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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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.

A184873 a(n) = n + floor(nr/t) + floor(ns/t), where r=log(2), s=log(3), t=log(5).

Original entry on oeis.org

1, 3, 6, 7, 10, 12, 14, 16, 18, 20, 22, 25, 26, 29, 31, 32, 35, 37, 39, 41, 44, 46, 47, 50, 52, 54, 56, 59, 60, 62, 65, 66, 69, 71, 73, 75, 77, 79, 81, 84, 85, 88, 90, 92, 94, 96, 99, 100, 103, 105, 106, 109, 111, 113, 115, 118, 119, 121, 124, 125, 128, 130, 133, 134, 136, 139, 140, 143, 145, 147, 149, 152, 153, 155, 158, 159, 162, 164, 166, 168, 170, 172, 174, 177, 179, 181, 183, 185, 187, 189, 192, 193, 196, 198, 199, 202, 204, 206, 208, 211, 212, 214, 217, 218, 221, 223, 226, 227, 229, 232, 233, 236, 238, 240, 242, 244, 246, 248, 251, 252
Offset: 1

Views

Author

Clark Kimberling, Jan 23 2011

Keywords

Comments

See A184812.

Crossrefs

Cf. A184871, A184872, A184876 (primes in A184873).

Programs

  • Mathematica
    r=Log[2]; s=Log[3]; t=Log[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}]  (* A184871 *)
Table[b[n], {n, 1, 120}]  (* A184872 *)
Table[c[n], {n, 1, 120}]  (* A184873 *)

Extensions

Name corrected by Harvey P. Dale, Jan 26 2011

A184913 n+[rn/s]+[tn/s]+[un/s], where []=floor and r=2^(1/5), s=r^2, t=r^3, u=r^4.

Original entry on oeis.org

3, 7, 11, 16, 20, 24, 30, 33, 37, 42, 46, 50, 55, 60, 64, 68, 72, 76, 81, 85, 90, 95, 99, 102, 106, 111, 116, 120, 125, 129, 132, 137, 141, 146, 151, 155, 159, 164, 167, 171, 177, 181, 185, 190, 194, 198, 202, 207, 211, 215, 220, 224, 228, 234, 237, 241, 246, 250, 254, 259, 264, 267, 272, 276, 280, 285, 289, 294, 299, 302, 306, 311, 315, 320, 324, 329, 333, 336, 341, 345, 350, 355, 359, 363, 367, 371, 375, 381, 385, 389, 394, 398, 401, 406, 411, 415, 419, 424, 428, 432, 437, 441, 445, 450, 454, 458, 463, 468, 471, 476, 480, 484, 489, 493, 498, 502, 506, 510, 515, 519
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2011

Keywords

Comments

The sequences A184912-A184915 partition the positive integers:
A184912: 4,9,13,19,23,28,34,...
A184913: 3,7,11,16,20,24,30,...
A184914: 2,6,10,14,17,21,26,...
A184915: 1,5,8,12,15,18,22,...
The joint ranking method of A184812 is extended here to four numbers r,s,t,u, as follows: jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, h>=1, i>=1, j>=1, k>=1.
The position of n*s in the joint ranking is
n+[rn/s]+[tn/s]+[un/s], and likewise for the
positions of n*r, n*t, and n*u.

Crossrefs

Cf. A184812, A184913, A184914, A184915.

Programs

  • Mathematica
    r=2^(1/5); s=2^(2/5); t=2^(3/5); u=2^(4/5);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r]+Floor[n*u/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s]+Floor[n*u/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t]+Floor[n*u/t];
d[n_]:=n+Floor[n*r/u]+Floor[n*s/u]+Floor[n*t/u];
Table[a[n],{n,1,120}]  (* A184912 *)
Table[b[n],{n,1,120}]  (* A184913 *)
Table[c[n],{n,1,120}]  (* A184914 *)
Table[d[n],{n,1,120}]  (* A184915 *)

A184914 n+[rn/t]+[sn/t]+[un/t], where []=floor and r=2^(1/5), s=r^2, t=r^3, u=r^4.

Original entry on oeis.org

2, 6, 10, 14, 17, 21, 26, 29, 32, 36, 40, 44, 47, 52, 56, 59, 62, 66, 70, 74, 78, 82, 86, 89, 92, 96, 101, 105, 108, 112, 115, 119, 123, 127, 131, 135, 139, 142, 145, 149, 154, 157, 161, 165, 169, 172, 175, 180, 184, 187, 191, 195, 199, 203, 206, 210, 214, 217, 221, 225, 230, 232, 236, 240, 244, 248, 251, 256, 260, 263, 266, 270, 274, 279, 282, 286, 290, 293, 296, 300, 305, 309, 312, 316, 319, 323, 326, 331, 335, 339, 342, 346, 349, 353, 357, 361, 365, 369, 373, 376, 380, 384, 388, 391, 395, 399, 403, 407, 410, 414, 418, 421, 425, 429, 434, 436, 440, 444, 448, 451
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2011

Keywords

Comments

The sequences A184912-A184915 partition the positive integers:
A184912: 4,9,13,19,23,28,34,...
A184913: 3,7,11,16,20,24,30,...
A184914: 2,6,10,14,17,21,26,...
A184915: 1,5,8,12,15,18,22,...
The joint ranking method of A184812 is extended here to four numbers r,s,t,u, as follows: jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, h>=1, i>=1, j>=1, k>=1.
The position of n*t in the joint ranking is
n+[rn/t]+[sn/t]+[un/t], and likewise for the
positions of n*r, n*s, and n*u.

