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 1-5 of 5 results.

A184835 a(n) = n + floor(n/t) + floor(n/t^2) + floor(n/t^3) + floor(n/t^4), where t is the pentanacci constant.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 11, 15, 16, 18, 19, 22, 23, 25, 27, 31, 32, 34, 35, 38, 39, 41, 43, 46, 47, 49, 50, 53, 54, 57, 60, 62, 63, 65, 67, 69, 70, 73, 75, 77, 78, 80, 82, 84, 86, 89, 91, 93, 94, 96, 98, 100, 101, 104, 106, 108, 109, 112, 114, 116, 119, 121, 123, 124, 126, 128, 130, 131, 134, 136, 138, 139, 141, 143, 146, 148, 150, 152, 154, 155, 157, 159, 161, 163, 165, 167, 169, 170, 173, 175, 177, 179, 182, 183, 185, 186, 189, 190, 193, 194, 197, 198, 200, 201, 205, 206, 209, 210, 213, 214, 216, 217, 220, 222, 224, 227, 228
Offset: 1

Views

Author

Paul D. Hanna, Jan 23 2011

Keywords

Comments

This is one of five sequences that partition the positive integers.
Given t is the pentanacci constant, then the following sequences are disjoint:
. A184835(n) = n + [n/t] + [n/t^2] + [n/t^3] + [n/t^4],
. A184836(n) = n + [n*t] + [n/t] + [n/t^2] + [n/t^3],
. A184837(n) = n + [n*t] + [n*t^2] + [n/t] + [n/t^2],
. A184838(n) = n + [n*t] + [n*t^2] + [n*t^3] + [n/t],
. A184839(n) = n + [n*t] + [n*t^2] + [n*t^3] + [n*t^4], where []=floor.
This is a special case of Clark Kimberling's results given in A184812.

Examples

			Given t = pentanacci constant, then t = 1 + 1/t + 1/t^2 + 1/t^3 + 1/t^4,
t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...

Crossrefs

Cf. A184836, A184837, A184838, A184839; A184820, A184823, A184812, A103814.

Programs

  • Mathematica
    With[{pc=x/.FindRoot[x^5-x^4-x^3-x^2-x-1==0,{x,1.96},WorkingPrecision-> 100]}, Table[n+Total[Table[Floor[n/pc^i],{i,4}]],{n,150}]] (* Harvey P. Dale, Jun 21 2011 *)
  • PARI
    {a(n)=local(t=real(polroots(1+x+x^2+x^3+x^4-x^5)[1])); n+floor(n/t)+floor(n/t^2)+floor(n/t^3)+floor(n/t^4)}

Formula

Limit a(n)/n = t = 1.9659482366454853371899373...
a(n) = n + floor(n*p/u) + floor(n*q/u) + floor(n*r/u) + floor(n*s/u), where p=t, q=t^2, r=t^3, s=t^4, u=t^5, and t is the pentanacci constant.

A184823 a(n) = n + floor(n/t) + floor(n/t^2) + floor(n/t^3), where t is the tetranacci constant.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 11, 15, 16, 18, 19, 22, 23, 25, 28, 30, 31, 33, 35, 37, 38, 41, 43, 45, 46, 48, 51, 52, 55, 57, 59, 60, 62, 64, 66, 68, 70, 72, 74, 75, 78, 79, 82, 83, 86, 87, 89, 90, 93, 94, 97, 98, 101, 103, 104, 107, 108, 111, 112, 115, 116, 118, 119, 122, 124, 126, 128, 130, 131, 133, 135, 138, 139, 141, 143, 145, 146, 148, 151, 153, 155, 157, 159, 160, 162, 165, 167, 168, 170, 172, 174, 175, 178, 180, 182, 183, 186, 187, 189, 190, 194, 195, 197, 198, 201, 202, 204, 208, 209, 211, 212, 215, 216, 218, 220, 223, 224
Offset: 1

Views

Author

Paul D. Hanna, Jan 23 2011

Keywords

Comments

This is one of four sequences that partition the positive integers.
Given t is the tetranacci constant, then the following sequences are disjoint:
. A184823(n) = n + [n/t] + [n/t^2] + [n/t^3],
. A184824(n) = n + [n*t] + [n/t] + [n/t^2],
. A184825(n) = n + [n*t] + [n*t^2] + [n/t],
. A184826(n) = n + [n*t] + [n*t^2] + [n*t^3], where []=floor.
This is a special case of Clark Kimberling's results given in A184812.

