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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A184415 Lower s(n)-Wythoff sequence, where s(n)=floor[(n+2)/3]. Complement of A184416.

Original entry on oeis.org

1, 3, 5, 7, 8, 11, 12, 14, 16, 18, 20, 21, 23, 26, 27, 29, 30, 33, 34, 37, 38, 40, 42, 43, 46, 47, 49, 52, 53, 54, 57, 59, 60, 61, 65, 66, 67, 69, 72, 74, 75, 76, 79, 81, 83, 84, 86, 87, 91, 92, 93, 95, 97, 99, 101, 104, 105, 106, 107, 111, 112, 114, 116, 118, 119, 121, 122, 125, 128, 129, 130, 132, 134, 136, 138, 139, 142, 144, 146, 147, 149, 150, 152, 155, 157, 158, 160, 162, 164, 166, 167, 169, 171, 172, 175, 177, 179, 181, 182, 184
Offset: 1

Views

Author

Clark Kimberling, Jan 13 2011

Keywords

Examples

			s=(1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,...)=A002264.
a=(1,3,5,7,8,11,12,14,16,18,20,...)=A184415.
b=(2,4,6,9,10,13,15,17,19,22,24,...)=A184416.
Briefly: s=a+b, where a=mex="least missing".

Crossrefs

Cf. A184117, A184416.

Programs

  • Mathematica
    mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
s[n_]:=Floor[(n+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,20}]
Table[a[n],{n,100}]
Table[b[n],{n,100}]

A184427 Lower s-Wythoff sequence of A000290 (the squares). Complement of A184428.

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109
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, A184428.

Programs

  • Mathematica
    mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
s[n_]:=n^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}]

A184428 Upper s-Wythoff sequence of A000290 (the squares). Complement of A184427.

Original entry on oeis.org

2, 7, 13, 21, 31, 44, 58, 74, 92, 112, 135, 159, 185, 213, 243, 275, 309, 346, 384, 424, 466, 510, 556, 604, 654, 706, 761, 817, 875, 935, 997, 1061, 1127, 1195, 1265, 1337, 1411, 1487, 1566, 1646, 1728, 1812, 1898, 1986, 2076, 2168, 2262, 2358, 2456, 2556, 2658, 2763, 2869, 2977, 3087, 3199, 3313, 3429, 3547, 3667, 3789, 3913, 4039, 4167, 4297, 4429, 4564, 4700, 4838, 4978, 5120
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, A184427.

Programs

  • Mathematica
    mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
s[n_]:=n^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}]

A184482 Lower s-Wythoff sequence, where s(n)=3n-1. Complement of A184483.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 97, 99, 100, 101, 103, 104, 105, 107, 108, 109, 110, 112, 113, 114, 116, 117, 118, 120, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 134, 135, 137, 138, 139, 140, 142, 143, 144, 146, 147, 148, 150, 151, 152, 153, 155, 156
Offset: 1

Views

Author

Clark Kimberling, Jan 15 2011

Keywords

Crossrefs

Cf. A184117, A184483.

Programs

  • 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}]

A001956 Beatty sequence of (5+sqrt(13))/2.

Original entry on oeis.org

4, 8, 12, 17, 21, 25, 30, 34, 38, 43, 47, 51, 55, 60, 64, 68, 73, 77, 81, 86, 90, 94, 98, 103, 107, 111, 116, 120, 124, 129, 133, 137, 141, 146, 150, 154, 159, 163, 167, 172, 176, 180, 185, 189, 193, 197, 202, 206, 210, 215, 219, 223, 228, 232, 236, 240, 245, 249
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Inserting a=3 into the Fraenkel formula, a scale factor alpha = (2-a+sqrt(a^2+4))/2 = (sqrt(13)-1)/2 is obtained, which defines the Beatty sequence A184480. The complementary beta parameter, 1/beta+1/alpha=1, is beta = (5+sqrt(13))/2 = 3+alpha, and defines this sequence here, which is the complement in the positive integers. - R. J. Mathar, Feb 12 2011
Upper s-Wythoff sequence, where s(n)=3n. See A184117 for the definition of lower and upper s-Wythoff sequences. - Clark Kimberling, Jan 15 2011

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Complement of A184480.
Cf. A184117, A184482.

