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.

User: Takumi Sato

Takumi Sato's wiki page.

Takumi Sato has authored 3 sequences.

A217575 Numbers n such that floor(sqrt(n)) = floor(n/floor(sqrt(n)))-1.

Original entry on oeis.org

2, 6, 7, 12, 13, 14, 20, 21, 22, 23, 30, 31, 32, 33, 34, 42, 43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 72, 73, 74, 75, 76, 77, 78, 79, 90, 91, 92, 93, 94, 95, 96, 97, 98, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 132, 133, 134, 135, 136
Offset: 1

Views

Author

Takumi Sato, Oct 07 2012

Keywords

Comments

One of four sequences given by classifying natural numbers according to the value of floor(sqrt(n)). See the paper in Link lines and A005563, A217570, A217571.
Can be interpreted as a triangle read by rows: T(n,k) = n*(n+1)+k-1 with n>0, k=1..n. - Bruno Berselli, Oct 11 2012

Examples

			As a triangle (see the second comment) this begins:
2;
6, 7;
12, 13, 14;
20, 21, 22, 23;
30, 31, 32, 33, 34;
42, 43, 44, 45, 46, 47;
56, 57, 58, 59, 60, 61, 62;
72, 73, 74, 75, 76, 77, 78, 79;
90, 91, 92, 93, 94, 95, 96, 97, 98; etc.
- _Bruno Berselli_, Oct 11 2012

Links

Crossrefs

Cf. A005563, A217570, A217571.
Cf. A063657.

Programs

  • Haskell
    a217575 = subtract 1 . a063657  -- Reinhard Zumkeller, Jun 20 2015
  • Magma
    [n: n in [1..150] | Isqrt(n) eq Floor(n/Isqrt(n))-1]; // Bruno Berselli, Oct 08 2012
  • Mathematica
    Select[Range[200],Floor[Sqrt[#]]==Floor[#/Floor[Sqrt[#]]]-1&] (* Harvey P. Dale, Oct 06 2018 *)
  • PARI
    is_A217575(n)=n\(n=sqrtint(n))-1==n  \\ - M. F. Hasler, Oct 09 2012

Formula

a(n) = A063657(n) - 1. - Reinhard Zumkeller, Jun 20 2015

A217570 Numbers n such that floor(sqrt(n)) = floor(n/(floor(sqrt(n))-1))-1.

Original entry on oeis.org

9, 16, 17, 25, 26, 27, 36, 37, 38, 39, 49, 50, 51, 52, 53, 64, 65, 66, 67, 68, 69, 81, 82, 83, 84, 85, 86, 87, 100, 101, 102, 103, 104, 105, 106, 107, 121, 122, 123, 124, 125, 126, 127, 128, 129, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 169, 170, 171, 172, 173
Offset: 1

Views

Author

Takumi Sato, Oct 07 2012

Keywords

Comments

The sequence consists of numbers n^2+k, 0<=k<=n-3, n=3,4,5,... - M. F. Hasler, Oct 09 2012
One of four sequences given by classifying natural numbers according to the value of floor(sqrt(n)). See the paper in Link lines and A005563, A217571, A217575. - Takumi Sato, Oct 09 2012

Examples

			As a triangle (see the first comment) this begins:
9;
16, 17;
25, 26, 27;
36, 37, 38, 39;
49, 50, 51, 52, 53;
64, 65, 66, 67, 68, 69;
81, 82, 83, 84, 85, 86, 87;
100, 101, 102, 103, 104, 105, 106, 107; etc.
[_Bruno Berselli_, Oct 12 2012]

Links

Crossrefs

Cf. A005563, A217571, A217575.

Programs

A217571 a(n) = (2*n*(n+5) + (2*n+1)*(-1)^n - 1)/8.

Original entry on oeis.org

1, 4, 5, 10, 11, 18, 19, 28, 29, 40, 41, 54, 55, 70, 71, 88, 89, 108, 109, 130, 131, 154, 155, 180, 181, 208, 209, 238, 239, 270, 271, 304, 305, 340, 341, 378, 379, 418, 419, 460, 461, 504, 505, 550, 551, 598, 599, 648, 649, 700, 701, 754, 755, 810, 811, 868
Offset: 1

Views

Author

Takumi Sato, Oct 07 2012

Keywords

Comments

One of four sequences given by classifying natural numbers according to the value of floor(sqrt(n)). See Sato link and sequences A005563, A217570, A217575.
Numbers n such that floor(sqrt(n)) = floor(n/floor(sqrt(n))) = floor(n/(floor(sqrt(n)) + 2)) + 1.

Examples

			From _Stefano Spezia_, Dec 14 2019: (Start)
Illustration of the initial terms:
o      o        o        o           o
     o o o    o o o    o o o       o o o
                o        o           o
                     o o o o o   o o o o o
                                     o
(1)   (4)      (5)     (10)        (11)
(End)

Links

Crossrefs

Cf. A005563, A028387, A028552, A217570, A217575.

Programs

  • GAP
    List([1..60], n-> (2*n^2 +10*n -1 +(-1)^n*(2*n+1))/8 ); # G. C. Greubel, Dec 19 2019
  • Magma
    [n: n in [1..900] | Floor(n/Isqrt(n)) eq Floor(n/(Isqrt(n)+2))+1]; // Bruno Berselli, Oct 10 2012
  • Magma
    I:=[1, 4, 5, 10, 11]; [n le 5 select I[n] else Self(n-1) + 2*Self(n-2) - 2*Self(n-3) - Self(n-4) + Self(n-5): n in [1..60]]; // Vincenzo Librandi, Dec 15 2012
  • Maple
    seq( (2*n^2 +10*n -1 +(-1)^n*(2*n+1))/8, n=1..60); # G. C. Greubel, Dec 19 2019
  • Mathematica
    CoefficientList[Series[(1 + 3*x - x^2 - x^3)/((1 + x)^2*(1 - x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 15 2012 *)
a[1]=1;a[n_]:=If[EvenQ[n],a[n-1]+1+n,a[n-1]+1]; Array[a,56] (* Stefano Spezia, Dec 18 2019 *)
  • Maxima
    makelist((2*n*(n+5)+(2*n+1)*(-1)^n-1)/8, n, 1, 56); /* Martin Ettl, Oct 15 2012 */
  • PARI
    vector(60, n, (2*n^2 +10*n -1 +(-1)^n*(2*n+1))/8 ) \\ G. C. Greubel, Dec 19 2019
  • Sage
    [(2*n^2 +10*n -1 +(-1)^n*(2*n+1))/8 for n in (1..60)] # G. C. Greubel, Dec 19 2019

Formula

G.f.: x*(1+3*x-x^2-x^3)/((1+x)^2*(1-x)^3). - Bruno Berselli, Oct 11 2012
From Stefano Spezia, Dec 14 2019: (Start)
E.g.f.: (x*(5+x)*cosh(x) - (1-7*x-x^2)*sinh(x))/4.
a(n) = a(n-1) + 1 for n odd.
a(n) = a(n-1) + n + 1 for n even.
a(2*n) = A028552(n).
a(2*n+1) = A028387(n).
(End)

Extensions

Definition by Bruno Berselli, Oct 11 2012

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