C be the union of c(m) for all m in S. C is the closure under multiplication by 3 of the scaled and translated Cantor ternary set spanning [0.5, 1.0].
(End)
Positive numbers whose balanced ternary expansions contain exactly one digit 1. - Rémy Sigrist, May 08 2022

G. C. Greubel, Table of n, a(n) for n = 1..1000
Peter Munn, Edges of Sierpiński arrowhead aligned with Sierpiński gasket.
Eric Weisstein's World of Mathematics, Cantor Set
Eric Weisstein's World of Mathematics, Closure

Cf. A000244, A007051, A005836, A147992, A153773, A153774, A306556, A307744.

See also the related sequences listed in A191106.

nxt[n_] := Flatten[3 # + {-1, 0} & /@ n]; Union[Flatten[NestList[nxt,{1},5]]] (* G. C. Greubel, Aug 28 2016 *)

From Peter Munn, Feb 04 2022: (Start)
For k >= 0, 2^k <= n <= 2^(k+1)-1, a(n) = A005836(n+1) - (3^k-1)/2.
For n >= 1, A307744(4*a(2n)-1) = A307744(4*a(2n+1)-1) = A307744(4*a(n)-1) + 1.
(End)

A153774 a(2n) = 3a(2n-1), a(2n+1) = 3a(2n) - 1, with a(1) = 1.

Original entry on oeis.org

1, 3, 8, 24, 71, 213, 638, 1914, 5741, 17223, 51668, 155004, 465011, 1395033, 4185098, 12555294, 37665881, 112997643, 338992928, 1016978784, 3050936351, 9152809053, 27458427158, 82375281474, 247125844421, 741377533263, 2224132599788, 6672397799364, 20017193398091

Offset: 1

Clark Kimberling, Jan 01 2009

Let A be the Hessenberg matrix of order n, defined by: A[1,j] = 1, A[i,i] := 11, (i>1), A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n>=1, a(2n-1)=(-1)^(n-1)*charpoly(A,2). - Milan Janjic, Feb 21 2010

			a(2) = 3*1 = 3.
a(3) = 3*a(2)-1 = 8.
a(4) = 3*a(3) = 24.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3, 1, -3).

Cf. A153773, A153775.

I:=[1,3,8]; [n le 3 select I[n] else 3*Self(n-1) + Self(n-2) - 3*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 28 2016

A153774 := proc(n) 1/4+(-1)^n/8+7*3^(n-1)/8 ; end proc: seq(A153774(n),n=1..80) ; # R. J. Mathar, Mar 13 2010

LinearRecurrence[{3,1,-3},{1,3,8},30] (* or *) Rest[ CoefficientList[ Series[x (-1+2x^2)/((1-x)(3x-1)(1+x)),{x,0,30}],x]] (* Harvey P. Dale, Jun 08 2011 *)
RecurrenceTable[{a[1] == 1, a[2] == 3, a[3] == 8, a[n] == 3 a[n-1] + a[n-2] - 3 a[n-3]}, a, {n, 30}] (* Vincenzo Librandi, Aug 28 2016 *)

a(n)=(3^(n-1)*7)\/8 \\ Charles R Greathouse IV, Aug 28 2016

From R. J. Mathar, Mar 13 2010: (Start)
a(n) = (7*3^(n - 1) + 2 + (-1)^n)/8.
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3).
G.f.: x*(-1 + 2*x^2)/ ((1-x) * (3*x-1) * (1+x)). (End)
E.g.f.: (1/24)*(3*exp(-x) - 16 + 6*exp(x) + 7*exp(3*x)). - G. C. Greubel, Aug 27 2016

A153777 Sequence S such that 1 is in S and if x is in S, then 5x-1 and 5x+1 are in S.

Original entry on oeis.org

1, 4, 6, 19, 21, 29, 31, 94, 96, 104, 106, 144, 146, 154, 156, 469, 471, 479, 481, 519, 521, 529, 531, 719, 721, 729, 731, 769, 771, 779, 781, 2344, 2346, 2354, 2356, 2394, 2396, 2404, 2406, 2594, 2596, 2604, 2606, 2644, 2646, 2654, 2656, 3594, 3596, 3604

Offset: 1

Clark Kimberling, Jan 02 2009

nonn

Subsequences include A003463, A083065.
1st generation: 1
2nd generation: 4, 6
3rd generation: 19, 21, 29, 31
4th generation: 94, 96, 104, 106, 144, 146, 154, 156
Does every generation contain p or 2p for some prime p?

