A167762 - OEIS

%I #46 Dec 12 2023 07:45:15

%S 0,0,1,2,7,14,37,74,175,350,781,1562,3367,6734,14197,28394,58975,

%T 117950,242461,484922,989527,1979054,4017157,8034314,16245775,

%U 32491550,65514541,131029082,263652487,527304974,1059392917,2118785834,4251920575,8503841150

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

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

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

%C 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

%C From _Gus Wiseman_, Oct 06 2023: (Start)

%C 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:

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

%C          {1,2,3}  {2,3}      {2,4}

%C                   {1,2,3}    {1,2,4}

%C                   {1,2,4}    {1,2,5}

%C                   {1,3,4}    {1,3,5}

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

%C                   {1,2,3,4}  {2,3,4}

%C                              {2,4,5}

%C                              {1,2,3,4}

%C                              {1,2,3,5}

%C                              {1,2,4,5}

%C                              {1,3,4,5}

%C                              {2,3,4,5}

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

%C The complement is counted by A038754.

%C Allowing twins gives A167936, complement A108411.

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

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

%C (End)

%H Vincenzo Librandi, <a href="/A167762/b167762.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-6).

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

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

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

%F G.f.: x^2/((2*x-1)*(3*x^2-1)). a(n) = 2^n - A038754(n). - _R. J. Mathar_, Nov 12 2009

%F G.f.: x^2/(1-2*x-3*x^2+6*x^3). - _Philippe Deléham_, Nov 11 2009

%t LinearRecurrence[{2,3,-6},{0,0,1},40] (* _Harvey P. Dale_, Sep 17 2013 *)

%t CoefficientList[Series[x^2/((2 x - 1) (3 x^2 - 1)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Sep 17 2013 *)

%t Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#,{2}],n+1]&]],{n,0,10}] (* _Gus Wiseman_, Oct 06 2023 *)

%Y Cf. A024495, A167710.

%Y First differences are A167936, complement A108411.

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

%K nonn,easy

%O 0,4

%A _Paul Curtz_, Nov 11 2009

%E Edited and extended by _R. J. Mathar_, Nov 12 2009