A131556 -id:A131556 - cp's OEIS frontend

A131531 Period 6: repeat [0, 0, 1, 0, 0, -1].

0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0

Offset: 1

Paul Curtz, Aug 26 2007

Also: partial sums of A092220 shifted by two indices. - R. J. Mathar, Feb 08 2008
From Paul Curtz, Jun 05 2011: (Start)
The square array of this sequence in the top row and further rows defined as first differences of preceding rows starts (see A167613):
.   0,   0,   1,   0,   0,  -1, ...
.   0,   1,  -1,   0,  -1,   1, ...  = A092220,
.   1,  -2,   1,  -1,   2,  -1, ...  = A131556,
.  -3,   3,  -2,   3,  -3    2, ...
.   6,  -5,   5,  -6,   5,  -5, ...
. -11,  10, -11,  11, -10,  11, ...
.  21, -21,  22, -21,  21, -22, ...
. -42,  43, -43,  42, -43,  43, ...
The main diagonal in this array is A001045; the first superdiagonal is the negated elements of A001045, the second superdiagonal is A078008.
The left column of the array is basically the inverse binomial transform, (-1)^n * A024495(n), assuming offset 0.
The second column of the array is A131708 with alternating signs, and the third column is A024493 with alternating signs (both assuming offset 0). (End)

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

Cf. A001045, A024493, A024495, A078008, A092220, A131556, A131708, A167613.

&cat[[0, 0, 1, 0, 0, -1]^^20]; // Wesley Ivan Hurt, Jun 20 2016

A131531:=n->[0, 0, 1, 0, 0, -1][(n mod 6)+1]: seq(A131531(n), n=0..100); # Wesley Ivan Hurt, Jun 20 2016

PadRight[{}, 120, {0,0,1,0,0,-1}] (* Harvey P. Dale, Nov 11 2012 *)

a(n)=[0,0,1,0,0,-1][n%6+1] \\ Charles R Greathouse IV, Jun 01 2011

G.f.: x^3/(x+1)/(x^2-x+1). - R. J. Mathar, Nov 14 2007
a(n) = (-A057079(n+1) - (-1)^n)/3. - R. J. Mathar, Jun 13 2011
a(n) = -cos(Pi*(n-1)/3)/3 + sin(Pi*(n-1)/3)/sqrt(3) - (-1)^n/3. - R. J. Mathar, Oct 08 2011
a(n) = ( (-1)^n - (-1)^floor((n+2)/3) )/2. - Bruno Berselli, Jul 09 2013
a(n) + a(n-3) = 0 for n > 3. - Wesley Ivan Hurt, Jun 20 2016

Edited by N. J. A. Sloane, Sep 15 2007

A131090 First differences of A131666.

Original entry on oeis.org

0, 1, 0, 1, 1, 4, 7, 15, 28, 57, 113, 228, 455, 911, 1820, 3641, 7281, 14564, 29127, 58255, 116508, 233017, 466033, 932068, 1864135, 3728271, 7456540, 14913081, 29826161, 59652324, 119304647, 238609295, 477218588, 954437177, 1908874353

Offset: 0

Paul Curtz, Sep 24 2007

The first differences b(n)=a(n+1)-a(n) obey the recurrence b(n+1)-2b(n) = (-3,3,-2,3,-3,2), continued with period 6.
The 2nd differences c(n)=b(n+1)-b(n) obey the recurrence c(n+1)-2c(n) = (6,-5,5,-6,5,-5), periodically continued with period 6.
The hexaperiodic coefficients in these recurrences for A113405, A131666 and their higher order differences define a table,
0, 0, 1, 0, 0, -1 <- A113405
0, 1, -1, 0, -1, 1 <- A131666
1, -2, 1, -1, 2, -1 <- a(n)
-3, 3, -2, 3, -3, 2 <- b(n)
6, -5, 5, -6, 5, -5 <- c(n)
-11,10,-11, 11,-10, 11
21,-21,22,-21, 21,-22
...
in which the first three columns are A024495, A131708 and A024493, multiplied by a checkerboard pattern of signs.

