A211605 a(n) is the nearest integer to x(n), where x(1) = 1, x(2) = 2, x(n) = operation (n-2 mod 4) applied to the pair (x(n-1), x(n-2)), where operation 1 is addition, operation 2 is multiplication, operation 3 is subtraction and operation 0 is division.

1, 2, 3, 6, 3, 1, 4, 2, -2, -1, -3, 3, 6, 2, 8, 15, 8, 1, 8, 4, -4, -1, -5, 5, 10, 2, 12, 24, 12, 1, 13, 6, -6, -1, -7, 7, 15, 2, 17, 33, 17, 1, 17, 9, -9, -1, -10, 10, 19, 2, 21, 42, 21, 1, 22, 11, -11, -1, -12, 12, 24, 2, 26, 51, 26, 1, 26, 13, -13, -1, -14

Offset

1,2

Comments

This sequence generated by operations [+ * - /] which is slowest increasing among total 24 permutation operations sequences.

The total 24 sequences can be grouped as 6 patterns of as following:

1: [+ - * /], [- * / +], [* / + -], [/ + - *]

2: [+ - / *], [- / * +], [/ * + -], [* + - /]

3: [+ * - /], [* - / +], [- / + *], [/ + * -]

4: [+ * / -], [* / - +], [/ - + *], [- + * /]

5: [+ / - *], [/ - * +], [- * + /], [* + / -]

6: [+ / * -], [/ * - +], [* - + /], [- + / *]

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

G.f.: -(-x^56 +x^55 +x^54 +x^53 -x^52 -2*x^50 +2*x^49 +3*x^48 +3*x^47 -4*x^46 +x^43 -3*x^41 +2*x^40 +4*x^39 +3*x^38 +3*x^37 +5*x^36 +2*x^35 -3*x^34 +x^31 -2*x^29 +x^28 +5*x^27 +2*x^26 +3*x^25 +4*x^24 +2*x^23 -3*x^22 +x^19 -2*x^17 +x^16 +4*x^15 +2*x^14 +4*x^13 +5*x^12 +3*x^11 -4*x^10 +x^7 -x^6 -2*x^5 +x^4 +5*x^3 +3*x^2 +2*x +1)*x / (-x^57 +x^54 -x^51 +x^48 +x^9 -x^6 +x^3 -1). - Alois P. Heinz, Feb 16 2013

Example

x(3) = x(2) + x(1) = 2 + 1   = 3,       a(3) = 3,
x(4) = x(3) * x(2) = 3 * 2   = 6,       a(4) = 6,
x(5) = x(4) - x(3) = 6 - 3   = 3,       a(5) = 3,
x(6) = x(5) / x(4) = 3 / 6   = 0.5,     a(6) = 1,
x(7) = x(6) + x(5) = 0.5 + 3 = 3.5,     a(7) = 4,...

Maple

x:= proc(n) option remember; `if`(n<3, n,
       [(x, y)->x/y, (x, y)->x+y, (x, y)->x*y, (x, y)->x-y]
       [1+irem(n-2, 4)](x(n-1), x(n-2)))
    end:
a:= n-> round(x(n)):
seq(a(n), n=1..100);   # Alois P. Heinz, Feb 16 2013

Mathematica

x[n_] := x[n] = If[n<3, n, {#1/#2, #1+#2, #1*#2, #1-#2}[[1+Mod[n-2, 4]]]&[ x[n-1], x[n-2]]];
a[n_] := With[{q = x[n]}, If[Denominator[q] == 2, q+Sign[q]/2, Round[q]]];
Array[a, 100] (* Jean-François Alcover, Mar 18 2021 *)

Prog

(Small Basic)
a[1]=1
a[2]=2
For n=3 To 100
  nn = math.Remainder(n, 4)
  If nn = 3 Then ' operation +
    a[n]=a[n-1]+a[n-2]
  Else
  EndIf
  If nn = 0 Then ' operation *
    a[n]=a[n-1]*a[n-2]
  Else
  EndIf
  If nn=1 Then ' operation -
    a[n]=a[n-1]-a[n-2]
  Else
  EndIf
  If nn=2 Then ' operation  /
    a[n]=a[n-1]/a[n-2]
  Else
EndIf
EndFor
For j = 1 to 100
  TextWindow.Write(j+" ")
  TextWindow.Write(math.Round(a[j]))
  TextWindow.WriteLine(" ")
EndFor

Crossrefs

Sequence in context: A214630 A093396 A189971 * A316610 A322229 A245886

Adjacent sequences: A211602 A211603 A211604 * A211606 A211607 A211608

Keyword

sign,easy

Author

Kival Ngaokrajang, Feb 10 2013

Status

approved