A122581 -id:A122581 - cp's OEIS frontend

A122584 Expansion of x(1+x)(1-2x)/(1 - 2x - x^2 + 2*x^3 - x^4).

1, 1, 1, 1, 2, 4, 9, 19, 41, 87, 186, 396, 845, 1801, 3841, 8189, 17462, 37232, 79389, 169275, 360937, 769603, 1640982, 3498968, 7460649, 15907905, 33919505, 72324585, 154213514, 328820508, 701124865, 1494967795, 3187632953

Offset: 1

Roger L. Bagula, Sep 19 2006

A. Messiah, Quantum mechanics, vol. 2, pp. 608-609, eq.(XIV.57), North Holland, 1969.

Muniru A Asiru, Table of n, a(n) for n = 1..3000
Miklos Bona and Rebecca Smith, Pattern avoidance in permutations and their squares, arXiv:1901.00026 [math.CO], 2018. See p. 6.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,1).

Cf. A122581, A122582, A122583.

[n le 5 select 1 else 2*Self(n-1) +Self(n-2) -2*Self(n-3) +Self(n-4): n in [1..50]]; // G. C. Greubel, Nov 28 2021

seq(coeff(series((x*(1+x)*(2*x-1))/(x^4-2*x^3+x^2+2*x-1),x,n+1), x, n), n = 1 .. 40); # Muniru A Asiru, Jan 03 2019

a[n_]:= a[n]= If[n<4, 1, 2*a[n-1] +a[n-2] -2*a[n-3] +a[n-4]];
Table[a[n], {n, 50}] (* modified by G. C. Greubel, Nov 28 2021 *)

Vec(x*(1+x)*(2*x-1)/(-1+2*x+x^2-2*x^3+x^4)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

@CachedFunction # a=A122584
def a(n): return 1 if (n<5) else 2*a(n-1) +a(n-2) -2*a(n-3) +a(n-4)
[a(n) for n in (1..50)] # G. C. Greubel, Nov 28 2021

G.f.: x*(1+x)*(1-2*x)/(1 - 2*x - x^2 + 2*x^3 - x^4).

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

A122582 a(n) = a(n - 1) - 2a(n - 2) + a(n - 3) - 2a(n - 4) + a(n - 5).

Original entry on oeis.org

1, 1, 1, 1, 1, -1, -3, -1, 3, 5, 3, -5, -13, -7, 13, 27, 15, -25, -61, -37, 57, 135, 81, -119, -297, -191, 257, 661, 431, -549, -1455, -991, 1169, 3225, 2257, -2497, -7115, -5145, 5299, 15725, 11715, -11261, -34709, -26623, 23829, 76603, 60479, -50361, -168997, -137173, 106105, 372655, 310905, -222951

Offset: 1

Roger L. Bagula, Sep 19 2006

sign

This recursion is inspired by Ulam's early experiments in derivative recursions.

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

Cf. A122581, A122583, A122584.

[n le 5 select 1 else Self(n-1) -2*Self(n-2) +Self(n-3) -2*Self(n-4) +Self(n-5): n in [1..50]]; // G. C. Greubel, Nov 28 2021

a[n_]:= a[n]= If[n<6, 1, a[n-1] -2*a[n-2] +a[n-3] -2*a[n-4] +a[n-5]];
Table[a[n], {n, 60}]
Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{1,-2,1,-2,1},#]}]&, {1,1,1,1,1},60]][[1]]  (* Harvey P. Dale, Mar 21 2011 *)

@CachedFunction # a=A122582
def a(n): return 1 if (n<6) else a(n-1) -2*a(n-2) +a(n-3) -2*a(n-4) +a(n-5)
[a(n) for n in (1..50)] # G. C. Greubel, Nov 28 2021

G.f.: x*(1+2*x^2+x^3+3*x^4)/(1-x+2*x^2-x^3+2*x^4-x^5). - R. J. Mathar, May 12 2013

Edited by N. J. A. Sloane, Oct 01 2006, Jan 01 2007

A122583 a(n) = a(n - 1) - 2a(n - 2) + a(n - 3) - 6a(n - 4) + 3*a(n - 5).

Original entry on oeis.org

1, 1, 1, 1, 1, -3, -7, -3, 5, 25, 45, -3, -107, -191, -175, 253, 1045, 1189, -171, -3547, -7527, -4603, 11497, 33945, 40869, -10487, -141071, -248407, -120131, 421141, 1227961, 1332777, -726439, -5051271, -8369959, -3306635, 16738977, 43110597, 41391949, -33360335, -183387403, -283721435

Offset: 1

Roger L. Bagula, Sep 19 2006

sign

This recursion is inspired by Ulam's early experiments in derivative recursions.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,-2,1,-6,3).

Cf. A122581, A122582, A122584.

