A147617 -id:A147617 - 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

A147606 Expansion of g.f.: 1/((1 - x - x^2 + x^4 - x^6)*(1 - x^2 + x^4 + x^5 - x^6)).

Original entry on oeis.org

1, 1, 3, 4, 6, 8, 12, 15, 25, 35, 56, 84, 130, 192, 294, 432, 654, 972, 1466, 2192, 3308, 4953, 7463, 11185, 16820, 25224, 37906, 56868, 85445, 128239, 192643, 289196, 434364, 652124, 979372, 1470436, 2208192, 3315556, 4978892, 7475948, 11226252

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 (1,2,-1,-3,0,5,0,-3,-1,2,1,-1).

Cf. A109537, A147607, A147617.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x-x^2+x^4-x^6)*(1-x^2+x^4+x^5-x^6)) )); // G. C. Greubel, Oct 24 2022

f[x_]= -1+x+x^2-x^4+x^6;
CoefficientList[Series[-1/(x^6*f[x]*f[1/x]), {x, 0, 50}], x] (* G. C. Greubel, Oct 24 2022 *)

def A147606_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x-x^2+x^4-x^6)*(1-x^2+x^4+x^5-x^6)) ).list()
A147606_list(50) # G. C. Greubel, Oct 24 2022

G.f.: 1/(1 - x - 2*x^2 + x^3 + 3*x^4 - 5*x^6 + 3*x^8 + x^9 - 2*x^10 - x^11 + x^12).

G.f.: -1/(x^6*f(x)*f(1/x)), where f(x) = -1 + x + x^2 - x^4 + x^6. - G. C. Greubel, Oct 24 2022

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

A147605 Expansion of g.f.: 1/((1 - x^2 - x^3 - x^4 - x^5 - x^6 - x^7)*(1 + x + x^2 + x^3 + x^4 + x^5 - x^7)).

Original entry on oeis.org

1, -1, 1, 0, 1, 1, 3, 3, 3, 10, 11, 21, 32, 52, 77, 128, 206, 320, 517, 817, 1297, 2060, 3290, 5220, 8298, 13205, 20980, 33360, 53056, 84366, 134114, 213263, 339086, 539123, 857240, 1363034, 2167197, 3445840, 5478951, 8711511, 13851359

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 (-1,0,1,2,3,5,7,5,3,2,1,0,-1,-1).

Cf. A107479, A147606, A147607, A147617.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x^2-x^3-x^4-x^5-x^6-x^7)*(1+x+x^2+x^3+x^4+x^5-x^7)) )); // G. C. Greubel, Oct 24 2022

f[x_]= -1 +x^2 +x^3 +x^4 +x^5 +x^6 +x^7;
CoefficientList[Series[-1/(x^7*f[x]*f[1/x]), {x,0,50}], x] (* G. C. Greubel, Oct 24 2022 *)

Vec(1/(1 +x -x^3 -2*x^4 -3*x^5 -5*x^6 -7*x^7 -5*x^8 -3*x^9 -2*x^10 -x^11 + x^13 +x^14) + O(x^40)) \\ Jinyuan Wang, Mar 10 2020

def A147605_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^2-x^3-x^4-x^5-x^6-x^7)*(1+x+x^2+x^3+x^4+x^5-x^7)) ).list()
A147605_list(50) # G. C. Greubel, Oct 24 2022

G.f.: 1/(1 + x - x^3 - 2*x^4 - 3*x^5 - 5*x^6 - 7*x^7 - 5*x^8 - 3*x^9 - 2*x^10 - x^11 + x^13 + x^14).

G.f.: -1/(x^7*f(x)*f(1/x)), where f(x) = -1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7. - G. C. Greubel, Oct 24 2022

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

A147620 Expansion of g.f.: 1/((1 - x - x^2 + x^6 - x^8)*(1 - x^2 + x^6 + x^7 - x^8)).

Original entry on oeis.org

1, 1, 3, 4, 8, 12, 19, 29, 46, 70, 111, 170, 271, 422, 668, 1048, 1655, 2603, 4104, 6453, 10167, 15989, 25175, 39599, 62329, 98064, 154335, 242845, 382183, 601399, 946451, 1489366, 2343847, 3688412, 5804459, 9134287, 14374533, 22620800, 35597998

Offset: 0

Roger L. Bagula, Nov 08 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1,0,-2,0,5,0,-2,0,-1,-1,2,1,-1).

