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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A184429 Lower s-Wythoff sequence of A184429 (quarter-squares). Complement of A184430.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 118
Offset: 1

Views

Author

Clark Kimberling, Jan 14 2011

Keywords

Comments

See A184117 for the definition of lower and upper s-Wythoff sequences.

Crossrefs

Cf. A184117, A184430.

Programs

  • Mathematica
    mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
s[n_]:=Floor[((n+1)/2)^2];a[1]=1;b[n_]:=b[n]=s[n]+a[n];
a[n_]:=a[n]=mex[Flatten[Table[{a[i],b[i]},{i,1,n-1}]]];
Table[s[n],{n,40}]
Table[a[n],{n,100}]
Table[b[n],{n,100}]~

A184431 Lower s-Wythoff sequence, where s=A184431 (eighth-cubes). Complement of A184432.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 106, 107, 108
Offset: 1

Views

Author

Clark Kimberling, Jan 14 2011

Keywords

Comments

See A184117 for the definition of lower and upper s-Wythoff sequences.

Crossrefs

Cf. A184117, A184432.

Programs

  • Mathematica
    mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
s[n_]:=Floor[((n+1)/2)^3];a[1]=1;b[n_]:=b[n]=s[n]+a[n];
a[n_]:=a[n]=mex[Flatten[Table[{a[i],b[i]},{i,1,n-1}]]];
Table[s[n],{n,40}]
Table[a[n],{n,100}]
Table[b[n],{n,100}]

A184478 Lower s-Wythoff sequence, where s(n) = 3n + 1. Complement of A184479.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 90, 92, 93, 94, 96, 97, 98, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 117, 118, 119, 120, 122, 123, 124, 126, 127, 128, 130, 131, 132, 133, 135, 136, 137, 139, 140, 141, 143, 144, 145, 146, 148, 149, 150, 152, 153, 154, 156
Offset: 1

Views

Author

Clark Kimberling, Jan 15 2011

Keywords

Comments

See A184117 for the definition of lower and upper s-Wythoff sequences.
The sequence is defined by a(1) = 1 and for n > 1, a(n) is the smallest positive integer not in {a(k), a(k) + s(k); k < n}. - M. F. Hasler, Jan 07 2019

Crossrefs

Cf. A184117, A184479, A184480.

Programs

  • Magma
    [(Floor(n*(-1+Sqrt(13))/2))+1: n in [0..120]]; // Vincenzo Librandi, Jan 08 2019
  • Maple
    a:=n->floor(n*(-1+sqrt(13))/2+1): seq(a(n),n=0..120); # Muniru A Asiru, Jan 08 2019
  • Mathematica
    k=3; r=-1; d=Sqrt[4+k^2];
a[n_]:=Floor[(1/2)(d+2-k)(n+r/(d+2))];
b[n_]:=Floor[(1/2)(d+2+k)(n-r/(d+2))];
Table[a[n],{n,120}]
Table[b[n],{n,120}]
Table[(Floor[n (-1 + Sqrt[13]) / 2]) + 1, {n, 0, 120}] (* Vincenzo Librandi, Jan 08 2019 *)
  • PARI
    A184478_upto(N, s(n)=3*n+1, a=[1], U=a)={while(a[#a]1&&U[2]==U[1]+1, U=U[^1]); a=concat(a, U[1]+1)); a} \\ M. F. Hasler, Jan 07 2019

Formula

a(n) = A184479(n) - s(n). - M. F. Hasler, Jan 07 2019

A184486 Lower s-Wythoff sequence, where s(n)=4n+1. Complement of A184487.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96, 97, 98, 99, 101, 102, 103, 104, 106, 107, 108, 109, 111, 112, 113, 114, 115, 117, 118, 119, 120, 122, 123, 124, 125, 127, 128, 129, 130, 132, 133, 134, 135, 137, 138, 139, 140, 141, 143, 144, 145, 146, 148
Offset: 1

Views

Author

Clark Kimberling, Jan 15 2011

Keywords

Comments

See A184117 for the definition of lower and upper s-Wythoff sequences.

Crossrefs

Cf. A184117,A184487.

Programs

  • Mathematica
    k=4; r=-1; d=Sqrt[4+k^2];
a[n_]:=Floor[(1/2)(d+2-k)(n+r/(d+2))];
b[n_]:=Floor[(1/2)(d+2+k)(n-r/(d+2))];
Table[a[n],{n,120}]
Table[b[n],{n,120}]

A184487 Upper s-Wythoff sequence, where s(n)=4n+1. Complement of A184486.