Harvey P. Dale, Table of n, a(n) for n = 1..10000
Gevorg Hmayakyan, Trig identity for a(n)

Cf. A003463, A030300, A083065, A153773, A153776.

Column k=5 of A360099.

nxt[n_]:=Flatten[5#+{1,-1}&/@n]; Union[Flatten[NestList[nxt,{1},5]]] (* Harvey P. Dale, Dec 25 2012 *)

Product_{j=0..n-1} cos(5^j) = 2^(-n+1)*Sum_{i=2^(n-1)..2^n-1} cos(a(i)). - Gevorg Hmayakyan, Jan 15 2017
Sum_{i=2^(n-1)..2^n-1} cos(a(i)/5^(n-1)*Pi/2) = 0. - Gevorg Hmayakyan, Jan 15 2017
a(n) mod 2 = A030300(n). - Alois P. Heinz, Jan 29 2023

A153776 Sequence S such that 1 is in S and if x is in S, then 5x-3 and 5x-1 are in S.

Original entry on oeis.org

1, 2, 4, 7, 9, 17, 19, 32, 34, 42, 44, 82, 84, 92, 94, 157, 159, 167, 169, 207, 209, 217, 219, 407, 409, 417, 419, 457, 459, 467, 469, 782, 784, 792, 794, 832, 834, 842, 844, 1032, 1034, 1042, 1044, 1082, 1084, 1092, 1094, 2032, 2034, 2042, 2044, 2082, 2084

Offset: 1

Clark Kimberling, Jan 02 2009

nonn

Subsequences include A047850, A083065.
First generation: 1
2nd generation: 2, 4
3rd generation: 7, 9, 17, 19
4th generation: 32, 34, 42, 44, 82, 84, 92, 94
Does every generation contain p or 2p for some prime p?

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A047850, A083065, A153773, A153775.

nxt[n_] := Flatten[5 # + {-3, -1} & /@ n]; Union[Flatten[NestList[nxt, {1}, 5]]] (* G. C. Greubel, Aug 28 2016 *)

A371275 a(n) is the number of runs in the balanced ternary expansion of n.

Original entry on oeis.org

0, 1, 2, 2, 1, 2, 3, 3, 3, 2, 3, 2, 2, 1, 2, 3, 3, 4, 3, 4, 4, 4, 3, 3, 4, 4, 3, 2, 3, 4, 4, 3, 2, 3, 3, 3, 2, 3, 2, 2, 1, 2, 3, 3, 4, 3, 4, 4, 4, 3, 4, 5, 5, 4, 3, 4, 5, 5, 4, 4, 5, 5, 5, 4, 5, 4, 4, 3, 3, 4, 4, 5, 4, 5, 5, 5, 4, 3, 4, 4, 3, 2, 3, 4, 4, 3, 4

Offset: 0

Rémy Sigrist, Mar 17 2024

Leading zeros are ignored.

Every positive integers occurs infinitely many times.

			The first terms, alongside the balanced ternary expansion of n, are:
  n    a(n)  bter(n)
  ---  ----  -------
    0     0        0
    1     1        1
    2     2       1T
    3     2       10
    4     1       11
    5     2      1TT
    6     3      1T0
    7     3      1T1
    8     3      10T
    9     2      100
   10     3      101
   11     2      11T
   12     2      110
   13     1      111
   14     2     1TTT
   15     3     1TT0

Cf. A005811, A043555, A134021, A153773, A297770.

a(n) = { my (r = 0, d); while (n, r++; d = centerlift(Mod(n, 3)); while (d==centerlift(Mod(n, 3)), n = (n-d)/3;);); return (r); }

a(n) <= A134021(n).

a(A153773(k)) = k for any k > 0.

cp's OEIS Frontend

A153775 Sequence S such that 1 is in S and if x is in S, then 3x-1 and 3x are in S.

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A153774 a(2*n) = 3*a(2*n-1), a(2*n+1) = 3*a(2*n) - 1, with a(1) = 1.

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A153777 Sequence S such that 1 is in S and if x is in S, then 5x-1 and 5x+1 are in S.

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A153776 Sequence S such that 1 is in S and if x is in S, then 5x-3 and 5x-1 are in S.

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A371275 a(n) is the number of runs in the balanced ternary expansion of n.

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Programs

PARI

Formula

A153774 a(2n) = 3a(2n-1), a(2n+1) = 3a(2n) - 1, with a(1) = 1.