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

LinearRecurrence[{2,0,-1,2},{0,1,0,1},40] (* Harvey P. Dale, Jan 15 2016 *)

a(n) = A131666(n+1)-A131666(n).
a(n+1)-2a(n) = A131556(n), a sequence with period length 6.
G.f.: -(x-1)^2*x / ((x+1)*(2*x-1)*(x^2-x+1)). - Colin Barker, Mar 04 2013

Edited by R. J. Mathar, Jun 28 2008

A167613 Array T(n,k) read by antidiagonals: the k-th term of the n-th difference of A131531.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, -1, -2, -3, 0, 0, 1, 3, 6, -1, -1, -1, -2, -5, -11, 0, 1, 2, 3, 5, 10, 21, 0, 0, -1, -3, -6, -11, -21, -42, 1, 1, 1, 2, 5, 11, 22, 43, 85, 0, -1, -2, -3, -5, -10, -21, -43, -86, -171, 0, 0, 1, 3, 6, 11, 21, 42, 85, 171, 342, -1, -1, -1, -2, -5, -11, -22, -43, -85, -170, -341, -683, 0, 1, 2, 3, 5, 10, 21, 43, 86, 171, 341, 682, 1365

Offset: 0

Paul Curtz, Nov 07 2009

The array contains A131708(0) in diagonal 0, then -A024495(0..1) in diagonal 1, then A024493(0..2) in diagonal 2, then -A131708(0..3), then A024495(0..4), then -A024493(0..5).

			The table starts in row n=0 with columns k >= 0 as:
0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0 A131531
0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1 A092220
1, -2, 1, -1, 2, -1, 1, -2, 1, -1, 2, -1, 1, -2, 1, -1, 2, -1, 1, -2 A131556
-3, 3, -2, 3, -3, 2, -3, 3, -2, 3, -3, 2, -3, 3, -2, 3, -3, 2, -3 A164359
6, -5, 5, -6, 5, -5, 6, -5, 5, -6, 5, -5, 6, -5, 5, -6, 5, -5, 6, -5
-11, 10, -11, 11, -10, 11, -11, 10, -11, 11, -10, 11, -11, 10, -11
21, -21, 22, -21, 21, -22, 21, -21, 22, -21, 21, -22, 21, -21, 22

Cf. A167617 (antidiagonal sums).

A131531 := proc(n) op((n mod 6)+1,[0,0,1,0,0,-1]) ; end proc:
A167613 := proc(n,k) option remember; if n= 0 then A131531(k); else procname(n-1,k+1)-procname(n-1,k) ; end if;end proc: # R. J. Mathar, Dec 17 2010

nmax = 13;
A131531 = Table[{0, 0, 1, 0, 0, -1}, {nmax}] // Flatten;
T[n_] := T[n] = Differences[A131531, n];
T[n_, k_] := T[n][[k]];
Table[T[n-k, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 20 2023 *)

T(0,k) = A131531(k). T(n,k) = T(n-1,k+1) - T(n-1,k), n > 0.
T(n,n) = A001045(n). T(n,n+1) = -A001045(n). T(n,n+2) = A078008(n).
T(n,0) = -T(n,3) = (-1)^(n+1)*A024495(n).
T(n,1) = (-1)^(n+1)*A131708(n).
T(n,2) = (-1)^n*A024493(n).
T(n,k+6) = T(n,k).
a(n) = A131708(0), -A024495(0,1), A024493(0,1,2), -A131708(0,1,2,3), A024495(0,1,2,3,4), -A024493(0,1,2,3,4,5).

A177702 Period 3: repeat [1, 1, 2].

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2

Offset: 0

Klaus Brockhaus, May 11 2010

Continued fraction expansion of (2+sqrt(10))/3.
Decimal expansion of 112/999.
a(n) = A131534(n+2) = |A132419(n)| = |A132367(n)| = |A131556(n+2)|= |A122876(n)|.

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

Cf. A022003, A131534, A177703.

Magma
```
&cat[ [1, 1, 2]: k in [1..35] ];
```

seq(op([1, 1, 2]), n=1..50); # Wesley Ivan Hurt, Jul 01 2016

PadRight[{},120,{1,1,2}] (* or *) LinearRecurrence[{0,0,1},{1,1,2},120] (* Harvey P. Dale, Dec 19 2014 *)

a(n)=max(n%3,1) \\ Charles R Greathouse IV, Jul 17 2016

a(n) = a(n-3) for n > 2, with a(0) = 1, a(1) = 1, a(2) = 2.
G.f.: (1+x+2*x^2)/(1-x^3).
a(n) = 4/3 - cos(2*Pi*n/3)/3 - sin(2*Pi*n/3)/sqrt(3). - R. J. Mathar, Oct 08 2011
a(n) = 1 + A022003(n). - Wesley Ivan Hurt, Jul 01 2016

A131714 Period 6: repeat [1, -2, 2, -1, 2, -2].

