A167762 a(n) = 2*a(n-1)+3*a(n-2)-6*a(n-3) starting a(0)=a(1)=0, a(2)=1.

0, 0, 1, 2, 7, 14, 37, 74, 175, 350, 781, 1562, 3367, 6734, 14197, 28394, 58975, 117950, 242461, 484922, 989527, 1979054, 4017157, 8034314, 16245775, 32491550, 65514541, 131029082, 263652487, 527304974, 1059392917, 2118785834, 4251920575, 8503841150

Offset: 0

Author

Paul Curtz, Nov 11 2009

Keywords

• nonn
• easy

Comments

Inverse binomial transform yields two zeros followed by A077917 (a signed variant of A127864).

a(n) mod 10 is zero followed by a sequence with period length 8: 0, 1, 2, 7, 4, 7, 4, 5 (repeat).

a(n) is the number of length n+1 binary words with some prefix w such that w contains three more 1's than 0's and no prefix of w contains three more 0's than 1's. - Geoffrey Critzer, Dec 13 2013

From Gus Wiseman, Oct 06 2023: (Start)

Also the number of subsets of {1..n} with two distinct elements summing to n + 1. For example, the a(2) = 1 through a(5) = 14 subsets are:

{1,2} {1,3} {1,4} {1,5}

{1,2,3} {2,3} {2,4}

{1,2,3} {1,2,4}

{1,2,4} {1,2,5}

{1,3,4} {1,3,5}

{2,3,4} {1,4,5}

{1,2,3,4} {2,3,4}

{2,4,5}

{1,2,3,4}

{1,2,3,5}

{1,2,4,5}

{1,3,4,5}

{2,3,4,5}

{1,2,3,4,5}

The complement is counted by A038754.

Allowing twins gives A167936, complement A108411.

For n instead of n + 1 we have A365544, complement A068911.

The version for all subsets (not just pairs) is A366130.

(End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

Crossrefs

Cf. A024495, A167710.

First differences are A167936, complement A108411.

Cf. A004526, A004737, A008967, A038754, A046663, A068911, A088809, A093971, A365376, A365544, A366130.

Programs

Mathematica

LinearRecurrence[{2,3,-6},{0,0,1},40] (* Harvey P. Dale, Sep 17 2013 *)
CoefficientList[Series[x^2/((2 x - 1) (3 x^2 - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 17 2013 *)
Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#,{2}],n+1]&]],{n,0,10}] (* Gus Wiseman, Oct 06 2023 *)

Formula

a(n) mod 9 = A153130(n), n>3 (essentially the same as A154529, A146501 and A029898).

a(n+1)-2*a(n) = 0 if n even, = A000244((1+n)/2) if n odd.

a(2*n) = A005061(n). a(2*n+1) = 2*A005061(n).

G.f.: x^2/((2*x-1)*(3*x^2-1)). a(n) = 2^n - A038754(n). - R. J. Mathar, Nov 12 2009

G.f.: x^2/(1-2*x-3*x^2+6*x^3). - Philippe Deléham, Nov 11 2009

Extensions

Edited and extended by R. J. Mathar, Nov 12 2009