Crossrefs

Cf. A184812, A184912, A184913, A184915.

Programs

  • Mathematica
    r=2^(1/5); s=2^(2/5); t=2^(3/5); u=2^(4/5);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r]+Floor[n*u/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s]+Floor[n*u/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t]+Floor[n*u/t];
d[n_]:=n+Floor[n*r/u]+Floor[n*s/u]+Floor[n*t/u];
Table[a[n],{n,1,120}]  (* A184912 *)
Table[b[n],{n,1,120}]  (* A184913 *)
Table[c[n],{n,1,120}]  (* A184914 *)
Table[d[n],{n,1,120}]  (* A184915 *)

A184917 n+[rn/s]+[tn/s]+[un/s], where []=floor and r=1, s=2^(1/4), t=s^2, u=s^3.

Original entry on oeis.org

3, 7, 12, 16, 21, 26, 29, 34, 38, 43, 48, 52, 56, 60, 65, 70, 75, 79, 82, 87, 91, 97, 101, 105, 110, 113, 119, 123, 128, 132, 136, 141, 145, 150, 154, 158, 164, 167, 172, 176, 180, 185, 190, 194, 198, 203, 207, 212, 217, 221, 225, 229, 234, 239, 243, 248, 251, 256, 261, 265, 270, 274, 278, 283, 287, 292, 296, 301, 306, 309, 314, 318, 323, 328, 333, 336, 340, 345, 349, 355, 359, 362, 367, 371, 377, 381, 386, 389, 393, 399, 403, 408, 412, 416, 420, 425, 430, 434, 439, 443, 447, 452, 456, 461, 465, 470, 474, 478, 483, 487, 492, 497, 501, 505, 509, 514, 519, 523, 528, 531
Offset: 1

Views

Author

Clark Kimberling, Jan 26 2011

Keywords

Comments

The sequences A184916-A184919 partition the positive integers:
A184916: 4,9,15,19,25,31,35,41,...
A184917: 3,7,12,16,21,26,29,34,...
A184918: 2,6,10,13,17,22,24,28,...
A184919: 1,5,8,11,14,18,20,23,27,...
The joint ranking method of A184812 is extended here to four numbers r,s,t,u, as follows: jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, h>=1, i>=1, j>=1, k>=1.
The position of n*s in the joint ranking is
n+[rn/s]+[tn/s]+[un/s], and likewise for the
positions of n*r, n*t, and n*u.

Crossrefs

Cf. A184912, A184916, A184918, A184919.

Programs

  • Mathematica
    r=1; s=2^(1/4); t=2^(1/2); u=2^(3/4);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r]+Floor[n*u/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s]+Floor[n*u/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t]+Floor[n*u/t];
d[n_]:=n+Floor[n*r/u]+Floor[n*s/u]+Floor[n*t/u];
Table[a[n],{n,1,120}]  (* A184916 *)
Table[b[n],{n,1,120}]  (* A184917 *)
Table[c[n],{n,1,120}]  (* A184918 *)
Table[d[n],{n,1,120}]  (* A184919 *)

A184918 n+[rn/t]+[sn/t]+[un/t], where []=floor and r=1, s=2^(1/4), t=s^2, u=s^3.

Original entry on oeis.org

2, 6, 10, 13, 17, 22, 24, 28, 32, 36, 40, 44, 47, 50, 54, 59, 63, 66, 69, 73, 76, 81, 85, 88, 92, 95, 100, 103, 107, 111, 114, 118, 122, 126, 129, 133, 138, 140, 144, 148, 151, 155, 160, 163, 166, 170, 174, 178, 182, 186, 189, 192, 197, 201, 204, 208, 211, 215, 219, 223, 227, 230, 233, 238, 241, 245, 249, 253, 257, 260, 264, 267, 271, 276, 280, 282, 286, 290, 293, 298, 302, 304, 308, 312, 317, 320, 324, 327, 330, 335, 339, 343, 346, 350, 353, 357, 361, 365, 369, 372, 376, 380, 383, 387, 391, 395, 398, 402, 406, 409, 414, 418, 421, 424, 428, 432, 436, 440, 444, 446
Offset: 1

Views

Author

Clark Kimberling, Jan 26 2011

Keywords

Comments

The sequences A184916-A184919 partition the positive integers:
A184916: 4,9,15,19,25,31,35,41,...
A184917: 3,7,12,16,21,26,29,34,...
A184918: 2,6,10,13,17,22,24,28,...
A184919: 1,5,8,11,14,18,20,23,27,...
The joint ranking method of A184812 is extended here to four numbers r,s,t,u, as follows: jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, h>=1, i>=1, j>=1, k>=1.
The position of n*t in the joint ranking is
n+[rn/t]+[sn/t]+[un/t], and likewise for the
positions of n*r, n*s, and n*u.

