A089077 -id:A089077 - cp's OEIS frontend

A089074 Expansion of x(1 + x + x^2)/(1 - 2x + x^5).

0, 1, 3, 7, 14, 28, 55, 107, 207, 400, 772, 1489, 2871, 5535, 10670, 20568, 39647, 76423, 147311, 283952, 547336, 1055025, 2033627, 3919943, 7555934, 14564532, 28074039, 54114451, 104308959, 201061984, 387559436, 747044833, 1439975215

Offset: 0

Roger L. Bagula, Dec 04 2003

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

Cf. A089075, A089076, A089077.

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( x*(1+x+x^2)/(1-2*x+x^5) )); // G. C. Greubel, Feb 19 2021

CoefficientList[Series[x*(1+x+x^2)/(1-2*x+x^5), {x, 0, 50}], x] (* G. C. Greubel, Feb 19 2021 *)

def A089074_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x+x^2)/(1-2*x+x^5) ).list()
a=A089074_list(51); a[1:] # G. C. Greubel, Feb 19 2021

From N. J. A. Sloane, Dec 05 2003: (Start)
G.f.: x*(1+x+x^2)/(1-2*x+x^5).
a(n) = 2*a(n-1) - a(n-5) for n >= 6. (End)
a(n) = A000078(n+4) - 1. - G. C. Greubel, Feb 19 2021

Title and offset changed by G. C. Greubel, Feb 19 2021

A089075 A nonsense sequence.

Original entry on oeis.org

-1, 1, -1, 2, -2, 4, -4, 7, -7, 12, -12, 22, -21, 39, -39, 68, -70, 119, -127, 210, -229, 369, -413, 649, -742, 1143, -1334, 2017, -2393, 3561, -4289, 6293, -7680, 11129, -13739, 19696, -24559, 34879, -43871, 61801, -78324, 109555, -139764, 194291, -249295, 344694, -444496, 611723, -792285

Offset: 1

Roger L. Bagula, Dec 04 2003

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A089074, A089076, A089077.

(* k is a root of x^4 - x^3 - 1: *)
k=1.38027756909761411567330169182273187781662670155876302541177133121124957411;
q=k^2-k-1/k-1/k^2;
m0={{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 1, 1, q}};
m[n_]=MatrixPower[m0, n];
Table[Floor[Re[m[n][[4, 4]]]], {n, 1, 1000}]

Edited by G. C. Greubel, Feb 19 2021

A089076 Expansion of -x - x^3(2 -2x^4 +x^5)/((1-x^2)*(1+x+x^4)).

Original entry on oeis.org

-1, 0, -2, 2, -4, 4, -6, 7, -11, 14, -20, 26, -37, 50, -70, 95, -132, 181, -251, 345, -477, 657, -908, 1252, -1729, 2385, -3293, 4544, -6273, 8657, -11950, 16493, -22766, 31422, -43372, 59864, -82630, 114051, -157423, 217286, -299916, 413966, -571389, 788674, -1088590, 1502555, -2073944, 2862617

Offset: 1

Roger L. Bagula, Dec 04 2003

sign

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

Cf. A089074, A089075, A089077.

R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( -x -x^3*(2-2*x^4+x^5)/((1-x^2)*(1+x-x^4)) )); // G. C. Greubel, Feb 19 2021

Rest@CoefficientList[Series[-x -x^3*(2-2*x^4+x^5)/((1-x^2)*(1+x-x^4)), {x,0,50}], x] (* G. C. Greubel, Feb 19 2021 *)
LinearRecurrence[{-1,1,1,1,0,-1},{-1,0,-2,2,-4,4,-6,7},50] (* Harvey P. Dale, Aug 11 2021 *)

def A089076_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( -x -x^3*(2-2*x^4+x^5)/((1-x^2)*(1+x-x^4)) ).list()
a=A089076_list(51); a[1:] # G. C. Greubel, Feb 19 2021

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

Edited by G. C. Greubel, Feb 19 2021

cp's OEIS Frontend

A089074 Expansion of x(1 + x + x^2)/(1 - 2x + x^5).

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A089075 A nonsense sequence.

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A089076 Expansion of -x - x^3(2 -2x^4 +x^5)/((1-x^2)*(1+x+x^4)).

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Mathematica

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cp's OEIS Frontend

A089074 Expansion of x*(1 + x + x^2)/(1 - 2*x + x^5).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A089075 A nonsense sequence.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A089076 Expansion of -x - x^3*(2 -2*x^4 +x^5)/((1-x^2)*(1+x+x^4)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A089074 Expansion of x(1 + x + x^2)/(1 - 2x + x^5).

A089076 Expansion of -x - x^3(2 -2x^4 +x^5)/((1-x^2)*(1+x+x^4)).