A278039 -id:A278039 - cp's OEIS frontend

A276798 Partial sums of A276791.

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15

Offset: 0

N. J. A. Sloane, Oct 28 2016

a(n+1) - 1 = z_C(n), where z_C(n) is the number of C numbers A276798 not exceeding n, for n >= 0, and z_C(-1) = 0. - Wolfdieter Lang, Dec 05 2018

Conjecture: 3*n - A140101(n) = a(n-1). - N. J. A. Sloane, Oct 26 2016 (added Mar 21 2019). This is true - see the Dekking et al. paper. - N. J. A. Sloane, Jul 22 2019

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
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.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
Jeffrey Shallit, Some Tribonacci conjectures, arXiv:2210.03996 [math.CO], 2022.

A276793(n) + A276794(n) + A276791(n) = 1;
A276796(n) + A276797(n) + A276798(n) = n + 1.
Cf. A276798, A278039.

M:=12;
S[1]:=`0`; S[2]:=`01`; S[3]:=`0102`;
for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:
t0:=S[M]: # has 927 terms of tribonacci ternary word A080843
# get numbers of 0's, 1's, 2's
N0:=[]: N1:=[]: N2:=[]: c0:=0: c1:=0: c2:=0:
L:=length(t0);
for i from 1 to L do
js := substring(t0, i..i);
j:=convert(js,decimal,10);
if j=0 then c0:=c0+1; elif j=1 then c1:=c1+1; else c2:=c2+1; fi;
N0:=[op(N0),c0]; N1:=[op(N1),c1]; N2:=[op(N2),c2];
od:
N0; N1; N2; # prints A276796, A276797, A276798 (except A276798 is off by 1 because it does not count the initial 0 in A003146). # N. J. A. Sloane, Jun 08 2018

a(n) = Sum_{k=0..n} A276791(k), for n >= 0.
a(n) = n + 1 - (A276796(n) + A276797(n)).
a(n) = 2*n + 1 - B(n), where B(n) = A278039(n), n >= 0. For a proof see the comment on z_C and Proposition 7, eq. 43, of the W. Lang link given in A080843. - Wolfdieter Lang, Dec 05 2018

A276796 Partial sums of A276793.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 25, 25, 26, 26, 27, 27, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 40, 41, 42, 42, 43, 43, 44

Offset: 0

N. J. A. Sloane, Oct 28 2016

a(n+1) = z_B(n), the number of B numbers A278039 not exceeding n. - Wolfdieter Lang, Dec 05 2018

Conjecture: A140100(n) - n = a(n-1). - N. J. A. Sloane, Oct 26 2016 (added Mar 21 2019). This is true: see the Dekking et al. paper. - N. J. A. Sloane, Jul 22 2019

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
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.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
Jeffrey Shallit, Some Tribonacci conjectures, arXiv:2210.03996 [math.CO], 2022.

Cf. A003144, A140100, A276793 (first differences), A278039, A278040.
A276793(n) + A276794(n) + A276791(n) = 1;
A276796(n) + A276797(n) + A276798(n) = n + 1.

M:=12;
S[1]:=`0`; S[2]:=`01`; S[3]:=`0102`;
for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:
t0:=S[M]: # has 927 terms of tribonacci ternary word A080843
# get numbers of 0's, 1's, 2's
N0:=[]: N1:=[]: N2:=[]: c0:=0: c1:=0: c2:=0:
L:=length(t0);
for i from 1 to L do
js := substring(t0, i..i);
j:=convert(js,decimal,10);
if j=0 then c0:=c0+1; elif j=1 then c1:=c1+1; else c2:=c2+1; fi;
N0:=[op(N0),c0]; N1:=[op(N1),c1]; N2:=[op(N2),c2];
od:
N0; N1; N2; # prints A276796, A276797, A276798 (except A276798 is off by 1 because it does not count the initial 0 in A003146). # N. J. A. Sloane, Jun 08 2018

a(n) = Sum_{k=0..n} A276793(k), n >= 0.
a(n) = n + 1 - (A276797(n) + A276798(n)).
a(n) = A(n) - B(n) - (n + 1), where A(n) = A278040(n) and B(n) = A278039(n), n >= 0. For a proof see the W. Lang link in A278040, Proposition 7, eq. (42). - Wolfdieter Lang, Dec 05 2018

A276788 First differences of A003144.