Original entry on oeis.org

1, -2, 2, -1, 2, -2, 1, -2, 2, -1, 2, -2, 1, -2, 2, -1, 2, -2, 1, -2, 2, -1, 2, -2, 1, -2, 2, -1, 2, -2, 1, -2, 2, -1, 2, -2, 1, -2, 2, -1, 2, -2, 1, -2, 2, -1, 2, -2, 1, -2, 2, -1, 2, -2, 1, -2, 2, -1, 2, -2, 1, -2, 2, -1, 2, -2, 1, -2, 2, -1, 2, -2, 1, -2

Offset: 0

Paul Curtz, Sep 14 2007

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

Cf. A130196, A131556.

&cat[[1, -2, 2, -1, 2, -2]^^20]; // Wesley Ivan Hurt, Jun 19 2016

A131714:=n->[1, -2, 2, -1, 2, -2][(n mod 6)+1]: seq(A131714(n), n=0..100); # Wesley Ivan Hurt, Jun 19 2016

PadRight[{}, 200, {1, -2, 2, -1, 2, -2}] (* Wesley Ivan Hurt, Jun 19 2016 *)

G.f.: (1-2*x+2*x^2)/(x+1)/(x^2-x+1). - R. J. Mathar, Nov 14 2007
From Wesley Ivan Hurt, Jun 19 2016: (Start)
a(n) + a(n-3) = 0 for n>2.
a(n) = (5*cos(n*Pi)-2*cos(n*Pi/3))/3. (End)

A135261 a(n) = 3*A131090(n) - A131090(n+1).

Original entry on oeis.org

-1, 3, -1, 2, -1, 5, 6, 17, 27, 58, 111, 229, 454, 913, 1819, 3642, 7279, 14565, 29126, 58257, 116507, 233018, 466031, 932069, 1864134, 3728273, 7456539, 14913082, 29826159, 59652325, 119304646, 238609297, 477218587, 954437178, 1908874351, 3817748709, 7635497414

Offset: 0

Paul Curtz, Dec 01 2007

sign

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

a:=[-1,2,-1,2];; for n in [5..40] do a[n]:=2*a[n-1] -a[n-3] +2*a[n-4]; od; a; # G. C. Greubel, Nov 21 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)^2*(1-3*x)/((2*x-1)*(1+x)*(x^2-x+1)) )); // G. C. Greubel, Nov 21 2019

seq(coeff(series((1-x)^2*(1-3*x)/((2*x-1)*(1+x)*(x^2-x+1)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Nov 21 2019

LinearRecurrence[{2,0,-1,2}, {-1,3,-1,2}, 40] (* G. C. Greubel, Oct 07 2016 *)

my(x='x+O('x^40)); Vec((1-x)^2*(1-3*x)/((2*x-1)*(1+x)*(x^2-x+1))) \\ G. C. Greubel, Nov 21 2019

def A135261_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)^2*(1-3*x)/((2*x-1)*(1+x)*(x^2-x+1))).list()
A135261_list(40) # G. C. Greubel, Nov 21 2019

A131090(n) - a(n) = A131556(n).
O.g.f.: (1-x)^2*(1-3*x)/((2*x-1)*(1+x)*(x^2-x+1)). - R. J. Mathar, Jul 22 2008
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4). - G. C. Greubel, Oct 07 2016

Edited and extended by R. J. Mathar, Jul 22 2008

cp's OEIS Frontend

A131531 Period 6: repeat [0, 0, 1, 0, 0, -1].

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A131090 First differences of A131666.

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A167613 Array T(n,k) read by antidiagonals: the k-th term of the n-th difference of A131531.

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A177702 Period 3: repeat [1, 1, 2].

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A131714 Period 6: repeat [1, -2, 2, -1, 2, -2].

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A135261 a(n) = 3*A131090(n) - A131090(n+1).

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