A131531 -id:A131531 - cp's OEIS frontend

A132951 Period 6: repeat [1, 3, 1, -1, -3, -1].

1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1, 1, 3, 1, -1, -3, -1

Offset: 0

Paul Curtz, Nov 22 2007

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

Cf. A109007, A130151, A131531.

&cat [[1, 3, 1, -1, -3, -1]: n in [0..20]]; // Wesley Ivan Hurt, Nov 18 2022

PadRight[{},120,{1,3,1,-1,-3,-1}] (* Harvey P. Dale, Feb 26 2024 *)

a(n)=[1,3,1,-1,-3,-1][n%6+1] \\ Charles R Greathouse IV, Jun 02 2011

a(n) = 3*a(n-1)-a(n-3)+3*a(n-4).
O.g.f.: (1+3*x+x^2)/((x+1)*(x^2-x+1)) = -(1/3)/(x+1)+(1/3)*(4*x+4)/(x^2-x+1). - R. J. Mathar, Nov 28 2007
a(n) = -(1/3)*(-1)^n+(4/3)*cos(Pi*n/3)+(4*3^0.5/3)*sin(Pi*n/3). - Richard Choulet, Jan 02 2008
a(n) = a(n-6) = A131531(n+3)+A131531(n+1)+3*A131531(n+2). - R. J. Mathar, Apr 04 2008
a(n) = A109007(n+2) * A130151(n). - Wesley Ivan Hurt, Jun 22 2013

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar.

A167617 G.f.: x^2(3+3x+x^2) / ( (2x+1) (1+x) * (1+x+x^2) * (x^2-x+1) ) .

Original entry on oeis.org

0, 0, 3, -6, 10, -21, 42, -84, 171, -342, 682, -1365, 2730, -5460, 10923, -21846, 43690, -87381, 174762, -349524, 699051, -1398102, 2796202, -5592405, 11184810, -22369620, 44739243, -89478486, 178956970, -357913941, 715827882, -1431655764, 2863311531

Offset: 0

Paul Curtz, Nov 07 2009

The derived sequence a(n+1) + 2*a(n) reads 0,3,0,-2,-1,0 (and repeat with period 6).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-3, -3, -3, -3, -3, -2).

Cf. A167613.

CoefficientList[Series[x^2(3+3x+x^2)/((2x+1)(1+x)(1+x+x^2)(x^2-x+1)), {x,0,40}],x] (* or *) LinearRecurrence[{-3,-3,-3,-3,-3,-2},{0,0,3,-6,10,-21},40] (* Harvey P. Dale, Sep 08 2011 *)

a(3*k+2) + a(3*k+3) + a(3*k+4) = (-1)^(k+1)*A024088(k+1).
a(n) = (-1)^n*A024495(n+1) + A131531(n+1).
a(n) = -3*a(n-1) -3*a(n-2) -3*a(n-3) -3*a(n-4) -3*a(n-5) -2*a(n-6).

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

A135450 a(n) = 3a(n-1) + 4a(n-2) - a(n-3) + 3a(n-4) + 4a(n-5).

Original entry on oeis.org

0, 0, 0, 1, 4, 16, 63, 252, 1008, 4033, 16132, 64528, 258111, 1032444, 4129776, 16519105, 66076420, 264305680, 1057222719, 4228890876, 16915563504, 67662254017, 270649016068, 1082596064272, 4330384257087, 17321537028348

Offset: 0

Paul Curtz, Dec 14 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,-1,4).

