A113497 Ascending descending base exponent transform of sequence A000034(n) = 1 + n mod 2.

1, 3, 6, 6, 11, 9, 16, 12, 21, 15, 26, 18, 31, 21, 36, 24, 41, 27, 46, 30, 51, 33, 56, 36, 61, 39, 66, 42, 71, 45, 76, 48, 81, 51, 86, 54, 91, 57, 96, 60, 101, 63, 106, 66, 111, 69, 116, 72, 121, 75, 126, 78, 131, 81, 136, 84, 141, 87, 146, 90, 151, 93, 156, 96, 161, 99, 166, 102, 171

Offset: 1

Author

Jonathan Vos Post, Jan 10 2006

Keywords

• nonn
• easy

Comments

A000034 = 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ... = continued fraction for (sqrt(3)+1)/2 (cf. A040001) = base 3 digital root of n+1. In general, the ascending descending base exponent transform of any simple periodic sequence can be written as a periodic set of interleaved sequences.

Examples

			a(1) = 1^1 = 1.
a(2) = 1^2 + 2^1 = 3.
a(3) = 1^1 + 2^2 + 1^1 = 6.
a(4) = 1^2 + 2^1 + 1^2 + 2^1 = 6.
a(5) = 1^1 + 2^2 + 1^1 + 2^2 + 1^1 = 11.
a(6) = 1^2 + 2^1 + 1^2 + 2^1 + 1^2 + 2^1 = 9.
		

Links

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

Crossrefs

Cf. A000034, A113320, A005408, A113122, A113153, A113154, A113336, A113271, A113258, A113257, A113231, A087316, A113208.

Programs

Mathematica

Table[(-3 + 3*(-1)^n + 8*n - 2*(-1)^n*n)/4, {n,1,50}] (* G. C. Greubel, Mar 12 2017 *)

PARI

x='x +O('x^50); Vec(x*(1+3*x+4*x^2)/((1-x)^2*(1+x)^2)) \\ G. C. Greubel, Mar 12 2017

Formula

a(n) = Sum_{i=1..n} A000034(i)^A000034(n-i+1).

a(2*n) = 3*n; a(2*n+1) = 5*n+1.

From Colin Barker, Jun 16 2012: (Start)

a(n) = (-3+3*(-1)^n+8*n-2*(-1)^n*n)/4.

a(n) = 2*a(n-2)-a(n-4).

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

E.g.f.: (1/2)*(3*(x-1)*sinh(x) + 5*x*cosh(x)). - G. C. Greubel, Mar 12 2017

Extensions

Definition improved by M. F. Hasler, Jan 13 2012