A276799 -id:A276799 - cp's OEIS frontend

A003145 Positions of letter b in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).

2, 6, 9, 13, 15, 19, 22, 26, 30, 33, 37, 39, 43, 46, 50, 53, 57, 59, 63, 66, 70, 74, 77, 81, 83, 87, 90, 94, 96, 100, 103, 107, 111, 114, 118, 120, 124, 127, 131, 134, 138, 140, 144, 147, 151, 155, 158, 162, 164, 168, 171, 175, 179, 182, 186, 188, 192, 195, 199, 202, 206, 208

Offset: 1

N. J. A. Sloane

nonn

A003144, A003145, A003146 may be defined as follows. Consider the map psi: a -> ab, b -> ac, c -> a. The image (or trajectory) of a under repeated application of this map is the infinite word a, b, a, c, a, b, a, a, b, a, c, a, b, a, b, a, c, ... (setting a = 1, b = 2, c = 3 gives A092782). The indices of a, b, c give respectively A003144, A003145, A003146. - Philippe Deléham, Feb 27 2009
The infinite word may also be defined as the limit S_oo where S_1 = a, S_n = psi(S_{n-1}). Or, by S_1 = a, S_2 = ab, S_3 = abac, and thereafter S_n = S_{n-1} S_{n-2} S_{n-3}. It is the unique word such that S_oo = psi(S_oo).
Also indices of b in the sequence closed under a -> abac, b -> aba, c -> ab; starting with a(1) = a. - Philippe Deléham, Apr 16 2004
Theorem: A number m is in this sequence iff the tribonacci representation of m-1 ends with 01. [Duchene and Rigo, Remark 2.5] - N. J. A. Sloane, Mar 02 2019

Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, Urban Larsson, Wythoff Visions, Games of No Chance, Vol. 5; MSRI Publications, Vol. 70 (2017), pages 101-153.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10609
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69. The present sequence is called b.
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Eric Duchêne and Michel Rigo, A morphic approach to combinatorial games: the Tribonacci case. RAIRO - Theoretical Informatics and Applications, 42, 2008, pp 375-393. doi:10.1051/ita:2007039. [Also available from Numdam]
A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Cf. A003144, A003146, A080843, A092782, A058265, A276799, A276800, A276794, A276797.
First differences give A276789. A278040 (subtract 1 from each term, and use offset 1).
For tribonacci representations of numbers see A278038.

M:=17; S[1]:=`a`; S[2]:=`ab`; S[3]:=`abac`;
for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:
t0:=S[M]: l:=length(t0); t1:=[];
for i from 1 to l do if substring(t0,i..i) = `b` then t1:=[op(t1),i]; fi; od: # N. J. A. Sloane

StringPosition[SubstitutionSystem[{"a" -> "ab", "b" -> "ac", "c" -> "a"}, "b", {#}][[1]], "b"][[All, 1]] &@9 (* Michael De Vlieger, Mar 30 2017, Version 10.2, after JungHwan Min at A003144 *)

It appears that a(n) = floor(n*t^2) + eps for all n, where t is the tribonacci constant A058265 and eps is 0, 1, or 2. See A276799. - N. J. A. Sloane, Oct 28 2016. This is true - see the Dekking et al. paper. - N. J. A. Sloane, Jul 22 2019

More terms from Philippe Deléham, Apr 16 2004
Corrected by T. D. Noe and N. J. A. Sloane, Nov 01 2006
Entry revised by N. J. A. Sloane, Oct 13 2016

A276800 Decimal expansion of t^2, where t is the tribonacci constant A058265.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 28 2016

The minimal polynomial of this constant is x^3 - 3*x^2 - x - 1, and it is its unique real root. - Amiram Eldar, May 27 2023

			3.38297576790623749412270853645503458694938204374857618201956267723537...

Index entries for algebraic numbers, degree 3

Cf. A058265, A276799, A276801.

A276800L[n_] := RealDigits[(1/3 (1 + (19 - 3 Sqrt[33])^(1/3) + (19 + 3 Sqrt[33])^(1/3)))^2, 10, n][[1]]; A276800L[107] (* JungHwan Min, Nov 06 2016 *)
RealDigits[x /. FindRoot[x^3 - 3*x^2 - x - 1, {x, 3}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, May 27 2023 *)

polrootsreal(x^3-3*x^2-x-1)[1] \\ Charles R Greathouse IV, Aug 21 2023

A277724 Intersection of A158919 and A277722.

Original entry on oeis.org

0, 3, 16, 20, 23, 27, 33, 40, 47, 57, 60, 64, 71, 77, 84, 91, 101, 104, 108, 115, 121, 125, 128, 145, 148, 152, 158, 165, 169, 172, 182, 189, 196, 202, 206, 209, 213, 226, 233, 240, 246, 250, 253, 257, 263, 270, 274, 277, 290, 294, 297, 301, 307, 314, 321, 331, 334, 338, 345, 351, 358, 375, 378, 382

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

See A277728 for discussion.

