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.

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A184821 a(n) = n + floor(n*t) + floor(n/t), where t is the tribonacci constant.

Original entry on oeis.org

2, 6, 9, 13, 16, 20, 22, 26, 29, 33, 36, 40, 43, 46, 50, 53, 57, 60, 63, 66, 70, 73, 77, 81, 83, 87, 90, 94, 97, 101, 104, 107, 110, 114, 118, 121, 125, 127, 131, 134, 138, 141, 145, 147, 151, 155, 158, 162, 165, 168, 171, 175, 178, 182, 185, 189, 191, 195, 199, 202, 206, 209, 212, 215, 219, 222, 226, 229, 232, 236, 239, 243, 246, 250, 252, 256, 259, 263, 266, 270, 273, 276, 280, 283, 287, 290, 294, 296, 300, 303, 307, 311, 314, 317, 320, 324, 327, 331, 334, 337, 340, 344, 347, 351, 355, 357, 361, 364, 368, 371, 375, 378, 381, 384, 388, 392, 395, 399, 401, 405, 408, 412, 415, 419, 421, 425, 429, 432
Offset: 1

Views

Author

Paul D. Hanna, Jan 22 2011

Keywords

Comments

This is one of three sequences that partition the positive integers.
Given t is the tribonacci constant, then the following sequences are disjoint:
. A184820(n) = n + [n/t] + [n/t^2],
. A184821(n) = n + [n*t] + [n/t],
. A184822(n) = n + [n*t] + [n*t^2], where []=floor.
This is a special case of Clark Kimberling's results given in A184812.

Examples

			Let t be the tribonacci constant, then t^2 = 1 + t + 1/t where:
t = 1.8392867552..., t^2 = 3.3829757679..., t^3 = 6.2222625231...

Crossrefs

Cf. A184820, A184822, A184812, A058265.

Programs

  • PARI
    {a(n)=local(t=real(polroots(1+x+x^2-x^3)[1]));n+floor(n*t)+floor(n/t)}

Formula

Limit a(n)/n = t^2 = 3.3829757679...
a(n) = n + floor(n*p/q) + floor(n*r/q), where p=t, q=t^2, r=t^3, and t is the tribonacci constant (see Clark Kimberling's formula in A184812).

A184822 a(n) = n + floor(n*t) + floor(n*t^2), where t is the tribonacci constant.

Original entry on oeis.org

5, 11, 18, 24, 30, 37, 42, 49, 55, 61, 68, 74, 79, 86, 92, 99, 105, 111, 117, 123, 130, 136, 142, 149, 154, 160, 167, 173, 180, 186, 192, 198, 204, 211, 217, 223, 230, 235, 241, 248, 254, 261, 267, 272, 279, 285, 291, 298, 304, 310, 316, 322, 329, 335, 342, 348, 353, 360, 366, 372, 379, 385, 391, 397, 403, 410, 416, 423, 428, 434, 441, 447, 453, 460, 465, 472, 478, 484, 491, 497, 503, 509, 515, 522, 528, 534, 541, 546, 553, 559, 565, 572, 578, 583, 590, 596, 603, 609, 615, 621, 627, 634, 640, 646, 653, 658, 664, 671, 677, 684, 690, 696, 702, 708, 715, 721, 727, 734, 739, 745, 752, 758, 765, 771
Offset: 1

Views

Author

Paul D. Hanna, Jan 22 2011

Keywords

Comments

This is one of three sequences that partition the positive integers.
Given t is the tribonacci constant, then the following sequences are disjoint:
. A184820(n) = n + [n/t] + [n/t^2],
. A184821(n) = n + [n*t] + [n/t],
. A184822(n) = n + [n*t] + [n*t^2], where []=floor.
This is a special case of Clark Kimberling's results given in A184812.

Examples

			Let t be the tribonacci constant, then t^3 = 1 + t + t^2 where:
t = 1.8392867552..., t^2 = 3.3829757679..., t^3 = 6.2222625231...

Crossrefs

Cf. A184820, A184821, A184812, A058265.

