A158919 -id:A158919 - cp's OEIS frontend

A276385 Defined by the properties that it starts with 2, and when you successively apply DIFF, RUNS, BISECT, RUNS you get (4,1,3,1) repeated infinitely often.

2, 5, 8, 11, 14, 17, 19, 22, 25, 28, 31, 34, 36, 39, 42, 45, 48, 51, 53, 56, 59, 62, 65, 68, 70, 73, 76, 79, 82, 85, 88, 90, 93, 96, 99, 102, 105, 107, 110, 113, 116, 119, 122, 124, 127, 130, 133, 136, 139, 141, 144, 147, 150, 153, 156, 159, 161, 164, 167, 170, 173, 176, 178, 181, 184, 187, 190, 193

Offset: 1

N. J. A. Sloane, Sep 03 2016

nonn

Here DIFF means take first differences, RUNS means list successive run lengths, and BISECT means take alternate terms.
This agree with the Beatty sequence for 1+t, where t is the tribonacci constant (A140099) for n <= 17160 but thereafter is different. In fact A140099(17161) = 48725, whereas a(17161) = 48724.
This arose in an attempt to find recurrences for A140099 and several related sequences. The moral is that without a proof, apparent recurrences are worthless.

			Seq. 2, 5, 8, 11, 14, 17, 19, 22, 25, 28, 31, 34, 36, 39, 42, 45, 48, 51, 53, 56, ...
DIFF 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, ...
RUNS 5, 1, 5, 1, 5, 1, 5, 1, 6, 1, 5, 1, 5, 1, 5, 1, 6, 1, 5, 1, 5, 1, 5, 1, 5, 1, ...
BISECT  5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 5, 5, 6, 5, 5, 5, ...
RUNS 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 4, ...

N. J. A. Sloane, Table of n, a(n) for n = 1..40000

Cf. A140099, A158919, A276384.

with(transforms): r1:=[]:
for n from 1 to 1000 do r1:=[op(r1), 5,1,5,1,5,1,5,1,6,1,5,1,5,1,5,1,6,1]; od:
r2:=[]: for n from 1 to nops(r1) do if r1[n]=1 then r2:=[op(r2),2]; else for i from 1 to r1[n] do r2:=[op(r2),3]; od: fi: od:
r3:=[2, op(map(x->x+2,PSUM(r2)))]:

For n >= 1, a(n) = A276384(n)+n.

A276384 Defined by the properties that it starts with 0, and when you successively apply DIFF, RUNS, BISECT, RUNS you get (4,1,3,1) repeated infinitely often.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 23, 25, 27, 29, 31, 33, 34, 36, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 57, 58, 60, 62, 64, 66, 68, 69, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 90, 91, 93, 95, 97, 99, 101, 103, 104, 106, 108, 110, 112, 114, 115, 117, 119, 121, 123, 125, 126, 128, 130, 132

Offset: 0

N. J. A. Sloane, Sep 03 2016

nonn

Here DIFF means take first differences, RUNS means list successive run lengths, and BISECT means take alternate terms.
This agrees with the Beatty sequence for the tribonacci constant (A158919) for n <= 17160 but thereafter is different. In fact A158919(17161) = 31564, whereas a(17161) = 31563.
This arose in an attempt to find recurrences for A158919 and several related sequences. The moral is that without a proof, apparent recurrences are worthless.

			Seq. 0, 1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 23, 25, 27, 29, ...
DIFF 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2 ...
RUNS 1, 5, 1, 5, 1, 5, 1, 5, 1, 6, 1, 5, 1, 5, 1, 5, 1, 6, 1, 5, 1, ...
BISECT  5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, ...
RUNS 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 4, ...

N. J. A. Sloane, Table of n, a(n) for n = 0..40000

Cf. A158919, A276385.

with(transforms): r1:=[]:
for n from 1 to 1000 do r1:=[op(r1), 1,5,1,5,1,5,1,5,1,6,1,5,1,5,1,5,1,6]; od:
r2:=[]: for n from 1 to nops(r1) do if r1[n]=1 then r2:=[op(r2),1]; else for i from 1 to r1[n] do r2:=[op(r2),2]; od: fi: od:
r3:=[0,op(PSUM(r2))]:

For n >= 1, a(n) = A276385(n)-n.

A316711 Decimal expansion of s:= t/(t - 1), with the tribonacci constant t = A058265.

