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.

A182643 A182642(n) - A107857(n-1) for n>=1.

Original entry on oeis.org

0, 1, 4, 6, 18, 31, 82, 132, 348, 565, 1480, 2394, 6270, 10147, 26566, 42984, 112536, 182089, 476716, 771342, 2019402, 3267463, 8554330, 13841196, 36236725, 58632253, 153501232, 248370211, 650241655
Offset: 1

Views

Author

Clark Kimberling, Nov 24 2010

Keywords

Comments

A182643(n) = distance from A107857(n-1) to A182642(n), where A182642(n) is the result of removing all but the outermost floor function in the formula for A107857 - so that n-1 floors are removed and the n-th remains.

Examples

			a(3)=floor(s*r*s)-floor(s*floor(r*floor(s))), where
r=(1+sqrt(5))/2 and s=r+1.

Crossrefs

Cf. A107857,A182642.

A328696 Rectangular array R read by descending antidiagonals: apply x -> (x+1)/2 to each odd term of the Wythoff array (A035513), and delete all others.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 7, 15, 8, 12, 11, 24, 20, 19, 9, 28, 62, 32, 49, 23, 10, 45, 100, 83, 79, 37, 16, 13, 117, 261, 134, 206, 96, 41, 21, 14, 189, 422, 350, 333, 155, 66, 54, 36, 25, 494, 1104, 566, 871, 405, 172, 87, 58, 40, 17, 799, 1786, 1481, 1409, 655
Offset: 1

Views

Author

Clark Kimberling, Oct 26 2019

Keywords

Comments

Every positive integer occurs exactly once in R, and every row of R is a linear recurrence sequence.

Examples

			Row 1 of the Wythoff array is (1,2,3,5,8,13,21,34,55,89,144,...), so that row 1 of R is (1,2,3,7,11,...) = A107857 (essentially).
_______________
Northwest corner of R:
   1,  2,  3,  7,  11,  28,  45,  117,  189,  494,   799
   4,  6, 15, 24,  62, 100, 261,  422, 1104, 1786,  4675
   5,  8, 20, 32,  83, 134, 350,  566, 1481, 2396,  6272
  12, 19, 49, 79, 206, 333, 871, 1409, 3688, 5967, 15621
   9, 23, 37, 96, 155, 405, 655, 1714, 2773, 7259, 11745
  10, 16, 41, 66, 172, 278, 727, 1176, 3078, 4980, 13037
  13, 21, 54, 87, 227, 367, 960, 1553, 4065, 6577, 17218

Crossrefs

Cf. A035513, A107857, A328695, A328697.

Programs

  • Mathematica
    w[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten;
q[n_, k_] := If[Mod[w[n, k], 2] == 1, (1 + w[n, k])/2, 0];
t[n_] := Union[Table[q[n, k], {k, 1, 50}]];
u[n_] := If[First[t[n]] == 0, Rest[t[n]], t[n]]
s = Select[Range[40], ! u[#] == {} &]; u1[n_] := u[s[[n]]];
Column[Table[u1[n], {n, 1, 10}]] (* A328696 array *)
v[n_, k_] := u1[n][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A328696 sequence *)

A107858 a(n) = b(k), where b(k) = Fibonacci(n-1) and k = floor( n*(1+sqrt(5))/2 ).

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 28, 45, 117, 189, 494, 799, 2091, 3383, 8856, 14329, 37513, 60697, 158906, 257115, 673135, 1089155, 2851444, 4613733, 12078909, 19544085, 51167078, 82790071, 216747219, 350704367, 918155952, 1485607537, 3889371025
Offset: 1

Views

Author

Roger L. Bagula, Jun 12 2005

Keywords

Comments

Limit_{n -> oo} a(n+1)/a(n) does not exist.
Apparently the same as A107857. - Georg Fischer, Nov 02 2018

Crossrefs

Cf. A000045, A107857, A185381.

Programs

  • Mathematica
    F[1] = 0; F[2] = 1; F[n__] := F[n] = F[n - 1] + F[n - 2]
Table[F[ Floor[(Sqrt[5] + 1)*n/2]], {n, 1, 50}] (* F[n] are the Fibonacci numbers, A000045, with offset 1 *)

Extensions

Edited by N. J. A. Sloane, May 06 2012

A182691 Composite Beatty sequence of sqrt(2).

Original entry on oeis.org

3, 4, 13, 18, 61, 86, 293, 414, 1413, 1998, 6821, 9646, 32933, 46574, 159013, 224878, 767781, 1085806, 3707173, 5242734, 17899813, 25314158, 86427941, 122227566, 417311013, 590166894, 2014955813, 2849577838, 9729067301
Offset: 1

Views

Author

Clark Kimberling, Nov 27 2010

Keywords

Comments

The bisection (4,18,86,...) is a subsequence of A001951.
The bisection (3,13,61,...) is a subsequence of A001952.
See the comment at A107857 regarding Beatty sequences.

Examples

			a(1)=floor(2+sqrt(2))=3, a(2)=floor(r*a(1))=4.

Links

Crossrefs

Cf. A001951, A001952, A107857.

Programs

  • Maple
    Digits := 16 ;
A182691 := proc(n) option remember; local r,s ; r := sqrt(2) ; s := 2+r ; if n = 1 then floor(s) ; elif type(n,'odd') then floor(s*procname(n-1)) ; else floor(r*procname(n-1)) ; end if; end proc:
seq(A182691(n),n=1..30) ;
  • Mathematica
    a[1]:= 3; a[n_]:= If[OddQ[n], Floor[(2+Sqrt[2])*a[n-1]], Floor[Sqrt[2]*a[n-1]]]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Sep 29 2018 *)

Formula

a(n) = floor(s*a(n-1)) if n odd, a(n)=floor(r*a(n-1)) if n even, where r=sqrt(2), s=2+r, a(1)=floor(s).

A182692 Composite Beatty sequence of sqrt(3).

Original entry on oeis.org

2, 3, 7, 12, 28, 48, 113, 195, 461, 798, 1888, 3270, 7736, 13399, 31702, 54909, 129916, 225021, 532405, 922152, 2181835, 3779049, 8941325, 15486829, 36642230, 63466204, 150162650, 260089339, 615377983, 1065865932
Offset: 1

Views

Author

Clark Kimberling, Nov 27 2010

Keywords

Comments

The bisection (3,12,48,...) is a subsequence of A022838.
The bisection (2,7,28,...) is a subsequence of A054406.
See the comment at A107857 regarding Beatty sequences.

Crossrefs

Cf. A022838, A054406, A107857, A182631.

Formula

a(n)=floor(s*a(n-1)) if n odd, a(n)=floor(r*a(n-1)) if n even, where r=sqrt(3), s=(r+3)/2, a(1)=floor(s).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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