Cf. A135343, A135345.

a = {0, 0, 0, 1, 4}; Do[AppendTo[a, 3*a[[ -1]] + 4*a[[ -2]] - a[[ -3]] + 3*a[[ -4]] + 4*a[[ -5]]], {25}]; a (* Stefan Steinerberger, Dec 31 2007 *)
LinearRecurrence[{3, 4, -1, 3, 4}, {0, 0, 0, 1, 4}, 25] (* G. C. Greubel, Oct 14 2016 *)
LinearRecurrence[{4,0,-1,4},{0,0,0,1},40] (* Harvey P. Dale, Jan 31 2021 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 4,-1,0,4]^n*[0;0;0;1])[1,1] \\ Charles R Greathouse IV, Oct 14 2016

a(n+1) - 4*a(n) = hexaperiodic 0, 0, 1, 0, 0, -1, A131531.
a(n) + a(n+3) = 1, 4, 16, 64 = 2^2n = A000302.
a(n) = (1/65)*4^n + (1/15)*(-1)^(n+1) + (2/39)*cos((Pi*n)/3) - (4*sqrt(3)/39) * sin((Pi*n)/3). Or, a(n) = (1/65)*(4^n + [ -1; -4; -16; 1; 4; 16]). - Richard Choulet, Dec 31 2007
O.g.f.: -x^3/[(4*x-1)*(1+x)*(x^2-x+1)]. - R. J. Mathar, Jan 07 2008

More terms from Stefan Steinerberger, Dec 31 2007

A191370 a(n) = 2(1+(-1)^n)/3 + 2A010892(n-1).

Original entry on oeis.org

1, 2, 4, 2, 4, 8, 22, 44, 88, 170, 340, 680, 1366, 2732, 5464, 10922, 21844, 43688, 87382, 174764, 349528, 699050, 1398100, 2796200, 5592406, 11184812, 22369624, 44739242

Offset: 0

Paul Curtz, Jun 01 2011

a(n) and successive differences define an infinite array:
    1,   2,   4,   2,   4,   8, ...
    1,   2,  -2,   2,   4,  14, ...
    1,  -4,   4,   2,  10,   8, ...
   -5,   8,  -2,   8,  -2,  14, ...
   13, -10,  10, -10,  16,   2, ...
  -23,  20, -20,  26, -14,  32, ...
  ...
Its main diagonal consists of the powers 2^n. The first upper diagonal is a signed sequence of 2's. The second upper diagonal contains essentially A135440.

Muniru A Asiru, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Cf. A010892, A131531, A007613, A001045.

Cf. A007283, A024495, A113405.

A010892 := proc(n) op( 1+(n mod 6),[1,1,0,-1,-1,0]) ; end proc:
A191370 := proc(n) 2^n/3+2*(-1)^n/3+2*A010892(n-1) ; end proc:
seq(A191370(n),n=0..30) ; # R. J. Mathar, Jun 06 2011

LinearRecurrence[{2,0,-1,2},{1,2,4,2},30] (* Harvey P. Dale, Sep 06 2022 *)

a(n+3) = 3*2^n - a(n), n >= 0.
a(n+1) - 2*a(n) = -6*A131531(n+1).
a(3*n) = A007613(n), a(1+3*n) = 2*A007613(n), a(2+3*n) = 4*A007613(n).
a(n+6) = a(n) + 21*2^n.
a(n) = ((2^n + 2*(-1)^n)*2^n - 2*i*sqrt(3)*((1+i*sqrt(3))^n - (1-i*sqrt(3))^n))/(3*2^n), where i=sqrt(-1); a(n+1) = 2*(A001045(n) + A010892(n)). - Bruno Berselli, Jun 06 2011
G.f.: ( -1+5*x^3 ) / ( (2*x-1)*(1+x)*(x^2-x+1) ). - R. J. Mathar, Jun 06 2011
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4). - Paul Curtz, Jun 07 2011
a(n) = A113405(n+3) - 5*A113405(n). - R. J. Mathar, Jun 24 2011

A227430 Expansion of x^2(1-x)^3/((1-2x)(1-x+x^2)(1-3*x+3x^2)).

