A138587 The union of all entries of A024495, A131708 and A024493 sorted into natural order.

0, 1, 2, 3, 5, 6, 10, 11, 21, 22, 42, 43, 85, 86, 170, 171, 341, 342, 682, 683, 1365, 1366, 2730, 2731, 5461, 5462, 10922, 10923, 21845, 21846, 43690, 43691, 87381, 87382, 174762, 174763, 349525, 349526, 699050, 699051, 1398101, 1398102, 2796202, 2796203, 5592405

Offset: 0

Author

Paul Curtz, May 13 2008

Keywords

• nonn

Comments

The three sequences of the definition share the same special recurrence which reflects that each equals its own sequence of third differences.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-1,1,1,2,2).

Programs

Mathematica

CoefficientList[Series[x*(3*x + 4*x^2 + 5*x^3 + 4*x^4 + 2*x^5 + 1)/((1 + x)*(1 - 2*x^2)*(1 + x^2)), {x,0,50}], x] (* G. C. Greubel, Oct 03 2017 *)
LinearRecurrence[{-1,1,1,2,2},{0,1,2,3,5,6,10},50] (* Harvey P. Dale, Feb 18 2023 *)

PARI

x='x+O('x^50); concat(0, Vec(x*(3*x+4*x^2+5*x^3+4*x^4 +2*x^5+ 1)/((1+x)*(1-2*x^2)*(1+x^2)))) \\ G. C. Greubel, Oct 03 2017

Formula

a(n+8) == a(n) (mod 10), n > 1.

a(2*n+1) - a(2*n) = 1.

a(2*n) = A000975(n+1), n>0 (bisection).

From R. J. Mathar, Nov 22 2009: (Start)

a(n) = -a(n-1) +a(n-2) +a(n-3) +2*a(n-4) +2*a(n-5), n>6.

G.f.: x*(3*x+4*x^2+5*x^3+4*x^4+2*x^5+1)/((1+x)*(1-2*x^2)*(1+x^2)). (End)

Extensions

Edited and extended by R. J. Mathar, Nov 22 2009