Original entry on oeis.org

2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2

Offset: 1

N. J. A. Sloane, Oct 14 2016

In A276790, leave 2's unchanged, but replace 1's by 2's and 0's by 1's, and then omit the initial 1.
If we prefixed A003144 with an initial 0, then its first differences would be a' := 1 followed by a, that is, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, ... If we now add 1 to every term of a' we get A305374 = first differences of A140101. - N. J. A. Sloane, Jul 17 2018
This relation between A003144 and A140101 is a conjecture - Michel Dekking, Mar 18 2019 [It has been a theorem since Mar 22 2019. - N. J. A. Sloane, Jun 25 2019. (See the Dekking et al. paper)]
(a(n)) is a morphic sequence: in the tribonacci word A092782 = 1,2,1,3,1,2,1,1,... map  1 -> 2, 2 -> 2, 3 -> 1. - Michel Dekking, Mar 21 2019

Robert Israel, Table of n, a(n) for n = 1..10608
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320. See page 316.
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.

Cf. A003144, A003145, A003146, A080843, A276790, A276791, A275925, A305374, A140101.

See A278039 for partial sums.

M:= 10: # to use M generations of strings
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:
P:=select(t -> S[M][t]="a", [$1..length(S[M])]):
P[2..-1]-P[1..-2]; # Robert Israel, Nov 01 2016

a(n) = A003144(n+1) - A003144(n), n >= 1.

a(n+1) = 2 - t(n)*(t(n) - 1)/2 = 2 - A276791(n+1), for n >= 0, where t(n) = A080843(n). See the W. Lang link in A080843, eq. (38). - Wolfdieter Lang, Dec 06 2018

A308199 The tribonacci representation of a(n) is obtained by appending 0,0 to the tribonacci representation of n (cf. A278038).

Original entry on oeis.org

0, 4, 7, 11, 13, 17, 20, 24, 28, 31, 35, 37, 41, 44, 48, 51, 55, 57, 61, 64, 68, 72, 75, 79, 81, 85, 88, 92, 94, 98, 101, 105, 109, 112, 116, 118, 122, 125, 129, 132, 136, 138, 142, 145, 149, 153, 156, 160, 162, 166, 169, 173, 177, 180, 184, 186, 190, 193, 197, 200, 204, 206, 210, 213, 217, 221, 224, 228

Offset: 0

N. J. A. Sloane, Jun 23 2019

From Michel Dekking, Oct 06 2019: (Start)
If w is a binary vector not containing 111, then w00 and w01 are also binary vectors not containing 111. So a(n) = A278040(n) - 1.
This sequence gives the positions of the word ab in the tribonacci word t, when t is given offset 0.
This sequence is the compound sequence A278039(A278039) of the three sequences A278039, A278040, A278041, which are the building blocks of the tribonacci world with offset 0. (End)

			u = abacabaabacaba.., then u(0)u(1) = ab, u(4)u(5) = ab, u(7)u(8) = ab, u(11)u(12) = ab.

Cf. A278038, A278039, A278040, A278041, A308200.

Essentially partial sums of A276789.

From Michel Dekking, Oct 06 2019: (Start)
a(n) = Sum_{k=1..n-1} d(k), where d is the tribonacci word on the alphabet {4,3,2}.
a(n) = A003144(A003144(n)) - 1. (End)

A319967 a(n) = A003145(A003144(n)) where A003144 and A003145 are positions of '1' and '2' in the tribonacci word A092782.

Original entry on oeis.org

2, 9, 15, 22, 26, 33, 39, 46, 53, 59, 66, 70, 77, 83, 90, 96, 103, 107, 114, 120, 127, 134, 140, 147, 151, 158, 164, 171, 175, 182, 188, 195, 202, 208, 215, 219, 226, 232, 239, 245, 252, 256, 263, 269, 276, 283, 289, 296, 300, 307, 313, 320, 327, 333, 340, 344

Offset: 1

N. J. A. Sloane, Oct 05 2018

By analogy with the Wythoff compound sequences A003622 etc., the nine compounds of A003144, A003145, A003146 might be called the tribonacci compound sequences. They are A278040, A278041, and A319966-A319972.