Programs

  • Maple
    A001956 := proc(n) local x ; x := (5+sqrt(13))/2 ; floor(n*x) ; end proc:
A184480 := proc(n) local x ; x := (sqrt(13)-1)/2 ; floor(n*x) ; end proc:
seq(A001956(n),n=1..100) ; # R. J. Mathar, Feb 12 2011
  • Mathematica
    Table[Floor[n*(5 + Sqrt[13])/2], {n, 100}] (* T. D. Noe, Aug 17 2012 *)

Formula

a(n) = floor(n*beta) with beta = (5+sqrt(13))/2 = 3+(sqrt(13)-1)/2 = 4.30277563773199...

A184118 Upper s(n)-Wythoff sequence, where s(n) = 2n + 1.

Original entry on oeis.org

4, 7, 10, 14, 17, 21, 24, 28, 31, 34, 38, 41, 45, 48, 51, 55, 58, 62, 65, 68, 72, 75, 79, 82, 86, 89, 92, 96, 99, 103, 106, 109, 113, 116, 120, 123, 127, 130, 133, 137, 140, 144, 147, 150, 154, 157, 161, 164, 168, 171, 174, 178, 181, 185, 188, 191, 195, 198, 202, 205, 208, 212, 215, 219, 222, 226, 229, 232, 236, 239, 243, 246, 249, 253, 256, 260, 263, 267, 270, 273, 277, 280, 284, 287, 290, 294, 297, 301, 304, 307, 311, 314, 318, 321, 325, 328, 331, 335, 338, 342
Offset: 1

Views

Author

Clark Kimberling, Jan 09 2011

Keywords

Comments

See A184117 (the lower s(n)-Wythoff sequence).

Links

Crossrefs

Cf. A000201, A001950, A001950, A001951, A136119, A184117.

Programs

  • Magma
    [Floor((2+Sqrt(2))*n+Sqrt(2)/2): n in [1..100]]; // Vincenzo Librandi, Jan 07 2019
  • Maple
    a:=n->floor((2+sqrt(2))*n+sqrt(2)/2): seq(a(n),n=1..80); # Muniru A Asiru, Jan 08 2019
  • Mathematica
    k=2; r=-1;
mex:=First[Complement[Range[1, Max[#1]+1], #1]]&;
S[n_]:=k n-r; 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, 30}]
Table[A[n], {n, 100}]
Table[B[n], {n, 100}]
Table[Floor[(2 + Sqrt[2]) n + Sqrt[2] / 2], {n, 80}] (* Vincenzo Librandi, Jan 07 2019 *)
  • PARI
    A184118_upto(N,s(n)=2*n+1,U=[0],b=[])={until(b[#b]>=N, b=concat(b,s(1+#b)+U[1]+=1); U=setunion(U,[b[#b]]); while(#U>1&&U[2]==U[1]+1,U=U[^1]));b} \\ M. F. Hasler, Jan 07 2019

Formula

a(n) = A184117(n) + s(n) for all n. - M. F. Hasler, Jan 07 2019

A184414 Upper s(n)-Wythoff sequence, where s(n)=floor[(n+1)/2].

Original entry on oeis.org

2, 4, 7, 8, 12, 13, 15, 18, 21, 22, 25, 26, 30, 31, 35, 36, 38, 41, 43, 44, 48, 50, 52, 54, 58, 59, 61, 63, 66, 68, 71, 72, 74, 77, 80, 82, 84, 86, 89, 90, 94, 96, 98, 100, 102, 104, 107, 109, 112, 113, 117, 118, 120, 122, 125, 127, 130, 132, 135, 136, 139, 141, 143, 146, 148, 149, 153, 155, 158, 159, 162, 164, 166, 168, 171, 172, 176, 177, 180, 182, 185, 186, 189, 192, 194, 195, 198, 200, 202, 205, 207, 209, 212, 214, 217, 218, 222, 223, 225, 228
Offset: 1

Views

Author

Clark Kimberling, Jan 13 2011

Keywords

Comments

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

Crossrefs

Cf. A184413, A184117.