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 29, 45, 90, 220, 561, 1365, 3095, 6555, 13110, 25126, 46971, 87381, 164921, 320001, 640002, 1309528, 2707629, 5592405, 11450531, 23166783, 46333566, 91869970, 181348455, 357913941, 708653429, 1410132405, 2820264810, 5662052980

Offset: 0

Paul Curtz, Jul 11 2013

Consider the binomial transform of 0, 0, 0, 0, 0, 1 (period 6) with its differences:
0, 0, 0, 0, 0,  1,  6, 21, 56, 126,...    d(n): after 0, it is A192080.
0, 0, 0, 0, 1,  5, 15, 35, 70, 126,...    e(n)
0, 0, 0, 1, 4, 10, 20, 35, 56,  85,...    f(n)
0, 0, 1, 3, 6, 10, 15, 21, 29,  45,...    a(n)
0, 1, 2, 3, 4,  5,  6,  8, 16,  45,...    b(n)
1, 1, 1, 1, 1,  1,  2,  8, 29,  85,...    c(n)
0, 0, 0, 0, 0,  1,  6, 21, 56, 126,...    d(n).
a(n) + d(n) = A024495(n),
b(n) + e(n) = A131708(n),
c(n) + f(n) = A024493(n).
a(n) - d(n) =  0,  0,  1,  3,  6,  9,   9,   0,...    A057083(n-2)
b(n) - e(n) =  0,  1,  2,  3,  3,  0,  -9, -27,...    A057682(n)
c(n) - f(n) =  1,  1,  1,  0, -3, -9, -18, -27,...    A057681(n)
d(n) - a(n) =  0,  0, -1, -3, -6, -9,  -9,   0,...   -A057083(n-2)
e(n) - b(n) =  0, -1, -2, -3, -3,  0,   9,  27,...   -A057682(n)
f(n) - c(n) = -1, -1, -1,  0,  3,  9,  18,  27,...   -A057681(n).
The first column is A131531(n).
The first two trisections are multiples of 3. Is the third (1, 10, 29,...) mod 9  A029898(n)?

			a(6)=6*10-15*6+20*3-15*1+6*0=15, a(7)=90-150+120-45+6=21.

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Join[{0},LinearRecurrence[{6,-15,20,-15,6},{0,1,3,6,10},40]] (* Harvey P. Dale, Dec 17 2014 *)

{a(n) = sum(k=0, n\6, binomial(n, 6*k+2))} \\ Seiichi Manyama, Mar 23 2019

a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) for n>5, a(0)=a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10.
a(n) = A024495(n) - A192080(n-5) for n>4.
G.f.: -(x^5 - 3*x^4 + 3*x^3 - x^2)/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)). - Ralf Stephan, Jul 13 2013
a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k+2). - Seiichi Manyama, Mar 23 2019

Definition uses the g.f. of Ralf Stephan.

More terms from Harvey P. Dale, Dec 17 2014

A290968 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) + a(n-5), with a(0)=a(1)=a(2)=1, a(3)=-1 and a(4)=1.

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 5, 5, 9, 11, 21, 33, 57, 89, 145, 231, 377, 609, 989, 1597, 2585, 4179, 6765, 10945, 17713, 28657, 46369, 75023, 121393, 196417, 317813, 514229, 832041, 1346267, 2178309, 3524577, 5702889, 9227465, 14930353, 24157815

Offset: 0

Jean-François Alcover and Paul Curtz, Aug 16 2017

The array of successive differences begins:
1,   1,   1,  -1,   1,   1,   5,   5,   9,  11,  21,  33,  57, ...
0,   0,  -2,   2,   0,   4,   0,   4,   2,  10,  12,  24,  32, ...
0,  -2,   4,  -2,   4,  -4,   4,  -2,   8,   2,  12,   8,  24, ...
-2,  6,  -6,   6,  -8,   8,  -6,  10,  -6,  10,  -4,  16,   6, ...
8, -12,  12, -14,  16, -14,  16, -16,  16, -14,  20, -10,  24, ...
...
First row is a(n) = 2*A141325(n) - A141325(n+1).
Main diagonal is A099430(n).
The first upper subdiagonal, 1, -2, -2, -8, -14, ..., has -3*A078008(n) as first differences.
The second upper subdiagonal is A000079(n) = 2^n.
a(n) is related to Fibonacci numbers a(n) = A000045(n-2) + period 6: repeat [2, 0, 1, -2, 0, -1].