Original entry on oeis.org

6, 11, 16, 21, 26, 32, 37, 42, 47, 53, 58, 63, 68, 74, 79, 84, 89, 95, 100, 105, 110, 116, 121, 126, 131, 136, 142, 147, 152, 157, 163, 168, 173, 178, 184, 189, 194, 199, 205, 210, 215, 220, 225, 231, 236, 241, 246, 252, 257, 262, 267, 273, 278, 283, 288, 294, 299, 304, 309, 314, 320, 325, 330, 335, 341, 346, 351, 356, 362, 367, 372, 377, 383, 388, 393, 398, 403, 409, 414, 419, 424, 430, 435, 440, 445, 451, 456, 461, 466, 472, 477, 482, 487, 492, 498, 503, 508
Offset: 1

Views

Author

Clark Kimberling, Jan 15 2011

Keywords

Crossrefs

Cf. A184117, A184486.

A184514 Lower s-Wythoff sequence, where s(n)=4n-1. Complement of A184515.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 55, 57, 58, 59, 60, 61, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 78, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 94, 95, 96, 97, 99, 100, 101, 102, 104, 105, 106, 107, 108, 110, 111, 112, 113, 115, 116, 117, 118, 120, 121, 122, 123, 125, 126, 127, 128, 129, 131, 132, 133, 134, 136, 137, 138, 139, 141, 142, 143, 144, 146, 147, 148
Offset: 1

Views

Author

Clark Kimberling, Jan 16 2011

Keywords

Comments

See A184117 for the definition of lower and upper s-Wythoff sequences.

Crossrefs

Cf. A184117, A184515.

Programs

  • Mathematica
    k=4; r=1; d=Sqrt[4+k^2];
a[n_]:=Floor[(1/2)(d+2-k)(n+r/(d+2))];
b[n_]:=Floor[(1/2)(d+2+k)(n-r/(d+2))];
Table[a[n],{n,120}]
Table[b[n],{n,120}]

A184515 Upper s-Wythoff sequence, where s=4n-1. Complement of A184514.

Original entry on oeis.org

4, 9, 14, 20, 25, 30, 35, 41, 46, 51, 56, 62, 67, 72, 77, 82, 88, 93, 98, 103, 109, 114, 119, 124, 130, 135, 140, 145, 151, 156, 161, 166, 171, 177, 182, 187, 192, 198, 203, 208, 213, 219, 224, 229, 234, 240, 245, 250, 255, 260, 266, 271, 276, 281, 287, 292, 297, 302, 308, 313, 318, 323, 329, 334, 339, 344, 350, 355, 360, 365, 370, 376, 381, 386, 391, 397, 402, 407, 412, 418, 423, 428, 433, 439, 444, 449, 454, 459, 465, 470, 475, 480, 486, 491, 496, 501
Offset: 1

Views

Author

Clark Kimberling, Jan 16 2011

Keywords

Comments

See A184117 for the definition of lower and upper s-Wythoff sequences.

Crossrefs

Cf. A184117, A184514.

Programs

  • Mathematica
    k=4; r=1; d=Sqrt[4+k^2];
a[n_]:=Floor[(1/2)(d+2-k)(n+r/(d+2))];
b[n_]:=Floor[(1/2)(d+2+k)(n-r/(d+2))];
Table[a[n],{n,120}]
Table[b[n],{n,120}]

A184517 Upper s-Wythoff sequence, where s=4n-2. Complement of A184516.

Original entry on oeis.org

3, 8, 14, 19, 24, 29, 35, 40, 45, 50, 55, 61, 66, 71, 76, 82, 87, 92, 97, 103, 108, 113, 118, 124, 129, 134, 139, 144, 150, 155, 160, 165, 171, 176, 181, 186, 192, 197, 202, 207, 213, 218, 223, 228, 234, 239, 244, 249, 254, 260, 265, 270, 275, 281, 286, 291, 296, 302, 307, 312, 317, 323, 328, 333, 338, 343, 349, 354, 359, 364, 370, 375, 380, 385, 391, 396, 401, 406, 412, 417, 422, 427, 432, 438, 443, 448, 453, 459, 464, 469, 474, 480, 485, 490, 495, 501, 506
Offset: 1

Views

Author

Clark Kimberling, Jan 16 2011

Keywords

Comments

See A184117 for the definition of lower and upper s-Wythoff sequences.