Programs

  • PARI
    {a(n)=local(t=real(polroots(1+x+x^2-x^3)[1]));n+floor(n*t)+floor(n*t^2)}

Formula

Limit a(n)/n = t^3 = 6.2222625231...
a(n) = n + floor(n*q/p) + floor(n*r/p), where p=t, q=t^2, r=t^3, and t is the tribonacci constant (see Clark Kimberling's formula in A184812).

A184904 n+floor(ns/r)+floor(nt/r), where r=2^(1/2), s=2^(1/3), t=2^(1/5).

Original entry on oeis.org

1, 4, 7, 10, 13, 15, 18, 21, 24, 26, 28, 31, 34, 37, 40, 42, 45, 48, 50, 53, 56, 58, 61, 64, 67, 70, 72, 74, 77, 80, 83, 85, 88, 91, 94, 97, 99, 101, 104, 107, 110, 113, 115, 118, 121, 123, 126, 128, 131, 134, 137, 140, 143, 145, 147, 150, 153, 156, 158, 161, 164, 167, 170, 172, 174, 177, 180, 183, 186, 188, 191, 194, 197, 199, 201, 204, 207, 210, 213, 215, 218, 221, 223, 226, 229, 231, 234, 237, 240, 243, 245, 247, 250, 253, 256, 258, 261, 264, 267, 270, 272, 274, 277, 280, 283, 286, 288, 291, 294, 296, 299, 301, 304, 307, 310, 313, 316, 318, 321, 323
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2011

Keywords

Comments

The sequences A184904, A184905, A184906, partition the positive integers:
A184904: 1,4,7,10,13,15,18,21,24,26,...
A184905: 2,5,8,11,14,17,20,23,27,30,...
A184906: 3,6,9,12,16,19,22,25,29,32,...
See A184812.

Crossrefs

Cf. A184812, A184905, A184906.

Programs

  • Mathematica
    r=2^(1/2); s=2^(1/3); t=2^(1/5);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t];
Table[a[n],{n,1,120}]  (* A184904 *)
Table[b[n],{n,1,120}]  (* A184905 *)
Table[c[n],{n,1,120}]  (* A184906 *)

A184905 n+floor(nr/s)+floor(nt/s), where r=2^(1/2), s=2^(1/3), t=2^(1/5).

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 20, 23, 27, 30, 33, 35, 38, 41, 44, 47, 51, 54, 57, 60, 63, 66, 68, 71, 75, 78, 81, 84, 87, 90, 93, 96, 100, 102, 105, 108, 111, 114, 117, 120, 124, 127, 130, 133, 136, 138, 141, 144, 148, 151, 154, 157, 160, 163, 166, 169, 171, 175, 178, 181, 184, 187, 190, 193, 196, 200, 203, 205, 208, 211, 214, 217, 220, 224, 227, 230, 233, 236, 239, 241, 244, 248, 251, 254, 257, 260, 263, 266, 269, 273, 275, 278, 281, 284, 287, 290, 293, 297, 300, 303, 306, 308, 311, 314, 317, 320, 324, 327, 330, 333, 336, 339, 342, 344, 348, 351, 354, 357, 360, 363
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2011

Keywords

Comments

The sequences A184904, A184905, A184906, partition the positive integers:
A184904: 1,4,7,10,13,15,18,21,24,26,...
A184905: 2,5,8,11,14,17,20,23,27,30,...
A184906: 3,6,9,12,16,19,22,25,29,32,...
See A184812.

Crossrefs

Cf. A184812, A184904, A184906.

Programs

  • Mathematica
    r=2^(1/2); s=2^(1/3); t=2^(1/5);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t];
Table[a[n],{n,1,120}]  (* A184904 *)
Table[b[n],{n,1,120}]  (* A184905 *)
Table[c[n],{n,1,120}]  (* A184906 *)

A184906 n+floor(nr/t)+floor(ns/t), where r=2^(1/2), s=2^(1/3), t=2^(1/5).

