A122584 -id:A122584 - cp's OEIS frontend

A147607 Expansion of g.f.: 1/((1 - 2x^2 + x^4 + 2x^6 - x^8)(1 - 2x^2 - x^4 + 2*x^6 - x^8)).

1, 0, 4, 0, 12, 0, 28, 0, 59, 0, 116, 0, 228, 0, 460, 0, 968, 0, 2092, 0, 4564, 0, 9908, 0, 21309, 0, 45444, 0, 96484, 0, 204700, 0, 434999, 0, 926440, 0, 1976344, 0, 4218936, 0, 9005328, 0, 19212728, 0, 40970200, 0, 87341032, 0, 186180665, 0, 396899620

Offset: 0

Roger L. Bagula, Nov 08 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4,0,-4,0,11,0,-4,0,-4,0,4,0,-1).

Cf. A122584, A147617.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-2*x^2+x^4 +2*x^6-x^8)*(1-2*x^2-x^4+2*x^6-x^8)) )); // G. C. Greubel, Oct 24 2022

CoefficientList[Series[1/(1-4 x^2+4 x^4+4 x^6-11 x^8+4 x^10+4 x^12-4 x^14+x^16),{x,0,60}],x] (* or *) LinearRecurrence[ {0,4,0,-4,0,-4,0,11,0,-4,0,-4,0,4,0,-1},{1,0,4,0,12,0,28,0,59,0,116,0,228,0,460,0},60] (* Harvey P. Dale, Apr 03 2013 *)

def A147607_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-2*x^2+x^4+2*x^6-x^8)*(1-2*x^2-x^4+2*x^6-x^8)) ).list()
A147607_list(60) # G. C. Greubel, Oct 24 2022

G.f.: 1/(1 - 4*x^2 + 4*x^4 + 4*x^6 - 11*x^8 + 4*x^10 + 4*x^12 - 4*x^14 + x^16).
a(n) = 4*a(n-2) - 4*a(n-4) - 4*a(n-6) + 11*a(n-8) - 4*a(n-10) - 4*a(n-12) + 4*a(n-14) - a(n-16) with a(0)=1, a(1)=0, a(2)=4, a(3)=0, a(4)=12, a(5)=0, a(6)=28, a(7)=0, a(8)=59, a(9)=0, a(10)=116, a(11)=0, a(12)=228, a(13)=0, a(14)=460, a(15)=0. - Harvey P. Dale, Apr 03 2013
G.f.: -1/(x^8*f(x)*f(1/x)), where f(x) = -1 + 2*x^2 - x^4 - 2*x^6 + x^8. - G. C. Greubel, Oct 24 2022

Definition corrected by N. J. A. Sloane, Nov 09 2008

A122581 a(n) = a(n - 1) - 2a(n - 2) + a(n - 3) - 4a(n - 4) + 2*a(n - 5).

Original entry on oeis.org

1, 1, 1, 1, 1, -2, -5, -2, 4, 13, 19, -5, -50, -65, -20, 118, 283, 187, -311, -914, -1001, 334, 3040, 4405, 835, -8273, -17030, -11189, 20068, 60178, 60427, -29165, -192491, -274310, -39845, 553798, 1070812, 635629, -1341437, -3836765, -3693914, 2237287, 12425356, 16921054, 1409755, -36343973

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,-4,2).

Cf. A122582, A122583, A122584.

A122581:= proc(n) option remember; if n <= 5 then 1; else A122581(n-1) -2*A122581(n-2)+A122581(n-3)+2*(-2*A122581(n-4)+A122581(n-5)); fi; end: seq(A122581(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] -2*(2*a[n-4] -a[n-5])];
Table[a[n], {n,50}]

@CachedFunction # a=A122581
def a(n): return 1 if (n<6) else a(n-1) -2*a(n-2) +a(n-3) -4*a(n-4) +2*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+5*x^4)/(1-x+2*x^2-x^3+4*x^4-2*x^5). - R. J. Mathar, Nov 18 2007

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

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

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

A322779 132-avoiding permutations whose squares are also 132-avoiding.

Original entry on oeis.org

1, 2, 5, 12, 24, 50, 101, 202, 398, 806, 1568, 3148, 6198, 12306, 24223, 48314

Offset: 1

N. J. A. Sloane, Jan 04 2019

Miklos Bona, Rebecca Smith, Pattern avoidance in permutations and their squares, arXiv:1901.00026 [math.CO], 2018. See Sect. 5.

Cf. A122584.

Name clarified by Colin Defant, Apr 30 2019

cp's OEIS Frontend

A147607 Expansion of g.f.: 1/((1 - 2*x^2 + x^4 + 2*x^6 - x^8)*(1 - 2*x^2 - x^4 + 2*x^6 - x^8)).

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A122581 a(n) = a(n - 1) - 2*a(n - 2) + a(n - 3) - 4*a(n - 4) + 2*a(n - 5).

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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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Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

Extensions

A322779 132-avoiding permutations whose squares are also 132-avoiding.

Views

Author

Keywords

Links

Crossrefs

Extensions

A147607 Expansion of g.f.: 1/((1 - 2x^2 + x^4 + 2x^6 - x^8)(1 - 2x^2 - x^4 + 2*x^6 - x^8)).

A122581 a(n) = a(n - 1) - 2a(n - 2) + a(n - 3) - 4a(n - 4) + 2*a(n - 5).

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).