A117903 Diagonal sums of number triangle A117901.

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

Author

Paul Barry, Apr 01 2006

Keywords

• easy
• sign

Links

• 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).

Crossrefs

Cf. A057079, A117901, A133851.

Programs

Magma

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

Mathematica

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

Sage

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

Formula

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