Original entry on oeis.org

3, 6, 9, 12, 16, 19, 22, 25, 29, 32, 36, 39, 43, 46, 49, 52, 55, 59, 62, 65, 69, 73, 76, 79, 82, 86, 89, 92, 95, 98, 103, 106, 109, 112, 116, 119, 122, 125, 129, 132, 135, 139, 142, 146, 149, 152, 155, 159, 162, 165, 168, 173, 176, 179, 182, 185, 189, 192, 195, 198, 202, 206, 209, 212, 216, 219, 222, 225, 228, 232, 235, 238, 242, 246, 249, 252, 255, 259, 262, 265, 268, 271, 276, 279, 282, 285, 289, 292, 295, 298, 302, 305, 309, 312, 315, 319, 322, 325, 328, 332, 335, 338, 341, 346, 349, 352, 355, 358, 362, 365, 368, 371, 375, 379, 382, 385, 389, 392, 395, 398
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2011

Keywords

Comments

The sequences A184904, A184905, A184906, partition the positive integers:
A184904: 1,4,7,10,13,15,18,21,24,26,...
A184905: 2,5,8,11,14,17,20,23,27,30,...
A184906: 3,6,9,12,16,19,22,25,29,32,...
See A184812.

Crossrefs

Cf. A184812, A184904, A184905.

Programs

  • Mathematica
    r=2^(1/2); s=2^(1/3); t=2^(1/5);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t];
Table[a[n],{n,1,120}]  (* A184904 *)
Table[b[n],{n,1,120}]  (* A184905 *)
Table[c[n],{n,1,120}]  (* A184906 *)

A379414 a(n) = n + floor(n*s/r) + floor(n*t/r), where r = 3^(1/4), s = 3^(1/2), t = 3^(3/4).

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 28, 31, 35, 40, 44, 47, 52, 56, 59, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 108, 112, 117, 120, 124, 129, 133, 136, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 194, 197, 201, 206, 210, 213, 218, 222, 225, 230
Offset: 1

Views

Author

Clark Kimberling, Jan 18 2025

Keywords

Comments

This sequence and A379415 and A379416 partition the positive integers; see A184812 for a proof.
For each k in A000027, write "a" if k=A379414(n) for some n, "b" if k=A379415(n) for some n, and "c" if k=A379416(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:
cbacbcabccabcbacbcacbcabcbcacbacbcabccbacbcabcacbcbacbcacbacbcbacbcacbcabcbaccbacbcabccabcbacbcacbcabcbcacbacbcabccbacbacbcabccbacbcabcacbcba

Crossrefs

Cf. A184812, A379415, A379416.
Cf. also A011002, A002194, A011022.

Programs

  • Mathematica
    r = 3^(1/4); s = 3^(1/2); t = 3^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379411 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379412 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379413 *)

Formula

a(n) = n + floor(n*r) + floor(n*r^2), where r = 3^(1/4).

A379415 a(n) = n + floor(n*r/s) + floor(n*t/s), where r = 3^(1/4), s = 3^(1/2), t = 3^(3/4).

Original entry on oeis.org

2, 5, 8, 12, 14, 17, 21, 24, 26, 30, 33, 36, 39, 42, 45, 49, 51, 54, 58, 61, 63, 66, 70, 73, 75, 79, 82, 85, 89, 91, 94, 98, 101, 103, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 138, 140, 143, 147, 150, 152, 156, 159, 162, 166, 168, 171, 175, 178, 180
Offset: 1

Views

Author

Clark Kimberling, Jan 18 2025

Keywords

Comments

This sequence and A379414 and A379416 partition the positive integers; see A184812 for a proof.

Crossrefs

Cf. A184812, A379415, A379416.
Cf. also A011002, A002194, A011022.

Programs

  • Mathematica
    r = 3^(1/4); s = 3^(1/2); t = 3^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379414 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379415 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379416 *)

Formula

a(n) = n + floor(n/r) + floor(n*r), where r = 3^(1/4).

A379416 a(n) = n + [n*r/t] + [n*s/t], where r = 3^(1/4); s = 3^(1/2); t = 3^(3/4) and [ ] = floor.