Crossrefs

Cf. A184912, A184916, A184917, A184919.

Programs

  • Mathematica
    r=1; s=2^(1/4); t=2^(1/2); u=2^(3/4);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r]+Floor[n*u/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s]+Floor[n*u/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t]+Floor[n*u/t];
d[n_]:=n+Floor[n*r/u]+Floor[n*s/u]+Floor[n*t/u];
Table[a[n],{n,1,120}]  (* A184916 *)
Table[b[n],{n,1,120}]  (* A184917 *)
Table[c[n],{n,1,120}]  (* A184918 *)
Table[d[n],{n,1,120}]  (* A184919 *)

A184919 n+[rn/u]+[sn/u]+[tn/u], where []=floor and r=1, s=2^(1/4), t=s^2, u=s^3.

Original entry on oeis.org

1, 5, 8, 11, 14, 18, 20, 23, 27, 30, 33, 37, 39, 42, 45, 49, 53, 55, 58, 61, 64, 68, 71, 74, 77, 80, 84, 86, 90, 93, 96, 99, 102, 106, 108, 112, 116, 117, 121, 124, 127, 130, 134, 137, 139, 143, 146, 149, 153, 156, 159, 161, 165, 169, 171, 175, 177, 181, 184, 187, 191, 193, 196, 200, 202, 206, 209, 213, 216, 218, 222, 224, 228, 232, 235, 237, 240, 244, 246, 250, 254, 255, 259, 262, 266, 269, 272, 275, 277, 281, 285, 288, 291, 294, 297, 300, 303, 307, 310, 313, 316, 319, 322, 325, 329, 332, 334, 338, 341, 344, 348, 351, 354, 356, 360, 363, 366, 370, 373, 375
Offset: 1

Views

Author

Clark Kimberling, Jan 26 2011

Keywords

Comments

The sequences A184916-A184919 partition the positive integers:
A184916: 4,9,15,19,25,31,35,41,...
A184917: 3,7,12,16,21,26,29,34,...
A184918: 2,6,10,13,17,22,24,28,...
A184919: 1,5,8,11,14,18,20,23,27,...
The joint ranking method of A184812 is extended here to four numbers r,s,t,u, as follows: jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, h>=1, i>=1, j>=1, k>=1.
The position of n*u in the joint ranking is
n+[rn/u]+[sn/u]+[tn/u], and likewise for the
positions of n*r, n*s, and n*t.

Crossrefs

Cf. A184912, A184916, A184917, A184918.

Programs

  • Mathematica
     r=1; s=2^(1/4); t=2^(1/2); u=2^(3/4);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r]+Floor[n*u/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s]+Floor[n*u/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t]+Floor[n*u/t];
d[n_]:=n+Floor[n*r/u]+Floor[n*s/u]+Floor[n*t/u];
Table[a[n],{n,1,120}]  (* A184916 *)
Table[b[n],{n,1,120}]  (* A184917 *)
Table[c[n],{n,1,120}]  (* A184918 *)
Table[d[n],{n,1,120}]  (* A184919 *)
Table[With[{s=Surd[2,4]},n+Floor[n/s^3]+Floor[(n*s)/s^3]+Floor[(n*s^2)/s^3]],{n,120}] (* Harvey P. Dale, Dec 01 2024 *)

A379411 a(n) = n + floor(n*s/r) + floor(n*t/r), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

Original entry on oeis.org

3, 7, 10, 15, 19, 22, 26, 31, 34, 38, 43, 46, 50, 54, 58, 62, 66, 70, 74, 77, 81, 86, 89, 93, 98, 101, 105, 109, 113, 117, 121, 125, 129, 133, 136, 141, 145, 148, 153, 156, 160, 164, 168, 172, 176, 180, 184, 188, 191, 196, 200, 203, 208, 212, 215, 219, 223
Offset: 1

Views

Author

Clark Kimberling, Jan 18 2025

Keywords

Comments

This sequence and A379412 and A379413 partition the positive integers; see A184812 for a proof. For each k in A000027, write "a" if k=A379411(n) for some n, "b" if k=A379412(n) for some n, and "c" if k=A379413(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:
cbacbcabcacbcbacbcabcacbcabcbcacbacbcabcbcacbacbcabccabcbacbcabccabcbacbcacbacbcabcbcacbacbcabccbacbacbcabccabcbacbcacbcabcbacbcacbcabcabccbacb...

Crossrefs

Cf. A184812, A379412, A379413.
Cf. A092042, A019774, A331501.

Programs

  • Mathematica
    r = E^(1/4); s = E^(1/2); t = E^(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 = e^(1/4).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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