This sequence gives the positions of the word bac in the tribonacci word t = abacabaa..., fixed point of the morphism a->ab, b->ac, c->a. This follows from the fact that the word ac is always preceded in t by the letter b, and the formula BA = C-2, where A := A003144, B := A003145, C := A003146. - Michel Dekking, Apr 09 2019

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320. Compare page 318.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69, Theorem 13.

Cf. A003144, A003145, A003146, A003622, A278039, A278040, A278041, and A319966-A319972.

Cf. A092782 (ternary tribonacci word).

a(n+1) = B(A(n)) = B(A(n) + 1) - 2 = A(n) + B(n) + n + 1, for n >= 0, where B = A278039 and A = A278040. For a proof see the W. Lang link in A278040, Proposition 9, eq. (51). - Wolfdieter Lang, Dec 13 2018

More terms from Rémy Sigrist, Oct 16 2018

A319198 Partial sums of the infinite self-similar tribonacci word, written in the form A080843.

Original entry on oeis.org

0, 1, 1, 3, 3, 4, 4, 4, 5, 5, 7, 7, 8, 8, 9, 9, 11, 11, 12, 12, 12, 13, 13, 15, 15, 16, 16, 18, 18, 19, 19, 19, 20, 20, 22, 22, 23, 23, 24, 24, 26, 26, 27, 27, 27, 28, 28, 30, 30, 31, 31, 31, 32, 32, 34, 34, 35, 35, 36, 36, 38, 38, 39, 39, 39, 40, 40, 42, 42, 43, 43

Offset: 0

Wolfdieter Lang, Oct 10 2018

This sequence produces a formula for the A-numbers A278040, specifying the positions (or indices) of 1's in A080843, namely A(n) = 4*n+1 - a(n-1), with a(-1) = 0.

Vincenzo Librandi, Table of n, a(n) for n = 0..10600
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Cf. A080843, A276797, A276798, A278039 (B-numbers), A278040 (A-numbers), A278041 (C-numbers).

a(n) = Sum_{j=0..n} A080843(n), n >= 0.

a(n) = z_A(n) + 2*z_C(n) = A276797(n+1) + 2*(A276798(n+1) - 1), where z_A(n) gives the number of A-numbers from A278040 not exceeding n, similarly for z_C(n) with the C-numbers from A278041. - Wolfdieter Lang, Dec 13 2018

A322409 Compound tribonacci sequence with a(n) = A278040(A278040(n)), for n >= 0.

Original entry on oeis.org

5, 18, 29, 42, 49, 62, 73, 86, 99, 110, 123, 130, 143, 154, 167, 178, 191, 198, 211, 222, 235, 248, 259, 272, 279, 292, 303, 316, 323, 336, 347, 360, 373, 384, 397, 404, 417, 428, 441, 452, 465, 472, 485, 496, 509, 522, 533, 546, 553, 566, 577, 590, 603, 614, 627, 634, 647, 658, 671, 682, 695

Offset: 0

Wolfdieter Lang, Jan 02 2019

(a(n+1)) = A319968(n)-1 = A003145(A003145(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking, Apr 04 2019

The nine sequences A308199, A319967, A319968, A322410, A322409, A322411, A322413, A322412, A322414 are based on defining the tribonacci ternary word to start with index 0 (in contrast to the usual definition, in A080843 and A092782, which starts with index 1). As a result these nine sequences differ from the compound tribonacci sequences defined in A278040, A278041, and A319966-A319972. - N. J. A. Sloane, Apr 05 2019

Cf. A278039, A278040, A322408.

Cf. A003144, A003145, A003146.

a(n) = A(A(n)) = A(A(n) + 1) - 3 = 2*(A(n) + B(n)) + n + 3, for n >= 0, where A = A278040 and B = A278039. For a proof see the W. Lang link in A278040, Proposition 9, eq. (48).

A322412 Compound tribonacci sequence with a(n) = A278041(A278040(n)), for n >= 0.