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

Cf. A000045, A078008, A099430, A131531, A141325.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2-2*x^3+x^4)/((1+x^3)*(1-x-x^2)) )); // G. C. Greubel, Jun 11 2019

LinearRecurrence[{1,1,-1,1,1}, {1,1,1,-1,1}, 40]

my(x='x+O('x^40)); Vec((1-x^2-2*x^3+x^4)/((1+x^3)*(1-x-x^2))) \\ G. C. Greubel, Jun 11 2019

((1-x^2-2*x^3+x^4)/((1+x^3)*(1-x-x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 11 2019

G.f.: (1-x^2-2*x^3+x^4)/((1+x)*(1-x+x^2)*(1-x-x^2)).
a(n) ~ phi^(n-2)/sqrt(5), where phi is the golden ratio.
a(n) = (1/2 + sqrt(5)/2)^n*(3*sqrt(5)/10-1/2) - (-1/2 + sqrt(5)/2)^n*(3*sqrt(5)/10 + 1/2)*(-1)^n + 2*sqrt(3)*sin(Pi*(n/3 + 1/3))/3 + (-1)^n. - Eric Simon Jacob, Jul 11 2024

A133511 a(n) = 3 A113405(n)- A113405(n+1).

Original entry on oeis.org

0, 0, -1, 1, 2, 5, 7, 14, 27, 57, 114, 229, 455, 910, 1819, 3641, 7282, 14565, 29127, 58254, 116507, 233017, 466034, 932069, 1864135, 3728270, 7456539, 14913081, 29826162, 59652325, 119304647, 238609294, 477218587, 954437177, 1908874354, 3817748709, 7635497415

Offset: 0

Paul Curtz, Nov 30 2007

sign

2a(n)-a(n+1)=A133513(n).
A113405(n)-a(n)=A131531(n).
O.g.f.: x^2(3x-1)/((1-2x)(1+x)(1-x+x^2)). - R. J. Mathar, Jul 22 2008
a(n+2)=4*(2^n-(-1)^n)/9+A117373(n+2)/3. [From R. J. Mathar, Jul 20 2009]

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

cp's OEIS Frontend

A132951 Period 6: repeat [1, 3, 1, -1, -3, -1].

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A167617 G.f.: x^2*(3+3*x+x^2) / ( (2*x+1) * (1+x) * (1+x+x^2) * (x^2-x+1) ) .

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A135450 a(n) = 3*a(n-1) + 4*a(n-2) - a(n-3) + 3*a(n-4) + 4*a(n-5).

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A191370 a(n) = 2*(1+(-1)^n)/3 + 2*A010892(n-1).

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A227430 Expansion of x^2*(1-x)^3/((1-2*x)*(1-x+x^2)*(1-3*x+3x^2)).

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A290968 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) + a(n-5), with a(0)=a(1)=a(2)=1, a(3)=-1 and a(4)=1.

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A133511 a(n) = 3 A113405(n)- A113405(n+1).

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A167617 G.f.: x^2(3+3x+x^2) / ( (2x+1) (1+x) * (1+x+x^2) * (x^2-x+1) ) .

A135450 a(n) = 3a(n-1) + 4a(n-2) - a(n-3) + 3a(n-4) + 4a(n-5).

A191370 a(n) = 2(1+(-1)^n)/3 + 2A010892(n-1).

A227430 Expansion of x^2(1-x)^3/((1-2x)(1-x+x^2)(1-3*x+3x^2)).