Programs

  • Mathematica
    mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
s[n_]:=Floor[(n+1)/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,20}]
Table[a[n],{n,100}]
Table[b[n],{n,100}]

A184416 Upper s(n)-Wythoff sequence, where s(n)=floor[(n+2)/3]. Complement of A184415.

Original entry on oeis.org

2, 4, 6, 9, 10, 13, 15, 17, 19, 22, 24, 25, 28, 31, 32, 35, 36, 39, 41, 44, 45, 48, 50, 51, 55, 56, 58, 62, 63, 64, 68, 70, 71, 73, 77, 78, 80, 82, 85, 88, 89, 90, 94, 96, 98, 100, 102, 103, 108, 109, 110, 113, 115, 117, 120, 123, 124, 126, 127, 131, 133, 135, 137, 140, 141, 143, 145, 148, 151, 153, 154, 156, 159, 161, 163, 165, 168, 170, 173, 174, 176, 178, 180, 183, 186, 187, 189, 192, 194, 196, 198, 200, 202, 204, 207, 209, 212, 214, 215, 218
Offset: 1

Views

Author

Clark Kimberling, Jan 13 2011

Keywords

Examples

			See A184415.

Crossrefs

Cf. A184117, A184415.

Programs

  • Mathematica
    mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
s[n_]:=Floor[(n+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,20}]
Table[a[n],{n,100}]
Table[b[n],{n,100}]

A184419 Lower s-Wythoff sequence, where s=lower Wythoff sequence. Complement of A184420.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 79, 81, 83, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 101, 103, 104, 106, 108, 109, 110, 112, 114, 115, 116, 118, 119, 121, 122, 124, 125, 127, 128, 130, 131, 133, 134, 135, 137, 139, 140, 141, 143, 144, 146, 148
Offset: 1

Views

Author

Clark Kimberling, Jan 13 2011

Keywords

Comments

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

Examples

			s=(1,3,4,6,8,9,11,12,...)=A000201=lower Wythoff sequence;
a=(1,3,4,5,7,9,10,12,...)=A184419;
b=(2,6,8,11,15,18,21,24,...)=A184420.
Briefly: b=s+a, where a=mex="least missing".

Crossrefs

Cf. A184117, A000201, A184420.

Programs

  • Mathematica
    mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
r=(1+5^(1/2))/2;s[n_]:=Floor[r*n];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}]

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

Original entry on oeis.org

5, 9, 13, 17, 22, 26, 30, 35, 39, 43, 48, 52, 56, 61, 65, 69, 73, 78, 82, 86, 91, 95, 99, 104, 108, 112, 116, 121, 125, 129, 134, 138, 142, 147, 151, 155, 159, 164, 168, 172, 177, 181, 185, 190, 194, 198, 202, 207, 211, 215, 220, 224, 228, 233, 237, 241, 246
Offset: 1

Views

Author

Clark Kimberling, Jan 15 2011

Keywords

Links

Crossrefs

Cf. A184117, A184478.

Programs

  • 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}]
  • PARI
    A184479_upto(N, s(n)=3*n+1, U=[0], b=[])={until(b[#b]>=N, b=concat(b, s(1+#b)+U[1]+=1); U=setunion(U, [b[#b]]); while(#U>1 && U[2]==U[1]+1, U=U[^1])); b} \\ M. F. Hasler, Jan 07 2019

Formula

a(n) = A184478(n) + s(n). - M. F. Hasler, Jan 07 2019
For n up to 1000 at least, a(n) = round(m*n + c) with m ~ 4.302774 and c ~ 0.268517. - M. F. Hasler, Jan 07 2019
Previous Showing 11-20 of 53 results. Next

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