cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

(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

Crossrefs

Cf. A278039, A278040, A322408.
Cf. A003144, A003145, A003146.

Formula

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).

A322410 Compound tribonacci sequence with a(n) = A278040(A278039(n)), for n >= 0.

Original entry on oeis.org

1, 8, 14, 21, 25, 32, 38, 45, 52, 58, 65, 69, 76, 82, 89, 95, 102, 106, 113, 119, 126, 133, 139, 146, 150, 157, 163, 170, 174, 181, 187, 194, 201, 207, 214, 218, 225, 231, 238, 244, 251, 255, 262, 268, 275, 282, 288, 295, 299, 306, 312, 319, 326, 332, 339, 343, 350, 356, 363, 369, 376
Offset: 0

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

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

Crossrefs

Cf. A278039, A278040, A322409.
Cf. A003144, A003145, A003146.

Formula

A(B(n)) = A(B(n) + 1) - 4 = A(n) + B(n) + n, for n >= 0, with A = A278040 and B = A278039. For a proof see the W. Lang link in A278040, Proposition 9, eq. (49).
a(n+1) = A319967(n)-1 = A003145(A003144(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking, Apr 04 2019

A322411 Compound tribonacci sequence with a(n) = A278040(A278041(n)), for n >= 0.

Original entry on oeis.org

12, 36, 56, 80, 93, 117, 137, 161, 185, 205, 229, 242, 266, 286, 310, 330, 354, 367, 391, 411, 435, 459, 479, 503, 516, 540, 560, 584, 597, 621, 641, 665, 689, 709, 733, 746, 770, 790, 814, 834, 858, 871, 895, 915, 939, 963, 983, 1007, 1020, 1044, 1064, 1088, 1112, 1132, 1156, 1169, 1193, 1213, 1237, 1257, 1281
Offset: 0

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

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

Links

Crossrefs

Cf. A278040, A278041, A322410.

Formula

a(n) = A(C(n)) = A(C(n) + 1) - 2 = 4*A(n) + 3*B(n) + 2*n + 8, for n >= 0, with A = A278040 and C = A278041. For a proof see the W. Lang link in A278040, Proposition 9, eq. (50).
This formula already follows from Theorem 15 in the 1972 paper by Carlitz et al., which gives that b(c(n)) = a(n) + 2b(n) + 2c(n), where a, b and c are the classical positional sequences of the letters in the tribonacci word. The connection is made by using that c(n) = a(n) + b(n) + n, and by making the translation B(n) = a(n+1)-1, A(n) = b(n+1)-1, C(n) = c(n+1)-1. (Note the switching of A and B!). - Michel Dekking, Apr 07 2019
a(n+1) = A319969(n)-1 = A003145(A003146(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking, Apr 04 2019

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

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

(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

Crossrefs

Cf. A278040, A278041, A322411.

Formula

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).

A322413 Compound tribonacci sequence with a(n) = A278041(A278039(n)), for n >= 0.

Original entry on oeis.org

3, 16, 27, 40, 47, 60, 71, 84, 97, 108, 121, 128, 141, 152, 165, 176, 189, 196, 209, 220, 233, 246, 257, 270, 277, 290, 301, 314, 321, 334, 345, 358, 371, 382, 395, 402, 415, 426, 439, 450, 463, 470, 483, 494, 507, 520, 531, 544, 551, 564, 575, 588, 601, 612, 625, 632, 645, 656, 669, 680, 693
Offset: 0

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

(a(n+1)) = A319970(n)-1 = A003146(A003144(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking, Apr 03 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

Crossrefs

Cf. A278039, A278040, A278041, A322412.
Cf. A003144, A003145, A003146.

Formula

a(n) = C(B(n)) = C(B(n) + 1) - 7 = 2*(A(n) + B(n)) + n + 1, for n >= 0, where A = A278040, B = A278039 and C = A278041. For a proof see the W. Lang link in A278040, Proposition 9, eq. (55).

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

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

(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

Crossrefs

Cf. A278039, A278040, A278041, A322413.
Cf. A003144, A003145, A003146.

Formula

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).

A322408 Compound sequence with a(n) = A319198(A278041(n)), for n >= 0.

Original entry on oeis.org

3, 7, 11, 15, 18, 22, 26, 30, 34, 38, 42, 45, 49, 53, 57, 61, 65, 68, 72, 76, 80, 84, 88, 92, 95, 99, 103, 107, 110, 114, 118, 122, 126, 130, 134, 137, 141, 145, 149, 153, 157, 160, 164, 168, 172, 176, 180, 184, 187, 191, 195, 199, 203, 207, 211, 214, 218, 222, 226, 230, 234
Offset: 0

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

Old name was: Compound tribonacci sequence a(n) = A319198(A278041(n)), for n >= 0.
a(n) gives the sum of the entries of the tribonacci word sequence t = A080843 not exceeding t(C(n)), with C(n) = A278041(n).
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
The difference sequence (a(n+1)-a(n)) is equal to a change of alphabet of the tribonacci word t = A092782. The alphabet is {4,4,3}. This follows from the formula a(n) = A278039(n) + 2*n + 3. - Michel Dekking, Oct 05 2019

Examples

			n = 2: C(2) = 16, t = {0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, ...} which sums to 11 = a(2) = 4 + 7, because B(2) = 4.

Crossrefs

Cf. A080843, A278039, A278041, A319198, A321333, A322407.

Formula

a(n) = z(C(n)) = Sum_{j=0..C(n)} t(j), n >= 0, with z = A319198, C = A278041 and t = A080843.
a(n) = B(n) + 2*n + 3, where B(n) = A278039(n). For a proof see the W. Lang link in A080843, Proposition 8, eq. (47).
a(n) = 3 + Sum_{k=1..n-1} d(k), where d is the tribonacci sequence on the alphabet {4,4,3}. - Michel Dekking, Oct 05 2019

Extensions

Name changed by Michel Dekking, Oct 08 2019
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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