Original entry on oeis.org

10, 34, 54, 78, 91, 115, 135, 159, 183, 203, 227, 240, 264, 284, 308, 328, 352, 365, 389, 409, 433, 457, 477, 501, 514, 538, 558, 582, 595, 619, 639, 663, 687, 707, 731, 744, 768, 788, 812, 832, 856, 869, 893, 913, 937, 961, 981, 1005, 1018, 1042, 1062, 1086, 1110, 1130, 1154, 1167, 1191, 1211, 1235

Offset: 0

Wolfdieter Lang, Jan 02 2019

(a(n+1)) = A319971(n)-1 = A003146(A003145(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking

The nine sequences A308199, A319967, A319968, A322410, A322409, A322411, A322413, A322412, A322414 are based on defining the tribonacci ternary word to start with index 0 (in contrast to the usual definition, in A080843 and A092782, which starts with index 1). As a result these nine sequences differ from the compound tribonacci sequences defined in A278040, A278041, and A319966-A319972. - N. J. A. Sloane, Apr 05 2019

Cf. A278040, A278041, A322411.

a(n) = C(A(n)) = C(A(n) + 1) - 6 = 4*A(n) + 3*B(n) + 2*(n+3). for n >= 0, where A = A278040, B = A278039 and C = A278041. For a proof see the W. Lang link in A278040, Proposition 9, eq. (54).

A322414 Compound tribonacci sequence with a(n) = A278041(A278041(n)), for n >= 0.

Original entry on oeis.org

23, 67, 104, 148, 172, 216, 253, 297, 341, 378, 422, 446, 490, 527, 571, 608, 652, 676, 720, 757, 801, 845, 882, 926, 950, 994, 1031, 1075, 1099, 1143, 1180, 1224, 1268, 1305, 1349, 1373, 1417, 1454, 1498, 1535, 1579, 1603, 1647, 1684, 1728, 1772, 1809, 1853, 1877, 1921, 1958, 2002, 2046, 2083, 2127, 2151, 2195, 2232, 2276, 2313, 2357

Offset: 0

Wolfdieter Lang, Jan 02 2019

(a(n+1)) = A319972(n)-1 = A003146(A003146(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking, Apr 04 2019

The nine sequences A308199, A319967, A319968, A322410, A322409, A322411, A322413, A322412, A322414 are based on defining the tribonacci ternary word to start with index 0 (in contrast to the usual definition, in A080843 and A092782, which starts with index 1). As a result these nine sequences differ from the compound tribonacci sequences defined in A278040, A278041, and A319966-A319972. - N. J. A. Sloane, Apr 05 2019

Cf. A278039, A278040, A278041, A322413.

Cf. A003144, A003145, A003146.

a(n) = C(C(n)) = C(C(n) + 1) - 4 = 7*A(n) + 6*B(n) + 4*(n + 4), for n >= 0, where A = A278040, B = A278039 and C = A278041. For a proof see the W. Lang link in A278040, Proposition 9, eq. (56).

A316713 Unique representation of nonnegative numbers by iterated tribonacci A, B and C sequences.

Original entry on oeis.org

1, 21, 121, 31, 1121, 221, 131, 11121, 2121, 1221, 321, 1131, 231, 111121, 21121, 12121, 3121, 11221, 2221, 1321, 11131, 2131, 1231, 331, 1111121, 211121, 121121, 31121, 112121, 22121, 13121, 111221, 21221, 12221, 3221, 11321, 2321, 111131, 21131, 12131, 3131, 11231, 2231, 1331, 11111121, 2111121, 1211121, 311121, 1121121, 221121, 131121, 1112121, 212121, 122121, 32121, 113121, 23121, 1111221, 211221, 121221, 31221, 112221, 22221, 13221, 111321, 21321, 12321

