A167936 -id:A167936 - cp's OEIS frontend

A167762 a(n) = 2a(n-1)+3a(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

Paul Curtz, Nov 11 2009

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)

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

Cf. A024495, A167710.
First differences are A167936, complement A108411.
Cf. A004526, A004737, A008967, A038754, A046663, A068911, A088809, A093971, A365376, A365544, A366130.

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 *)

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

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

A117855 Number of nonzero palindromes of length n (in base 3).

Original entry on oeis.org

2, 2, 6, 6, 18, 18, 54, 54, 162, 162, 486, 486, 1458, 1458, 4374, 4374, 13122, 13122, 39366, 39366, 118098, 118098, 354294, 354294, 1062882, 1062882, 3188646, 3188646, 9565938, 9565938, 28697814, 28697814, 86093442, 86093442, 258280326, 258280326, 774840978

Offset: 1

Martin Renner, May 02 2006

See A225367 for the sequence that counts all base 3 palindromes, including 0 (and thus also the number of n-digit terms in A006072). -- A nonzero palindrome of length L=2k-1 or of length L=2k is determined by the first k digits, which then determine the last k digits by symmetry. Since the first digit cannot be 0, there are 2*3^(k-1) possibilities. - M. F. Hasler, May 05 2013
From Gus Wiseman, Oct 18 2023: (Start)
Also the number of subsets of {1..n} with n not the sum of two subset elements (possibly the same). For example, the a(0) = 1 through a(4) = 6 subsets are:
  {}  {}   {}   {}     {}
      {1}  {2}  {1}    {1}
                {2}    {3}
                {3}    {4}
                {1,3}  {1,4}
                {2,3}  {3,4}
For subsets with no subset summing to n we have A365377.
Requiring pairs to be distinct gives A068911, complement A365544.
The complement is counted by A366131.
(End) [Edited by Peter Munn, Nov 22 2023]

			The a(3)=6 palindromes of length 3 are: 101, 111, 121, 202, 212, and 222. - _M. F. Hasler_, May 05 2013

Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A050683 and A070252.
Bisections are both A025192.
A093971/A088809/A364534 count certain types of sum-full subsets.
A108411 lists powers of 3 repeated, complement A167936.
Cf. A004526, A004737, A008967, A038754, A046663, A167762, A365376, A366130.

With[{c=NestList[3#&,2,20]},Riffle[c,c]] (* Harvey P. Dale, Mar 25 2018 *)
Table[Length[Select[Subsets[Range[n]],!MemberQ[Total/@Tuples[#,2],n]&]],{n,0,10}] (* Gus Wiseman, Oct 18 2023 *)

A117855(n)=2*3^((n-1)\2) \\ - M. F. Hasler, May 05 2013

def A117855(n): return 3**(n-1>>1)<<1 # Chai Wah Wu, Oct 28 2024

a(n) = 2*3^floor((n-1)/2).
a(n) = 2*A108411(n-1).
From Colin Barker, Feb 15 2013: (Start)
a(n) = 3*a(n-2).
G.f.: -2*x*(x+1)/(3*x^2-1). (End)

More terms from Colin Barker, Feb 15 2013

A366131 Number of subsets of {1..n} with two elements (possibly the same) summing to n.

Original entry on oeis.org

0, 0, 2, 2, 10, 14, 46, 74, 202, 350, 862, 1562, 3610, 6734, 14926, 28394, 61162, 117950, 249022, 484922, 1009210, 1979054, 4076206, 8034314, 16422922, 32491550, 66045982, 131029082, 265246810, 527304974, 1064175886, 2118785834, 4266269482, 8503841150, 17093775742, 34101458042, 68461196410, 136664112494

Offset: 0

Gus Wiseman, Oct 07 2023

			The a(0) = 0 through a(5) = 14 subsets:
  .  .  {1}    {1,2}    {2}        {1,4}
        {1,2}  {1,2,3}  {1,2}      {2,3}
                        {1,3}      {1,2,3}
                        {2,3}      {1,2,4}
                        {2,4}      {1,3,4}
                        {1,2,3}    {1,4,5}
                        {1,2,4}    {2,3,4}
                        {1,3,4}    {2,3,5}
                        {2,3,4}    {1,2,3,4}
                        {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}

Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

The complement is counted by A117855.
For pairs summing to n + 1 we have A167936.
A068911 counts subsets of {1..n} w/o two distinct elements summing to n.
A093971/A088809/A364534 count certain types of sum-full subsets.
Cf. A008967, A167762, A238628, A365376, A365377, A365381, A365541, A365544.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Tuples[#,2],n]&]],{n,0,10}]

def A366131(n): return (1<>1)<<1) if n else 0 # Chai Wah Wu, Nov 14 2023

From Chai Wah Wu, Nov 14 2023: (Start)
a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3) for n > 3.
G.f.: 2*x^2*(1 - x)/((2*x - 1)*(3*x^2 - 1)). (End)

A191582 Riordan matrix (1/(1-3*x^2),x/(1-x)).