Cf. A003144, A003145, A003146, A058265, A158919, A275926, A276799, A276800, A277722, A277723, A277724, A277725, A277726, A277727, A277728.

A277725 Intersection of A158919 and A277723.

Original entry on oeis.org

0, 12, 18, 31, 49, 62, 68, 80, 93, 99, 112, 130, 136, 143, 161, 174, 180, 211, 217, 224, 242, 248, 255, 261, 286, 292, 323, 329, 336, 342, 354, 360, 367, 373, 404, 410, 423, 435, 441, 448, 454, 472, 485, 491, 516, 522, 535, 553, 560, 566, 572, 584, 597, 603, 616, 634, 640, 647, 665, 678, 684, 709, 715

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

See A277728 for discussion.

Cf. A003144, A003145, A003146, A058265, A158919, A275926, A276799, A276800, A277722, A277723, A277724, A277725, A277726, A277727, A277728.

A277726 Intersection of A277722 and A277723.

Original entry on oeis.org

0, 6, 37, 43, 74, 87, 118, 155, 186, 192, 199, 230, 236, 267, 280, 304, 311, 317, 348, 385, 392, 416, 429, 460, 466, 497, 504, 510, 541, 578, 622, 659, 690, 696, 703, 734, 740, 771, 784, 808, 815, 852, 889, 896, 920, 933, 964, 970, 1001, 1008, 1014, 1045, 1082, 1126, 1163, 1194, 1200, 1207, 1238, 1244, 1275, 1288, 1312, 1319, 1356, 1387, 1393, 1400, 1424

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

See A277728 for discussion.

Cf. A003144, A003145, A003146, A058265, A158919, A275926, A276799, A276800, A277722, A277723, A277724, A277725, A277726, A277727, A277728.

Digits := 120;
isA277722 := proc(n)
    a276800 :=  3.3829757679062374941227085364550345869493820437485761820195626772353718960099402922235933340043661396041006 ;
    for x from floor((n-3)/a276800) to (n+3)/a276800 do
        if floor(x*a276800) = n then
            return true;
        end if;
    end do:
    return false;
end proc:
isA277723 := proc(n)
    a276801 :=  6.2222625231203986266745611011083211873735607898461684287983213166395751180919067179620287534326731537460804;
    for x from floor((n-3)/a276801) to (n+3)/a276801 do
        if floor(x*a276801) = n then
            return true;
        end if;
    end do:
    return false;
end proc:
isA277726 := proc(n)
    isA277722(n) and isA277723(n) ;
end proc:
for n from 0 to 8000 do
    if isA277726(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Nov 02 2016

Corrected by R. J. Mathar, Nov 01 2016

A277727 Union of A158919, A277722, A277723.

Original entry on oeis.org

0, 1, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 18, 20, 22, 23, 24, 25, 27, 29, 30, 31, 33, 34, 36, 37, 38, 40, 42, 43, 44, 45, 47, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 62, 64, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 82, 84, 86, 87, 88, 90, 91, 93, 94, 95, 97, 98, 99, 101, 103, 104, 105, 106

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

See A277728 for discussion.

Cf. A003144, A003145, A003146, A058265, A158919, A275926, A276799, A276800, A277722, A277723, A277724, A277725, A277726, A277727, A277728.

A277721 a(n) = floor(n*t^3) - A003146(n), where t = 1.8392867... is the tribonacci constant A058265.

Original entry on oeis.org

2, 1, 1, 0, 3, 2, 2, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 3, 2, 2, 1, 0, 1, 0, 2, 1, 2, 1, 3, 2, 2, 2, 1, 1, 0, 3, 2, 2, 1, 1, 1, 3, 2, 2, 2, 1, 1, 0, 2, 2, 2, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 2, 2, 2, 1, 0, 0, 0, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 0, 2, 2, 2, 1, 1, 0, 3, 2, 2, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 3, 2

Offset: 1

N. J. A. Sloane, Oct 28 2016

sign

Always in the set {-1,0,1,2,3}, but first occurrence of -1 is at n = 2047. - Jeffrey Shallit, Nov 19 2016

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A003144, A003145, A003146, A275926, A276799, A058265, A276800, A276801, A278353 (positions of -1's).

cp's OEIS Frontend

A003145 Positions of letter b in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).

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A276800 Decimal expansion of t^2, where t is the tribonacci constant A058265.

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A277724 Intersection of A158919 and A277722.

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A277725 Intersection of A158919 and A277723.

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A277726 Intersection of A277722 and A277723.

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A277727 Union of A158919, A277722, A277723.

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A277721 a(n) = floor(n*t^3) - A003146(n), where t = 1.8392867... is the tribonacci constant A058265.

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