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.

A184820 a(n) = n + floor(n/t) + floor(n/t^2), where t is the tribonacci constant.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 12, 14, 15, 17, 19, 21, 23, 25, 27, 28, 31, 32, 34, 35, 38, 39, 41, 44, 45, 47, 48, 51, 52, 54, 56, 58, 59, 62, 64, 65, 67, 69, 71, 72, 75, 76, 78, 80, 82, 84, 85, 88, 89, 91, 93, 95, 96, 98, 100, 102, 103, 106, 108, 109, 112, 113, 115, 116, 119, 120, 122, 124, 126, 128, 129, 132, 133, 135, 137, 139, 140, 143, 144, 146, 148, 150, 152, 153, 156, 157, 159, 161, 163, 164, 166, 169, 170, 172, 174, 176, 177, 179, 181, 183, 184, 187, 188, 190, 193, 194, 196, 197, 200, 201, 203, 205, 207, 208, 210, 213, 214, 216, 218, 220, 221, 224, 225, 227, 228, 231, 233, 234, 237, 238, 240, 242, 244, 245, 247
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 = 1 + 1/t + 1/t^2 where:
t = 1.8392867552..., t^2 = 3.3829757679..., t^3 = 6.2222625231...

Crossrefs

Cf. A184821, A184822, A184812, A058265; A184823, A184835.

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 = 1.8392867552141611325518525646532866...
a(n) = n + floor(n*p/r) + floor(n*q/r), where p=t, q=t^2, r=t^3, and t is the tribonacci constant (see Clark Kimberling's formula in A184812).

A184824 a(n) = n + floor(n*t) + floor(n/t) + floor(n/t^2), where t is the tetranacci constant.

Original entry on oeis.org

2, 6, 9, 14, 17, 21, 24, 29, 32, 36, 39, 44, 47, 50, 54, 58, 61, 65, 69, 73, 76, 80, 84, 88, 91, 95, 100, 102, 106, 110, 114, 117, 121, 125, 129, 132, 136, 140, 144, 147, 152, 154, 158, 161, 166, 169, 173, 176, 181, 184, 188, 191, 196, 200, 203, 207, 210, 214, 217, 222, 225, 229, 232, 237, 240, 244, 248, 252, 255, 258, 262, 266, 269, 273, 277, 281, 284, 288, 292, 296, 300, 304, 307, 310, 314, 318, 322, 325, 329, 333, 337, 340, 345, 348, 352, 355, 359, 362, 366, 369, 374, 377, 381, 384, 389, 392, 396, 401, 404, 408
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^2 = 1 + t + 1/t + 1/t^2 and:
t = 1.92756197548..., t^2 = 3.71549516932..., t^3 = 7.16184720848..., t^4 = 13.8049043532...

Crossrefs

Cf. A184823, A184825, A184826; 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)+floor(n/t^2)}

Formula

Limit a(n)/n = t^2 = 3.7154951693276375317543272...
a(n) = n + floor(n*p/r) + floor(n*q/r) + floor(n*s/r), where p=t, q=t^2, r=t^3, s=t^4, and t is the tetranacci constant.

A184825 a(n) = n + floor(n*t) + floor(n*t^2) + floor(n/t), where t is the tetranacci constant.

Original entry on oeis.org

5, 13, 20, 27, 34, 42, 49, 56, 63, 71, 77, 85, 92, 99, 105, 113, 120, 127, 134, 142, 149, 156, 163, 171, 177, 185, 193, 199, 206, 213, 221, 227, 235, 242, 250, 256, 264, 271, 278, 285, 293, 299, 306, 313, 321, 327, 335, 342, 350, 356, 364, 371, 378, 386, 393, 400, 406, 414, 421, 428, 435, 443, 450, 457, 464, 472, 478, 486, 493, 500, 506, 514, 521, 528, 535, 543, 550, 557, 564, 572, 579, 586, 593, 600, 607, 614, 622, 628, 636, 643, 651, 657, 665, 672, 679, 686, 693, 700, 707, 714, 722, 728, 736, 743, 751, 757
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^3 = 1 + t + t^2 + 1/t and:
t = 1.92756197548..., t^2 = 3.71549516932..., t^3 = 7.16184720848..., t^4 = 13.8049043532...

Crossrefs

Cf. A184823, A184824, A184826; A184812, A086088.

Programs

  • Mathematica
    With[{t=x/.Last[Solve[x^4==Total[x^Range[0,3]],x]]},Table[n+Floor[n t]+Floor[n t^2]+Floor[n/t],{n,120}]]  (* Harvey P. Dale, Feb 02 2011 *)
  • 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)}

Formula

Lim_{n->infinity} a(n)/n = t^3 = 7.1618472084864470579236869...
a(n) = n + floor(n*p/q) + floor(n*r/q) + floor(n*s/q), where p=t, q=t^2, r=t^3, s=t^4, and t is the tetranacci constant.

A184826 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

12, 26, 40, 53, 67, 81, 96, 109, 123, 137, 150, 164, 179, 192, 205, 219, 233, 246, 261, 275, 289, 302, 316, 330, 344, 358, 372, 385, 398, 412, 427, 440, 454, 468, 482, 495, 509, 524, 537, 551, 565, 578, 591, 606, 620, 633, 647, 661, 675, 689, 703, 717, 730, 744, 758, 772, 785, 799, 813, 826, 840, 855, 869, 882, 896, 910, 923, 938, 952, 965, 978, 992, 1006, 1019, 1034, 1048, 1062, 1075, 1089, 1103, 1117, 1131, 1144, 1158, 1171, 1185, 1200, 1213, 1227, 1241, 1255, 1268, 1283, 1297, 1310, 1324, 1337, 1351
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^4 = 1 + t + t^2 + t^3 and:
t = 1.92756197548..., t^2 = 3.71549516932..., t^3 = 7.16184720848..., t^4 = 13.8049043532...

Crossrefs

Cf. A184823, A184824, A184825; A184812, A086088.

Programs

  • Mathematica
    Module[{t=x/.FindRoot[x^4-x^3-x^2-x-1==0,{x,2},WorkingPrecision->200], t2,t3},t2=t^2;t3=t^3;Table[n+Floor[t*n]+Floor[t2*n]+Floor[t3*n], {n,100}]] (* Harvey P. Dale, Oct 18 2012 *)
  • 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^4 = 13.804904353297009893939920...
a(n) = n + floor(n*q/p) + floor(n*r/p) + floor(n*s/p), where p=t, q=t^2, r=t^3, s=t^4, and t is the tetranacci constant.
Showing 1-5 of 5 results.

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