A146501 -id:A146501 - 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

A154127 Period 6: repeat [1, 2, 5, 8, 7, 4].

Original entry on oeis.org

1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2, 5, 8, 7, 4, 1, 2

Offset: 0

Paul Curtz, Jan 05 2009

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A020806, A029898, A070366, A141425, A146501, A153130.

&cat[[1, 2, 5, 8, 7, 4]: n in [0..20]]; // Wesley Ivan Hurt, Jun 17 2016

A154127:=n->(27-cos(n*Pi)-20*cos(n*Pi/3)-4*sqrt(3)*sin(n*Pi/3))/6: seq(A154127(n), n=0..100); # Wesley Ivan Hurt, Jun 17 2016

Flatten[Table[{1, 2, 5, 8, 7, 4}, {20}]] (* Wesley Ivan Hurt, Jun 17 2016 *)

a(n)=[1,2,5,8,7,4][n%6+1] \\ Charles R Greathouse IV, Jun 02 2011

From R. J. Mathar, Feb 25 2009, Mar 09 2009: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
G.f.: (1+x+3*x^2+4*x^3)/((1-x)*(1+x)*(x^2-x+1)). (End)
a(n) = (27-cos(n*Pi)-20*cos(n*Pi/3)-4*sqrt(3)*sin(n*Pi/3))/6. - Wesley Ivan Hurt, Jun 17 2016

Corrected numerator in g.f R. J. Mathar, Mar 09 2009

A154870 Period 6: repeat [7, 5, 1, -7, -5, -1].

Original entry on oeis.org

7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1

Offset: 0

Paul Curtz, Jan 16 2009

The sequence b(n) = (-A153130(n)) mod 9 = A153130(n+3) = A146501(n-1) = 8, 7, 5, 1, 2, 4,... has period length 6. This here is a(n)=b(n)-A153130(n).

a(n) is (-1)^(n+1) * numerator of F(n) where F(n) = f(F(n-1)) starting from F(0) = -7/4 and step f(z) = z^2 -29/16. - Nicolas Bělohoubek, Nov 20 2024

Holly Krieger and Brady Haran, A Fascinating Thing about Fractions, Numberphile video, Dec 15 2019.
Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Cf. A146501, A153130.

&cat [[7, 5, 1, -7, -5, -1]^^20]; // Wesley Ivan Hurt, Jun 20 2016

A154870:=n->[7, 5, 1, -7, -5, -1][(n mod 6)+1]: seq(A154870(n), n=0..100); # Wesley Ivan Hurt, Jun 20 2016

PadRight[{}, 100, {7, 5, 1, -7, -5, -1}] (* Wesley Ivan Hurt, Jun 20 2016 *)

a(n) = -a(n-3) for n>2; G.f.: (7+5*x+x^2)/((1+x)*(1-x+x^2)). [R. J. Mathar, Jan 23 2009]

a(n) = cos(n*Pi) + 6*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3). - Wesley Ivan Hurt, Jun 20 2016

Edited and extended by R. J. Mathar, Jan 23 2009

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

A166577 Inverse binomial transform of A166517.

Original entry on oeis.org

1, 4, -5, 10, -20, 40, -80, 160, -320, 640, -1280, 2560, -5120, 10240, -20480, 40960, -81920, 163840, -327680, 655360, -1310720, 2621440, -5242880, 10485760, -20971520, 41943040, -83886080, 167772160, -335544320, 671088640, -1342177280, 2684354560, -5368709120

Offset: 0

Paul Curtz, Oct 17 2009

The definition assumes that the offset of A166517 is changed to 0.
A166517 mod 9 yields a periodic sequence with period 1, 5, 4, 8, 7, 2.
This set of numbers in the period appears also in A153130, A146501, and A166304.

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

Cf. A010716, A010692, A010859.

Join[{1,4},NestList[-2#&,-5,40]] (* Harvey P. Dale, Aug 02 2012 *)
Join[{1, 4}, LinearRecurrence[{-2}, {-5}, 48]] (* G. C. Greubel, May 17 2016 *)

a(n) = -2*a(n-1), n>2.
a(n) = (-1)^(n+1)*A020714(n-2), n>1.
From Colin Barker, Jan 07 2013: (Start)
a(n) = -5*(-1)^n*2^(n-2) for n>1.
G.f.: (3*x^2+6*x+1)/(2*x+1). (End)
E.g.f.: (9/4) + (3/2)*x - (5/4)*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Edited, comments not concerning this sequence removed, and extended by R. J. Mathar, Oct 21 2009

A154529 A090040 mod 9.

Original entry on oeis.org

1, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8

Offset: 0

Paul Curtz, Jan 11 2009

For n>2, equal to 2^(n-2) mod 9 [From Michael B. Porter, Feb 02 2010]

Apart from leading terms the same as A146501, A153130 and A029898. [From R. J. Mathar, Apr 13 2010]

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Join[{1},LinearRecurrence[{1,0,-1,1},{5,1,2,4},101]] (* Ray Chandler, Jul 15 2015 *)

a(n)=a(n-1)-a(n-3)+a(n-4), n>4. G.f.: (6*x^4+2*x^3+4*x+1-4*x^2)/((1-x)*(1+x)*(x^2-x+1)). [From R. J. Mathar, Feb 25 2009]

Edited by N. J. A. Sloane, Jan 12 2009

Extended by Ray Chandler, Jul 15 2015

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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A154127 Period 6: repeat [1, 2, 5, 8, 7, 4].

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A154870 Period 6: repeat [7, 5, 1, -7, -5, -1].

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

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A166577 Inverse binomial transform of A166517.

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A154529 A090040 mod 9.

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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.