A121496 Run lengths of successive numbers in A068225.

1, 2, 2, 1, 3, 4, 4, 3, 5, 6, 6, 5, 7, 8, 8, 7, 9, 10, 10, 9, 11, 12, 12, 11, 13, 14, 14, 13, 15, 16, 16, 15, 17, 18, 18, 17, 19, 20, 20, 19, 21, 22, 22, 21, 23, 24, 24, 23, 25, 26, 26, 25, 27, 28, 28, 27, 29, 30, 30, 29, 31, 32, 32, 31, 33, 34, 34, 33, 35, 36, 36, 35, 37, 38, 38

Offset: 1

Author

Rick L. Shepherd, Aug 03 2006

Keywords

• nonn
• easy

Comments

A000027 and A103889 are bisections.

Examples

			The fifth run of successive numbers in A068225 is 8, 9, 10 with run length three so a(5) = 3.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A000027, A068225, A103889.

Programs

Mathematica

Rest@ CoefficientList[Series[x (1 + x - x^3 + x^4)/((1 - x)^2*(1 + x) (1 + x^2)), {x, 0, 75}], x] (* Michael De Vlieger, Oct 02 2017 *)

PARI

a(n) = if(n%2==1,(n+1)/2,if(n%4==0,(n/2)-1,(n/2)+1))
for(n=1,80,print1(a(n),", "))

PARI

Vec(x*(1+x-x^3+x^4)/((1-x)^2*(1+x)*(1+x^2)) + O(x^100)) \\ Colin Barker, Apr 08 2016

Formula

a(2*k-1) = k, a(4*k) = 2*k-1, a(4*k-2) = 2*k, for k >= 1.

From Colin Barker, Apr 08 2016: (Start)

a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.

G.f.: x*(1+x-x^3+x^4) / ((1-x)^2*(1+x)*(1+x^2)). (End)

a(n) = (2*n+1-4*cos(n*Pi/2)-cos(n*Pi))/4. - Wesley Ivan Hurt, Oct 02 2017