Links

Crossrefs

Cf. A184117, A184516.

Programs

  • Magma
    [Floor((3+Sqrt(5))*(n - 1/(1+Sqrt(5)))): n in [1..100]]; // G. C. Greubel, Nov 16 2018
  • Mathematica
    k = 4; r = 2; d = Sqrt[4 + k^2];
a[n_] := Floor[(1/2) (d + 2 - k) (n + r/(d + 2))];
b[n_] := Floor[(1/2) (d + 2 + k) (n - r/(d + 2))];
Table[a[n], {n, 120}] (* A184516 *)
Table[b[n], {n, 120}] (* A184517 *)
(* alternate program *)
Table[Ceiling[(2 n - 1) GoldenRatio^2], {n, 1, 120}] (* Jon Maiga, Nov 15 2018 *)
  • PARI
    vector(100, n, floor((3+sqrt(5))*(n - 1/(1+sqrt(5))))) \\ G. C. Greubel, Nov 16 2018
  • Sage
    [floor((3+sqrt(5))*(n - 1/(1+sqrt(5)))) for n in (1..100)] # G. C. Greubel, Nov 16 2018

Formula

a(n) = ceiling((2*n-1)*phi^2), where phi = A001622. - Jon Maiga, Nov 15 2018

A184521 Upper s-Wythoff sequence, where s=5n+1. Complement of A184520.

Original entry on oeis.org

7, 13, 19, 25, 31, 37, 44, 50, 56, 62, 68, 75, 81, 87, 93, 99, 106, 112, 118, 124, 130, 137, 143, 149, 155, 161, 168, 174, 180, 186, 192, 199, 205, 211, 217, 223, 229, 236, 242, 248, 254, 260, 267, 273, 279, 285, 291, 298, 304, 310, 316, 322, 329, 335, 341, 347, 353, 360, 366, 372, 378, 384, 390, 397, 403, 409, 415, 421, 428, 434, 440, 446, 452, 459, 465, 471, 477
Offset: 1

Views

Author

Clark Kimberling, Jan 16 2011

Keywords

Comments

See A184117 for the definition of lower and upper s-Wythoff sequences.
The terms 7,13,19,25,31,37,44,50 appear as the initial values of the n-weight domination number gamma_n(P_3 X P_8) in Hare (1990). This may or may not be a coincidence. - N. J. A. Sloane, May 31 2012

References

  • Hare, E. O., k-weight domination and fractional domination of P_m X P_n. Proceedings of the Twenty-first Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1990). Congr. Numer. 78 (1990), 71--80. MR1140471 (92i:05201). - From N. J. A. Sloane, May 31 2012

Crossrefs

Cf. A184117, A184520.

Programs

  • Mathematica
    k = 5; r = -1; d = Sqrt[4 + k^2];
a[n_] := Floor[(1/2) (d + 2 - k) (n + r/(d + 2))];
b[n_] := Floor[(1/2) (d + 2 + k) (n - r/(d + 2))];
Table[a[n], {n, 120}]
Table[b[n], {n, 120}]

A184523 Upper s-Wythoff sequence, where s=5n. Complement of A184522.

Original entry on oeis.org

6, 12, 18, 24, 30, 37, 43, 49, 55, 61, 68, 74, 80, 86, 92, 99, 105, 111, 117, 123, 130, 136, 142, 148, 154, 161, 167, 173, 179, 185, 191, 198, 204, 210, 216, 222, 229, 235, 241, 247, 253, 260, 266, 272, 278, 284, 291, 297, 303, 309, 315, 322, 328, 334, 340
Offset: 1

Views

Author

Clark Kimberling, Jan 16 2011

Keywords

Comments

See A184117 for the definition of lower and upper s-Wythoff sequences.

Crossrefs

Cf. A184117, A184522.

Programs

  • Mathematica
    k = 5; r = -1; d = Sqrt[4 + k^2];
a[n_] := Floor[(1/2) (d + 2 - k) (n + r/(d + 2))];
b[n_] := Floor[(1/2) (d + 2 + k) (n - r/(d + 2))];
Table[a[n], {n, 120}]
Table[b[n], {n, 120}]

Formula

a(n) = floor((n/2)*(7+sqrt(29))). - Jason Yuen, Oct 16 2024
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