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 1-4 of 4 results.

A184912 n+[ns/r]+[nt/r]+[nu/r], where []=floor and r=2^(1/5), s=r^2, t=r^3, u=r^4.

Original entry on oeis.org

4, 9, 13, 19, 23, 28, 34, 39, 43, 49, 53, 58, 63, 69, 73, 79, 83, 88, 93, 98, 103, 109, 113, 118, 122, 128, 133, 138, 143, 148, 152, 158, 163, 168, 174, 178, 183, 188, 193, 197, 204, 208, 213, 218, 223, 227, 233, 238, 243, 247, 253, 257, 262, 268, 273, 277, 283, 287, 292, 297, 303, 307, 313, 318, 322, 328, 332, 338, 343, 348, 352, 358, 362, 368, 372, 378, 382, 387, 392, 397, 402, 408, 412, 417, 422, 427, 431, 438, 442, 447, 452, 457, 461, 467, 472, 477, 482, 487, 492, 496, 503, 507, 512, 517, 522, 526, 532, 537, 542, 547, 552, 556, 562, 566, 572, 577, 582, 586, 592, 596
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*r in the joint ranking is
n+[sn/r]+[tn/r]+[un/r], and likewise for the
positions of n*s, n*t, and n*u.

Crossrefs

Cf. A184312, 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 *)

Formula

a(n)=n+[ns/r]+[nt/r]+[nu/r], where []=floor and
r=2^(1/5), s=r^2, t=r^3, u=r^4.

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 *)

A185914 Array: T(n,k)=k-n+1 for k>=n; T(n,k)=0 for k

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 2, 0, 0, 5, 3, 1, 0, 0, 6, 4, 2, 0, 0, 0, 7, 5, 3, 1, 0, 0, 0, 8, 6, 4, 2, 0, 0, 0, 0, 9, 7, 5, 3, 1, 0, 0, 0, 0, 10, 8, 6, 4, 2, 0, 0, 0, 0, 0, 11, 9, 7, 5, 3, 1, 0, 0, 0, 0, 0, 12, 10, 8, 6, 4, 2, 0, 0, 0, 0, 0, 0, 13, 11, 9, 7, 5, 3, 1, 0, 0, 0, 0, 0, 0, 14, 12, 10, 8, 6, 4, 2, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Clark Kimberling, Feb 06 2011

Keywords

Comments

A member of the accumulation chain
...< A185916 < A185914 < A185915 < ...
(See A144112 for definitions of weight array and accumulation array.)

Examples

			Northwest corner:
1...2...3...4...5...6...7...8...9
0...1...2...3...4...5...6...7...8
0...0...1...2...3...4...5...6...7
0...0...0...1...2...3...4...5...6
0...0...0...0...1...2...3...4...5

Links

Crossrefs

Cf. A144112, A185915, A185916.

Programs

  • Mathematica
    (* This program generates the array A185914, its accumulation array A185915, and its weight array A185916. *)
f[n_,0]:=0;f[0,k_]:=0; (* needed for the weight array *)
f[n_,k_]:=k-n+1; f[n_,k_]:=0/;kA185914 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* A184915 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A184916 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

Formula

T(n,k) = k-n+1 for k>=n; T(n,k)=0 for k=1, n>=1.
Showing 1-4 of 4 results.

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