A047252 -id:A047252 - cp's OEIS frontend

A075249 x-value of the solution (x,y,z) to 5/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The y and z components are in A075250 and A075251.

1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 14, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18

Offset: 3

T. D. Noe, Sep 10 2002

See A075248 for more details.

Cf. A075248, A075250, A075251.

For[xLst={}; yLst={}; zLst={}; n=3, n<=100, n++, cnt=0; xr=n/5; If[IntegerQ[xr], x=xr+1, x=Ceiling[xr]]; While[yr=1/(5/n-1/x); If[IntegerQ[yr], y=yr+1, y=Ceiling[yr]]; cnt==0&&y>x, While[zr=1/(5/n-1/x-1/y); cnt==0&&zr>y, If[IntegerQ[zr], z=zr; cnt++; AppendTo[xLst, x]; AppendTo[yLst, y]; AppendTo[zLst, z]]; y++ ]; x++ ]]; xLst

Is a(n) = A047252(n-3)-n+4 ? - Ralf Stephan, Feb 24 2004

A296050 Number of permutations p of [n] such that min_{j=1..n} |p(j)-j| = 1.

Original entry on oeis.org

0, 0, 1, 2, 8, 40, 236, 1648, 13125, 117794, 1175224, 12903874, 154615096, 2007498192, 28075470833, 420753819282, 6726830163592, 114278495205524, 2055782983578788, 39039148388975552, 780412763620655061, 16381683795665956242, 360258256118419518680, 8283042472303599966974

Offset: 0

Alois P. Heinz, Jan 21 2019

nonn

			a(2) = 1: 21.
a(3) = 2: 231, 312.
a(4) = 8: 2143, 2341, 2413, 3142, 3421, 4123, 4312, 4321.
a(5) = 40: 21453, 21534, 23154, 23451, 23514, 24153, 24513, 24531, 25134, 25413, 25431, 31254, 31452, 31524, 34152, 34251, 35124, 35214, 35412, 35421, 41253, 41523, 41532, 43152, 43251, 43512, 43521, 45132, 45213, 45231, 51234, 51423, 51432, 53124, 53214, 53412, 53421, 54132, 54213, 54231.

Alois P. Heinz, Table of n, a(n) for n = 0..450

Column k=1 of A299789.

Cf. A000142, A000166, A001883, A002467, A016933, A047252, A323671.

b:= proc(s, k) option remember; (n-> `if`(n=0, `if`(k=1, 1, 0), add(
      `if`(n=j, 0, b(s minus {j}, min(k, abs(n-j)))), j=s)))(nops(s))
    end:
a:= n-> b({$1..n}, n):
seq(a(n), n=0..14);
# second Maple program:
a:= n-> (f-> f(1)-f(2))(k-> `if`(n=0, 1, LinearAlgebra[Permanent](
        Matrix(n, (i, j)-> `if`(abs(i-j)>=k, 1, 0))))):
seq(a(n), n=0..14);
# third Maple program:
g:= proc(n) g(n):= `if`(n<2, 1-n, (n-1)*(g(n-1)+g(n-2))) end:
h:= proc(n) h(n):= `if`(n<7, [1, 0$3, 1, 4, 29][n+1], n*h(n-1)+4*h(n-2)
      -3*(n-3)*h(n-3)+(n-4)*h(n-4)+2*(n-5)*h(n-5)-(n-7)*h(n-6)-h(n-7))
    end:
a:= n-> g(n)-h(n):
seq(a(n), n=0..25);

g[n_] := g[n] = If[n < 2, 1-n, (n-1)(g[n-1] + g[n-2])];
h[n_] := h[n] = If[n < 7, {1, 0, 0, 0, 1, 4, 29}[[n+1]],
     n h[n-1] + 4h[n-2] - 3(n-3)h[n-3] + (n-4)h[n-4] +
     2(n-5)h[n-5] - (n-7)h[n-6] - h[n-7]];
a[n_] := g[n] - h[n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 30 2021, after third Maple program *)

a(n) = A000142(n) - A001883(n) - A002467(n).
a(n) = A000166(n) - A001883(n).
a(n) = Sum_{k=1..n} A323671(n,k).
a(n) is odd <=> n in { A016933 }.
a(n) is even <=> n in { A047252 }.

A047248 Numbers that are congruent to {0, 2, 3, 4, 5} (mod 6).

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68

Offset: 1

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A047252.

Rest[CoefficientList[Series[x^2*(2 + x + x^2 + x^3 + x^4)/((x^4 + x^3 + x^2 + x + 1)*(x - 1)^2), {x, 0, 50}], x]] (* G. C. Greubel, Nov 02 2017 *)
DeleteCases[Range[0,70],?(Mod[#,6]==1&)] (* or *) Complement[ Range[ 0,70], Range[1,70,6]] (* _Harvey P. Dale, Dec 30 2017 *)

x='x+O('x^50); concat([0], Vec(x^2*(2+x+x^2+x^3+x^4)/((x^4 +x^3 +x^2 +x+1)*(x-1)^2))) \\ G. C. Greubel, Nov 02 2017

G.f.: x^2*(2+x+x^2+x^3+x^4) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

Sum_{n>=2} (-1)^n/a(n) = log(2+sqrt(3))/(2*sqrt(3)) + log(2)/6 - (9-4*sqrt(3))*Pi/36. - Amiram Eldar, Dec 17 2021

cp's OEIS Frontend

A075249 x-value of the solution (x,y,z) to 5/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z and having the largest z-value. The y and z components are in A075250 and A075251.

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A296050 Number of permutations p of [n] such that min_{j=1..n} |p(j)-j| = 1.

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A047248 Numbers that are congruent to {0, 2, 3, 4, 5} (mod 6).

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