Offset: 0

Wolfdieter Lang, Sep 11 2018

This representation is the tribonacci A000073 analog of the Wythoff representation of numbers (A189921 or A317208) for the Fibonacci case.
The complementary and disjoint sets A, B and C are given by the sequences A278040, A278039, and A278041, respectively.
The present representation uses 1 for B, 2 for A and 3 for C numbers. The brackets for sequence iteration and the final argument 0 have to be added. E.g.: a(0) = 1 for B(1),  a(1) = 21 for A(B(0)), a(2) = 121 for B(A(B(0))), a(3) = 31 for C(B(0)), ...
An equivalent such representation is given by A317206 using different complementary sequences A, B and C, related to our B = A278039,  A = A278040, and C = A278041: A(n) = A003144(n) = A278039(n-1) + 1, B(n) = A003145(n) =  A278040(n-1) + 1, C(n) = A003146(n)  =  A278041(n-1) + 1 with n >= 1.
The length of the string a(n) is A316714(n). The number of B, A and C sequences used for the ABC-representation of n (that is the number of 1s, 2s and 3s of a(n)) is A316715, A316716 and A316717, respectively.

			The complementary and disjoint sequences A, B, C begin, for n >= 0:
n: 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15  16  17  18  19  20  21  22 ...
A: 1  5  8 12 14 18 21 25 29 32 36 38 42 45 49 52  56  58  62  65  69  73  76 ...
B: 0  2  4  6  7  9 11 13 15 17 19 20 22 24 26 28  30  31  33  35  37  39  41 ...
C: 3 10 16 23 27 34 40 47 54 60 67 71 78 84 91 97 104 108 115 121 128 135 141 ...
---------------------------------------------------------------------------------
The ABC representations begin:
                                              #(1)   #(2)    #(3)   L(a(n))
           a(n)                             A316715 A316716 A316717 A316714
n = 0:       1                  B(0) =  0      1      0       0       1
n = 1:      21               A(B(0)) =  1      1      1       0       2
n = 2:     121            B(A(B(0))) =  2      2      1       0       3
n = 3:      31               C(B(0)) =  3      1      0       1       2
n = 4:    1121         B(B(A(B(0)))) =  4      3      1       0       4
n = 5:     221            A(A(B(0))) =  5      1      2       0       3
n = 6:     131            B(C(B(0))) =  6      2      0       1       3
n = 7:   11121      B(B(B(A(B(0))))) =  7      4      1       0       5
n = 8:    2121         A(B(A(B(0)))) =  8      2      2       0       4
n = 9:    1221         B(A(A(B(0)))) =  9      2      2       0       4
n = 10:    321            C(A(B(0))) = 10      1      1       1       3
n = 11:   1131         B(B(C(B(0)))) = 11      3      0       1       4
n = 12:    231            A(C(B(0))) = 12      1      1       1       3
n = 13: 111121   B(B(B(B(A(B(0)))))) = 13      5      1       0       6
n = 14:  21121      A(B(B(A(B(0))))) = 14      3      2       0       5
n = 15:  12121      B(A(B(A(B(0))))) = 15      3      2       0       5
n = 16:   3121         C(B(A(B(0)))) = 16      2      1       1       4
n = 17:  11221      B(B(A(A(B(0))))) = 17      3      2       0       5
n = 18:   2221         A(A(A(B(0)))) = 18      1      3       0       4
n = 19:   1321         B(C(A(B(0)))) = 19      2      1       1       4
n = 20:  11131      B(B(B(C(B(0))))) = 20      4      0       1       5
...
----------------------------------------------------------------------------

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

Cf. A000073, A003144, A003145, A003146, A189921, A317208, A278040, A278039, A278041, A316714, A316715, A316716, A316717, A317208.

cp's OEIS Frontend

A276798 Partial sums of A276791.

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A276796 Partial sums of A276793.

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A276788 First differences of A003144.

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A308199 The tribonacci representation of a(n) is obtained by appending 0,0 to the tribonacci representation of n (cf. A278038).

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A319967 a(n) = A003145(A003144(n)) where A003144 and A003145 are positions of '1' and '2' in the tribonacci word A092782.

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A319198 Partial sums of the infinite self-similar tribonacci word, written in the form A080843.

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A322409 Compound tribonacci sequence with a(n) = A278040(A278040(n)), for n >= 0.

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A322412 Compound tribonacci sequence with a(n) = A278041(A278040(n)), for n >= 0.

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Formula

A322414 Compound tribonacci sequence with a(n) = A278041(A278041(n)), for n >= 0.

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Formula

A316713 Unique representation of nonnegative numbers by iterated tribonacci A, B and C sequences.

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