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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

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OFFSET

0,4

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).

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

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 *)

CoefficientList[Series[(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)), {x, 0, 40}], x] (* Wesley Ivan Hurt, Feb 14 2026 *)

PROG

(Magma) R<x>:=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

(SageMath)

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

CROSSREFS

Cf. A057079, A117901, A133851.

Sequence in context: A279481 A233131 A202395 * A167685 A268740 A120493

Adjacent sequences: A117900 A117901 A117902 * A117904 A117905 A117906

KEYWORD

easy,sign

AUTHOR

Paul Barry, Apr 01 2006

STATUS

approved

URL: https://oeis.org/A117903

⇱ A117903 - OEIS