[n le 5 select 1 else Self(n-1) -2*Self(n-2) +Self(n-3) -6*Self(n-4) +3*Self(n-5): n in [1..50]]; // G. C. Greubel, Nov 28 2021

A122583:= proc(n) option remember; if n <= 5 then 1; else A122583(n-1) -2*A122583(n-2)+A122583(n-3)+3*(-2*A122583(n-4)+A122583(n-5)); fi; end: seq(A122583(n), n=1..50) ; # R. J. Mathar, Sep 18 2007

a[n_]:= a[n]= If[n<6, 1, a[n-1] -2*a[n-2] +a[n-3] -6*a[n-4] +3*a[n-5]];
Table[a[n], {n, 50}]
LinearRecurrence[{1,-2,1,-6,3},{1,1,1,1,1},50] (* Harvey P. Dale, Jun 09 2025 *)

@CachedFunction # a=A122583
def a(n): return 1 if (n<6) else a(n-1) -2*a(n-2) +a(n-3) -6*a(n-4) +3*a(n-5)
[a(n) for n in (1..50)] # G. C. Greubel, Nov 28 2021

G.f.: x*(1 +2*x^2 +x^3 +7*x^4)/(1 -x +2*x^2 -x^3 +6*x^4 -3*x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

Edited by N. J. A. Sloane, Oct 01 2006

More terms from R. J. Mathar, Sep 18 2007

A135564 a(n) defined by a(2n) = a(2n-2) - (a(n) - 2a(n-1) + a(n-2)) for n > 2, a(2n+1) = a(2n) - (a(n-2) - 2a(n-3) + a(n-4)), for n > 3, with a(0)=0, a(1)=1, a(2)=3, a(3)=-1, a(4)=-2, a(5)=-3, a(6)=4, a(7)=2.

Original entry on oeis.org

0, 1, 3, -1, -2, -3, 4, 2, 1, 0, 1, 7, -7, -10, 2, 2, 1, -7, 1, 10, -1, -2, -6, -6, 14, 12, 3, -2, -12, 8, 0, -11, 1, -14, 8, 20, -8, -7, -9, -2, 11, -5, 1, 0, 4, 24, 0, -10, -20, -17, 2, -2, 9, -11, 5, 27, 10, 17, -20, -24, 8, 13, 11, -19, -12, 16, 15, 18, -22, -45, -12, 15, 28, -9, -1, 9, 2, 42, -7, -36, -13, -10, 16, 7, -6, -12, 1, 30, -4

Offset: 0

Roger L. Bagula, Feb 23 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A122581, A135689, A135690, A135692.

a[0]:=0; a[1]:=1; a[2]:=3; a[3]:=-1; a[4]:=-2; a[5]:=-3; a[6]:=4; a[7]:=2;
a[n_]:= a[n]= If[Mod[n, 2]==0, a[n-2] -a[n/2] +2*a[n/2 -1] -a[n/2 -2], a[n-1] -a[(n-1)/2 -2] +2*a[(n-1)/2 -3] -a[(n-1)/2 -4]];
Table[a[n], {n, 0, 100}]

@CachedFunction
def a(n): # A135564
    if (n<8): return [0, 1, 3, -1, -2, -3, 4, 2][n]
    elif ((n%2)==0): return a(n-2) - a(n/2) + 2*a(n/2 - 1) - a(n/2 -2)
    else: return a(n-1) - a((n-1)/2 - 2) + 2*a((n-1)/2 - 3) - a((n-1)/2 -4)
[a(n) for n in (0..100)] # G. C. Greubel, Nov 26 2021

a(n) = a(n-2) - (a(floor(n/2)) - 2*a(abs(floor(n/2) -1)) + a(abs(floor(n/2) -2)) ) if (n mod 2) = 0, otherwise a(n-1) - (a(abs(floor(n/2) - 2)) - 2*a(abs(floor(n/2) - 3)) + a(abs(floor(n/2) - 4)).
a(2*n) = a(2*n-2) - (a(n) - 2*a(n-1) + a(n-2)), for n > 2.
a(2*n+1) = a(2*n) - (a(n-2) - 2*a(n-3) + a(n-4)), for n > 3.

Edited by G. C. Greubel, Nov 28 2021

cp's OEIS Frontend

A122584 Expansion of x*(1+x)*(1-2*x)/(1 - 2*x - x^2 + 2*x^3 - x^4).

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A122582 a(n) = a(n - 1) - 2*a(n - 2) + a(n - 3) - 2*a(n - 4) + a(n - 5).

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A122583 a(n) = a(n - 1) - 2*a(n - 2) + a(n - 3) - 6*a(n - 4) + 3*a(n - 5).

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A135564 a(n) defined by a(2*n) = a(2*n-2) - (a(n) - 2*a(n-1) + a(n-2)) for n > 2, a(2*n+1) = a(2*n) - (a(n-2) - 2*a(n-3) + a(n-4)), for n > 3, with a(0)=0, a(1)=1, a(2)=3, a(3)=-1, a(4)=-2, a(5)=-3, a(6)=4, a(7)=2.

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Mathematica

Sage

Formula

Extensions

A122584 Expansion of x(1+x)(1-2x)/(1 - 2x - x^2 + 2*x^3 - x^4).

A122582 a(n) = a(n - 1) - 2a(n - 2) + a(n - 3) - 2a(n - 4) + a(n - 5).

A122583 a(n) = a(n - 1) - 2a(n - 2) + a(n - 3) - 6a(n - 4) + 3*a(n - 5).

A135564 a(n) defined by a(2n) = a(2n-2) - (a(n) - 2a(n-1) + a(n-2)) for n > 2, a(2n+1) = a(2n) - (a(n-2) - 2a(n-3) + a(n-4)), for n > 3, with a(0)=0, a(1)=1, a(2)=3, a(3)=-1, a(4)=-2, a(5)=-3, a(6)=4, a(7)=2.