Cf. A147607, A147617, A147621.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x-x^2+x^6- x^8)*(1-x^2+x^6+x^7-x^8)) )); // G. C. Greubel, Oct 24 2022

f[x_]= -1+x^2-x^6-x^7+x^8;
CoefficientList[Series[-1/(x^8*f[x]*f[1/x]), {x, 0, 50}], x]

Vec(1/(1-x-2*x^2+x^3+x^4+2*x^6-5*x^8+2*x^10+x^12+x^13-2*x^14-x^15+x^16) + O(x^40)) \\ Jinyuan Wang, Mar 10 2020

def A147620_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x-x^2+x^6-x^8)*(1-x^2+x^6+x^7-x^8)) ).list()
A147620_list(40) # G. C. Greubel, Oct 24 2022

G.f.: 1/(1 - x - 2*x^2 + x^3 + x^4 + 2*x^6 - 5*x^8 + 2*x^10 + x^12 + x^13 - 2*x^14 - x^15 + x^16).

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

A147598 Expansion of g.f. 1/((1-x^2+x^3+x^4-x^5)*(1-x-x^2+x^3-x^5)).

Original entry on oeis.org

1, 1, 3, 2, 4, 3, 6, 9, 14, 23, 29, 45, 57, 88, 123, 184, 267, 382, 556, 787, 1149, 1643, 2392, 3444, 4978, 7184, 10348, 14956, 21550, 31152, 44924, 64881, 93611, 135101, 195000, 281382, 406201, 586164, 846121, 1221064, 1762399, 2543555, 3671003

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 (1,2,-3,-1,5,-1,-3,2,1,-1).

Cf. A147605, A147606, A147607, A147617, A147620.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x^2+x^3+x^4-x^5)*(1-x-x^2+x^3-x^5)) )); // G. C. Greubel, Oct 25 2022

f[x_]= x^5 -x^4 -x^3 +x^2 -1;
CoefficientList[Series[-1/(x^5*f[x]*f[1/x]), {x,0,50}],x]

def A147598_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^2+x^3+x^4-x^5)*(1-x-x^2+x^3-x^5)) ).list()
A147598_list(50) # G. C. Greubel, Oct 25 2022

G.f.: -1/(x^5*f(x)*f(1/x)), where f(x) = -1 +x^2 -x^3 -x^4 +x^5.

G.f.: 1/((x^5-x^4-x^3+x^2-1)*(x^5-x^3+x^2+x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

Better name (using g.f.) from Joerg Arndt, Apr 06 2018

A147593 Expansion of 1/(1 - x + x^3 - 3*x^4 + x^5 - x^7 + x^8).

Original entry on oeis.org

1, 1, 1, 0, 2, 3, 5, 3, 6, 8, 16, 16, 24, 28, 50, 61, 91, 109, 170, 220, 327, 415, 607, 800, 1164, 1536, 2192, 2928, 4172, 5616, 7921, 10705, 15049, 20460, 28638, 39027, 54453, 74451, 103662, 141996, 197288, 270704, 375632, 516096, 715258, 983661, 1362091

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 (1,0,-1,3,-1,0,1,-1).

Cf. A147598, A147605, A147606, A147607, A147617, A147620.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1+x^3-x^4)*(1-x-x^4)) )); // G. C. Greubel, Oct 25 2022

f[x_]= x^4-x^3-1; CoefficientList[Series[-1/(x^4*f[x]*f[1/x]), {x,0,50}], x]

def A147593_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x^3-x^4)*(1-x-x^4)) ).list()
A147593_list(50) # G. C. Greubel, Oct 25 2022

G.f.: -1/(x^4*f(x)*f(1/x)), where f(x) = -1 - x^3 + x^4.

G.f.: 1/((1+x^3-x^4)*(1-x-x^4)). - Colin Barker, Nov 04 2012

Edited by Joerg Arndt and Colin Barker, Nov 04 2012.

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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A147606 Expansion of g.f.: 1/((1 - x - x^2 + x^4 - x^6)*(1 - x^2 + x^4 + x^5 - x^6)).

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A147605 Expansion of g.f.: 1/((1 - x^2 - x^3 - x^4 - x^5 - x^6 - x^7)*(1 + x + x^2 + x^3 + x^4 + x^5 - x^7)).

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A147620 Expansion of g.f.: 1/((1 - x - x^2 + x^6 - x^8)*(1 - x^2 + x^6 + x^7 - x^8)).

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A147598 Expansion of g.f. 1/((1-x^2+x^3+x^4-x^5)*(1-x-x^2+x^3-x^5)).

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A147593 Expansion of 1/(1 - x + x^3 - 3*x^4 + x^5 - x^7 + x^8).

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Magma

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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)).