Examples

			Let t be the tetranacci constant, then t = 1 + 1/t + 1/t^2 + 1/t^3 and:
t = 1.92756197548..., t^2 = 3.71549516932..., t^3 = 7.16184720848..., t^4 = 13.8049043532...

Crossrefs

Cf. A184824, A184825, A184826; A184820, A184835, A184812, A086088.

Programs

  • PARI
    {a(n)=local(t=real(polroots(1+x+x^2+x^3-x^4)[2])); n+floor(n/t)+floor(n/t^2)+floor(n/t^3)}

Formula

Limit a(n)/n = t = 1.9275619754829253042619058...
a(n) = n + floor(n*p/s) + floor(n*q/s) + floor(n*r/s), where p=t, q=t^2, r=t^3, s=t^4, and t is the tetranacci constant.

A277728 Numbers not in any of A158919, A277722, A277723.

Original entry on oeis.org

2, 4, 8, 15, 17, 19, 21, 26, 28, 32, 35, 39, 41, 46, 48, 52, 59, 61, 63, 65, 70, 72, 76, 78, 83, 85, 89, 92, 96, 100, 102, 107, 109, 113, 116, 120, 122, 127, 129, 133, 140, 144, 146, 151, 153, 157, 159, 164, 166, 170, 173, 177, 181, 184, 188, 190, 195, 197, 201, 203, 208, 210, 212, 214, 221
Offset: 1

Views

Author

N. J. A. Sloane, Oct 30 2016

Keywords

Comments

Let tau be the tribonacci constant (A058265). Although 1/tau + 1/tau^2 + 1/tau^3 = 1, by Uspensky's 1927 theorem, the three sequences floor(n*tau) (A158919), floor(n*tau^2) (A277722), and floor(n*tau^3) (A277723) cannot form a partition of the nonnegative integers. (Compare Beatty's theorem.)
Entries A277724-A277727 investigate how these three sequences meet, and the present sequence gives the numbers not in any of the three sequences. Any two of the three sequences have a nontrivial intersection, while the intersection of all three is {0}.
On the other hand, the three sequences A003144, A003145, A003146, which arise from the tribonacci word, DO form a partition of the positive integers and are closely connected with the three sequences mentioned in the definition.
It would be nice to have b-files for all the sequences mentioned here. (Many do, but some do not.)

Links

Crossrefs

Cf. A003144, A003145, A003146, A058265, A158919, A184820, A277722, A277723, A277724, A277725, A277726, A277727.

Programs

  • Mathematica
    maxTerm = 10000;
a19[n_] := Floor[n*Root[#^3 - #^2 - # - 1&, 1]];
a22[n_] := Floor[n*Root[#^3 - 3 #^2 - # - 1&, 1]];
a23[n_] := Floor[n*Root[#^3 - 7 #^2 + 5 # - 1&, 1]];
A19 = Reap[k = 1; While[a19[k] <= maxTerm, Sow[a19[k++]]]][[2, 1]];
A22 = Reap[k = 1; While[a22[k] <= maxTerm, Sow[a22[k++]]]][[2, 1]];
A23 = Reap[k = 1; While[a23[k] <= maxTerm, Sow[a23[k++]]]][[2, 1]];
Select[Range[maxTerm], FreeQ[A19, #] && FreeQ[A22, #] && FreeQ[A23, #]&] (* Jean-François Alcover, Dec 06 2018 *)

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).
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.