A133851 -id:A133851 - cp's OEIS frontend

A117902 Expansion of (1-x^2-2x^3)/(1-4x^3).

1, 0, -1, 2, 0, -4, 8, 0, -16, 32, 0, -64, 128, 0, -256, 512, 0, -1024, 2048, 0, -4096, 8192, 0, -16384, 32768, 0, -65536, 131072, 0, -262144, 524288, 0, -1048576, 2097152, 0, -4194304, 8388608, 0, -16777216, 33554432, 0, -67108864, 134217728, 0, -268435456, 536870912, 0, -1073741824

Offset: 0

Paul Barry, Apr 01 2006

Row sums of number triangle A117901.

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

Cf. A117901, A133851.

[1] cat [n le 3 select (-1)^(n-1)*(n-1) else 4*Self(n-3): n in [1..50]]; // G. C. Greubel, Oct 09 2021

LinearRecurrence[{0,0,4}, {1,0,-1,2}, 50] (* G. C. Greubel, Oct 09 2021 *)

def A133851(n): return 4^(n/3) if (n%3==0) else 0
def A117902(n): return bool(n==0)/2 + 2*A133851(n-3) - A133851(n-2)
[A117902(n) for n in (0..50)] # G. C. Greubel, Oct 09 2021

a(n) = 0^n/2 - (2^(2*n/3)/12)*( 2*cos((2*n+1)*Pi*n/3) + 2*sqrt(3)*sin((2*n+1)*Pi*n/3) -(2^(2/3) + 8)*cos(2*Pi*n/3) - 2^(1/6)*sqrt(6)*sin(2*Pi*n/3) + 2^(2/3) - 2 ).

a(n) = (1/2)*[n=0] + 2*A133851(n-3) - A133851(n-2). - G. C. Greubel, Oct 09 2021

A117903 Diagonal sums of number triangle A117901.

Original entry on oeis.org

1, -1, 1, -2, 4, -2, -5, 14, -5, -26, 64, -26, -101, 254, -101, -410, 1024, -410, -1637, 4094, -1637, -6554, 16384, -6554, -26213, 65534, -26213, -104858, 262144, -104858, -419429

Offset: 0

Paul Barry, Apr 01 2006

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,-1,3,3,3,4,4,4).

Cf. A057079, A117901, A133851.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x^2-5*x^3+3*x^4-3*x^5-x^6-2*x^7)/((1-4*x^3)*(1+x+x^2+x^3+x^4+x^5)) )); // G. C. Greubel, Oct 09 2021

LinearRecurrence[{-1,-1,3,3,3,4,4,4},{1,-1,1,-2,4,-2,-5,14},40] (* Harvey P. Dale, Oct 04 2021 *)

def A133851(n): return 4^(n/3) if (n%3==0) else 0
def A057079(n): return chebyshev_U(n, 1/2) + chebyshev_U(n-1, 1/2)
def A117903(n): return (1/30)*(28*(-1)^n + (15*(-1)^n - 1)* A057079(n) - 6*(2*A133851(n) - 5*A133851(n-1) + 2*A133851(n-2)))
[A117903(n) for n in (0..50)] # G. C. Greubel, Oct 09 2021

G.f.: (1+x^2-5*x^3+3*x^4-3*x^5-x^6-2*x^7)/((1-4*x^3)*(1+x+x^2+x^3+x^4+x^5)).
a(n) = -a(n-1) -a(n-2) +3*a(n-3) +3*a(n-4) +3*a(n-5) +4*a(n-6) +4*a(n-7) +4*a(n-8).
a(n) = (1/30)*(28*(-1)^n + (15*(-1)^n - 1)*A057079(n) - 6*(2*A133851(n) - 5*A133851(n-1) + 2*A133851(n-2))). - G. C. Greubel, Oct 09 2021

cp's OEIS Frontend

A117902 Expansion of (1-x^2-2x^3)/(1-4x^3).

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