A247907 -id:A247907 - cp's OEIS frontend

A247918 Expansion of (1 + x) / ((1 - x^4) * (1 + x^4 - x^5)) in powers of x.

1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 2, 0, -1, 2, -1, 2, 1, -2, 4, -3, 1, 4, -5, 7, -4, -2, 10, -12, 11, -1, -11, 22, -23, 13, 11, -33, 45, -35, 3, 44, -78, 81, -37, -41, 122, -158, 119, 4, -163, 281, -276, 115, 167, -443, 558, -391, -52, 611, -1000, 949

Offset: 0

Michael Somos, Sep 26 2014

			G.f. = 1 + x + x^5 + x^6 + x^8 + x^11 + 2*x^13 - x^15 + 2*x^16 - x^17 + ...

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

Cf. A010892, A056594, A077905, A112553, A176971, A247907.

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!((1 + x)/((1-x^4)*(1+x^4-x^5)))); // G. C. Greubel, Aug 04 2018

CoefficientList[Series[(1+x)/((1-x^4)(1+x^4-x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)

{a(n) = if( n<0, n=-8-n; polcoeff( -1/((1-x)*(1-x+x^2)*(1+x^2)*(1 - x^2 - x^3)) + x * O(x^n), n), polcoeff( 1/((1-x)*(1-x+x^2)*(1+x^2)*(1+x-x^3)) + x * O(x^n), n))};

def A247918_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/((1-x^4)*(1+x^4-x^5)) ).list()
A247918_list(70) # G. C. Greubel, Aug 08 2022

G.f.: 1 / ((1 - x) * (1 - x + x^2) * (1 + x^2) * (1 + x - x^3)).
a(n) = a(n+1) + a(n+5) - mod(floor((n-1)/2), 2) for all n in Z.
a(n) = -A247907(-8-n) for all n in Z.
Convolution of A077905 and A112553.
a(n) = (35 + 7*(A056594(n) + 3*A056594(n-1)) + 10*(3*A010892(n) - A010892(n-1)) - 2*(A176971(n) + 4*A172971(n-1) + 12*A176971(n-2)))/70. - G. C. Greubel, Aug 08 2022

A248049 a(n) = (a(n-1) + a(n-2)) * (a(n-2) + a(n-3)) / a(n-4) with a(0) = 2, a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

2, 1, 1, 1, 2, 6, 24, 240, 3960, 184800, 33033000, 26125799700, 219429008298500, 31064340573760168675, 206377779224083011749949745, 245390990689739612867279321757020455, 230795626149641527446533813473152766756062242744

Offset: 0

Michael Somos, Sep 30 2014

nonn

It seems that degrees of factors when using [2,1,1,y] as initial condition are given by A233522. - F. Chapoton, May 21 2020
It seems also that degrees (w.r.t. x) of factors when using [2,1,x,y] as initial condition are given by A247907. - F. Chapoton, Jan 03 2021
Somos conjectures that log(a(n)) ~ 1.25255*c^n, where c = A060006. - Bill McEachen, Oct 11 2022

Robert Israel, Table of n, a(n) for n = 0..26
Math Stack Exchange, Is OEIS A248049 an Integer Sequence

Cf. A233522, A060006.

a[0]:= 2: a[1]:= 1: a[2]:= 1: a[3]:= 1:
for n from 4 to 20 do
a[n] := (a[n-1] + a[n-2]) * (a[n-2] + a[n-3]) / a[n-4]
od:
seq(a[i],i=0..20); # Robert Israel, Mar 18 2020

{a(n) = if( n<0, n=4-n); if( n<4, (n==0)+1, (a(n-1) + a(n-2)) * (a(n-2) + a(n-3)) / a(n-4))};

a(n) = a(4-n) for all n in Z.

a(n) * a(n+4) = (a(n+1) + a(n+2)) * (a(n+2) + a(n+3)) for all n in Z.

cp's OEIS Frontend

A247918 Expansion of (1 + x) / ((1 - x^4) * (1 + x^4 - x^5)) in powers of x.

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