Original entry on oeis.org

2, 1, 9, 1, 4, 8, 7, 8, 8, 3, 9, 5, 3, 1, 1, 8, 7, 4, 7, 0, 6, 1, 3, 5, 4, 2, 6, 8, 2, 2, 7, 5, 1, 7, 2, 9, 3, 4, 7, 4, 6, 9, 1, 0, 2, 1, 8, 7, 4, 2, 8, 8, 0, 9, 1, 0, 0, 9, 7, 8, 1, 3, 3, 8, 6, 1, 7, 6, 8, 5, 9, 4, 8, 0, 0, 4, 9, 7, 0, 1, 4, 6, 1, 1, 1, 7, 9, 6, 6, 6, 7, 0, 0, 2, 1, 8, 3, 0, 6

Offset: 1

Wolfdieter Lang, Sep 07 2018

Because the tribonacci constant t = A058265 > 1, with Beatty sequence At(n) := floor(n*t), n >= 1 (with At(0) = 0) given in A158919, has the companion sequence Bt := floor(n*s), n >= 1, (with Bt(0) = 0), with 1/t + 1/s = 1, and At and Bt are complementary, disjoint sequences for the positive integers. Note that Bt is not A172278. The first entries n = 0..161 coincide. A172278(162) = 354 but At(193) = A158919(193) = 354, hence A172278 is not complementary together with At. In fact, Bt(162) = 355, which is not a member of At.

s-1 = 1/(t-1) equals the real root of 2*x^3 - 2*x - 1. See the formulas below. - Wolfdieter Lang, Sep 15 2022

			s = 2.191487883953118747061354268227517293474691021874288091009781338617685...

Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
Wikipedia, Beatty Sequence

Cf. A317202, A058265, A158919, A172278, A276801, A357463.

Digits := 120: a := (1/4 + sqrt(33)/36)^(1/3): 1 + a + 1/(3*a): evalf(%)*10^98: ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 15 2022

With[{t=x/.Solve[x^3-x^2-x-1==0,x][[1]]},RealDigits[t/(t-1),10,120][[1]]] (* Harvey P. Dale, Sep 12 2021 *)

s = t/(t - 1) with the tribonacci constant t = A058265, the real root of the cubic x^3 - x^2 - x - 1.
s = (1 + (19 + 3*sqrt(33))^(1/3) + (19 - 3*sqrt(33))^(1/3)) / (-2 + (19 + 3*sqrt(33))^(1/3) + (19 - 3*sqrt(33))^(1/3)).
From Wolfdieter Lang, Sep 15 2022: (Start)
s = 1 + ((1 + (1/9)*sqrt(33))/4)^(1/3)+(1/3)*((1 + (1/9)*sqrt(33))/4)^(-1/3).
s = 1 + ((1 + (1/9)*sqrt(33))/4)^(1/3) + ((1 - (1/9)*sqrt(33))/4)^(1/3).
s = 1 + (2/3)*sqrt(3)*cosh((1/3)*arccosh((3/4)*sqrt(3))). (End)
From Dimitri Papadopoulos, Nov 07 2023: (Start)
s = 1 + t^3/(t^3 - 1) = 1 + A276801/(A276801 - 1).
s = 1 + t^2/(t+1). (End)

A108168 Padovan sequence for indices of the Beatty sequence of the tribonacci constant.

Original entry on oeis.org

0, 1, 2, 3, 5, 9, 12, 21, 37, 65, 114, 200, 265, 465, 816, 1432, 2513, 4410, 5842, 10252, 17991, 31572, 55405, 97229, 128801, 226030, 396655, 696081, 1221537, 2143648, 3761840, 4983377, 8745217, 15346786, 26931732, 47261895, 82938844, 109870576

Offset: 1

Roger L. Bagula, Jun 13 2005

Limit[a(n)/a(n-1),n->Infinity]={1.32472, 1.75488, 1.75488, 1.75488, 1.75488, 1.75488}

Cf. A000931.

NSolve[x^3 - x^2 - x - 1 = 0, x] beta = 1.8392867552141612 F[1] = 0; F[2] = 1; F[3] = 1; F[n__] := F[n] = F[n - 2] + F[n - 3] a1 = Table[F[Floor[beta*n]], {n, 1, 50}] Table[N[a1[[n]]/a1[[n - 1]]], {n, 3, Length[a1]}]

a(n) = A000931(3+A158919(n)). - R. J. Mathar, Sep 11 2011

cp's OEIS Frontend

A276385 Defined by the properties that it starts with 2, and when you successively apply DIFF, RUNS, BISECT, RUNS you get (4,1,3,1) repeated infinitely often.

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A276384 Defined by the properties that it starts with 0, and when you successively apply DIFF, RUNS, BISECT, RUNS you get (4,1,3,1) repeated infinitely often.

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A316711 Decimal expansion of s:= t/(t - 1), with the tribonacci constant t = A058265.

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A108168 Padovan sequence for indices of the Beatty sequence of the tribonacci constant.

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