Original entry on oeis.org

1, 0, 1, 3, 1, 1, 0, 4, 2, 1, 9, 4, 6, 3, 1, 0, 13, 10, 9, 4, 1, 27, 13, 23, 19, 13, 5, 1, 0, 40, 36, 42, 32, 18, 6, 1, 81, 40, 76, 78, 74, 50, 24, 7, 1, 0, 121, 116, 154, 152, 124, 74, 31, 8, 1, 243, 121, 237, 270, 306, 276, 198, 105, 39, 9, 1, 0, 364, 358, 507, 576, 582, 474, 303, 144, 48, 10, 1, 729, 364, 722, 865, 1083, 1158, 1056, 777, 447, 192, 58, 11, 1

Offset: 0

Emanuele Munarini, Jun 07 2011

Row sums = A167936(n+1).
Diagonal sums = A191584.
Central coefficients = A191585.
Alternated row sums: Sum_{k=0..n} (-1)^(n-k)*T(n,k) = 3^floor(n/2) (A167936).
Binomial row sums: Sum_{k=0..n} binomial(n,k)*T(n,k) = central coefficients.

			Triangle begins:
1
0, 1
3, 1, 1
0, 4, 2, 1
9, 4, 6, 3, 1
0, 13, 10, 9, 4, 1
27, 13, 23, 19, 13, 5, 1
0, 40, 36, 42, 32, 18, 6, 1
81, 40, 76, 78, 74, 50, 24, 7, 1

Cf. A167936, A191584, A191585.

Flatten[Table[Sum[Binomial[n-2i-1,n-k-2i]3^i,{i,0,((n-k))/2}],{n,0,20},{k,0,n}]]

create_list(sum(binomial(n-2*i-1,n-k-2*i)*3^i,i,0,(n-k)/2),n,0,20,k,0,n);

T(n,k) = Sum_{i=0..(n-k)/2} binomial(n-2*i-1,n-k-2*i)*3^i.

Recurrence: T(n+1,k+1) = T(n,k) + T(n,k+1).

A167784 a(n) = 2^n - (1 - (-1)^n)*3^((n-1)/2).

Original entry on oeis.org

1, 0, 4, 2, 16, 14, 64, 74, 256, 350, 1024, 1562, 4096, 6734, 16384, 28394, 65536, 117950, 262144, 484922, 1048576, 1979054, 4194304, 8034314, 16777216, 32491550, 67108864, 131029082, 268435456, 527304974, 1073741824, 2118785834, 4294967296, 8503841150

Offset: 0

Paul Curtz, Nov 12 2009

Binomial transform of A077917, the signed variant of A127864.

Robert Israel, Table of n, a(n) for n = 0..3318
Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

Cf. A154383.

seq(2^n - (1 - (-1)^n)*3^((n-1)/2), n=0..100); # Robert Israel, Apr 11 2019

LinearRecurrence[{2, 3, -6}, {1, 0, 4}, 40] (* Harvey P. Dale, Nov 29 2011 *)

a(n) = A167936(n+1) - A167936(n).
a(2n) = A000302(n). a(2n+1) = 2*A005061(n).
a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3).
G.f.: (x-1)^2/((2*x-1)*(3*x^2-1)).
a(n+4) mod 9 = A153130(n+4) = A146501(n+2), n>=0.
E.g.f.: exp(2*x) - (2/sqrt(3))*sinh(sqrt(3)*x). - G. C. Greubel, Jun 27 2016

Edited and extended by R. J. Mathar, Feb 27 2010

Incorrect b-file corrected by Robert Israel, Apr 11 2019

cp's OEIS Frontend

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

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A117855 Number of nonzero palindromes of length n (in base 3).

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A366131 Number of subsets of {1..n} with two elements (possibly the same) summing to n.

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A191582 Riordan matrix (1/(1-3*x^2),x/(1-x)).

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A167784 a(n) = 2^n - (1 - (-1)^n)*3^((n-1)/2).

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