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

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

1,2

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.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Takumi Sato, Classification of Natural Numbers [Wayback Machine link]

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

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)

Example

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)

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 *)

Prog

(Visual Basic in Excel)
Sub A217571()
Dim x As Long, n As Long, y As Long, i As Long
x = InputBox("Count to")
For n = 1 To x
y = Int(Sqr(n))
If y = Int(n / y) Then GoTo L1
GoTo L2
L1: If y = Int(n / (y + 2)) + 1 Then
i = i + 1
Cells(i, 1) = n
End If
L2: Next n
End Sub
(Magma) [n: n in [1..900] | Floor(n/Isqrt(n)) eq Floor(n/(Isqrt(n)+2))+1]; // Bruno Berselli, Oct 10 2012
(Maxima) makelist((2*n*(n+5)+(2*n+1)*(-1)^n-1)/8, n, 1, 56); /* Martin Ettl, Oct 15 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
(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
(GAP) List([1..60], n-> (2*n^2 +10*n -1 +(-1)^n*(2*n+1))/8 ); # G. C. Greubel, Dec 19 2019

Crossrefs

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

Sequence in context: A327311 A151735 A050039 * A376653 A058335 A222353

Adjacent sequences: A217568 A217569 A217570 * A217572 A217573 A217574

Keyword

nonn,easy

Author

Takumi Sato, Oct 07 2012

Extensions

Definition by Bruno Berselli, Oct 11 2012

Status

approved