Original entry on oeis.org

1, 4, 6, 9, 10, 13, 16, 18, 20, 22, 25, 27, 29, 32, 34, 37, 38, 41, 43, 46, 48, 50, 53, 55, 57, 60, 62, 65, 67, 69, 71, 74, 77, 78, 81, 83, 86, 87, 90, 93, 95, 97, 99, 102, 104, 106, 109, 111, 114, 115, 118, 121, 123, 126, 127, 130, 132, 135, 137, 139, 142
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2025

Keywords

Comments

This sequence and A379414 and A379415 partition the positive integers; see A184812 for a proof.

Crossrefs

Cf. A184812, A379414, A379415.

Programs

  • Mathematica
    r = 3^(1/4); s = 3^(1/2); t = 3^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379414 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379415 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379416 *)

Formula

a(n) = n + [n*r/t] + [n*s/t], where r = 3^(1/4); s = 3^(1/2); t = 3^(3/4) and [ ] = floor.
a(n) = n + [n/r] + [n/r^2], where r = 3^(1/4) and [ ] = floor.

A379417 a(n) = n + [n*s/r] + [n*t/r], where r = (3/2)^(1/4); s = (3/2)^(1/2); t = (3/2)^(3/4) and [ ] = floor.

Original entry on oeis.org

3, 6, 9, 12, 16, 19, 22, 25, 29, 33, 36, 39, 42, 46, 49, 52, 55, 59, 63, 66, 69, 72, 76, 79, 82, 85, 89, 92, 96, 99, 102, 106, 109, 112, 115, 119, 122, 126, 129, 132, 136, 139, 142, 145, 149, 152, 156, 159, 163, 166, 169, 172, 175, 179, 182, 185, 189, 193
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2025

Keywords

Comments

This sequence and A379418 and A379419 partition the positive integers; see A184812 for a proof. For each k in A000027, write "a" if k=A379417(n) for some n, "b" if k=A379418(n) for some n, and "c" if k=A379419(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:
cbacbacbacbacbcabcabcabcabccabcbacbacbacbacbcabcabcabcacbcabcbacbacbacbacbcabcabcabcacbcabcabcbacbacbacbcabcabcacbacbcabcabcbacbacbacbcabcacbacbacbcabcabcbacbacbcab...

Crossrefs

Cf. A184812, A379418, A379419.

Programs

  • Mathematica
    r = (3/2)^(1/4); s = (3/2)^(1/2); t = (3/2)^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379417 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379418 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379419 *)

Formula

a(n) = n + [n*s/r] + [n*t/r], where r = (3/2)^(1/4); s = (3/2)^(1/2); t = (3/2)^(3/4) and [ ] = floor.
a(n) = n + [n*r] + [n*r^2], where r = (3/2)^(1/4) and [ ] = floor.

A379418 a(n) = n + [n*r/s] + [n*t/s], where r = (3/2)^(1/4); s = (3/2)^(1/2); t = (3/2)^(3/4) and [ ] = floor.

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 20, 23, 26, 30, 32, 35, 38, 41, 44, 47, 50, 53, 57, 60, 62, 65, 68, 71, 74, 77, 80, 83, 87, 90, 93, 95, 98, 101, 104, 107, 110, 114, 117, 120, 123, 125, 128, 131, 134, 137, 141, 144, 147, 150, 153, 155, 158, 161, 164, 167, 171, 174, 177
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2025

Keywords

Comments

This sequence and A379417 and A379419 partition the positive integers; see A184812 for a proof.

Crossrefs

Cf. A184812, A379417, A379419.

Programs

  • Mathematica
    r = (3/2)^(1/4); s = (3/2)^(1/2); t = (3/2)^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379417 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379418 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379419 *)

Formula

a(n) = n + [n*r/s] + [n*t/s], where r = (3/2)^(1/4); s = (3/2)^(1/2); t = (3/2)^(3/4) and [ ] = floor.
a(n) = n + [n/r] + [n*r], where r = (3/2)^(1/4) and [ ] = floor.
Previous Showing 31-40 of 46 results. Next

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