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A110523
Expansion of (1 + x)/(1 + x + 3*x^2).
6
1, 0, -3, 3, 6, -15, -3, 48, -39, -105, 222, 93, -759, 480, 1797, -3237, -2154, 11865, -5403, -30192, 46401, 44175, -183378, 50853, 499281, -651840, -846003, 2801523, -263514, -8141055, 8931597, 15491568, -42286359, -4188345, 131047422, -118482387, -274659879, 630107040, 193872597
OFFSET
0,3
COMMENTS
Row sums of number triangle A110522.
The sequence a(n) is conjugate with A214733 since the following alternative relations: either ((-1 + i*sqrt(11))/2)^n = a(n) + A214733(n)*(-1 + i*sqrt(11))/2 or ((-1 - i*sqrt(11))/2)^n = a(n) + A214733(n)*(-1 - i*sqrt(11))/2. We have a(n+1) = -3*A214733(n), A214733(n+1) = a(n) - A214733(n). We note that sequences A110512 and A001607 are conjugated in a similar way. The above relations are connected with the Gauss sums, for example if e := exp(i*2Pi/11) then e + e^3 + e^4 + e^5 + e^9 = (-1 + i*sqrt(11))/2, and e^2 + e^6 + e^7 + e^8 + e^10 = (-1 - i*sqrt(11))/2, which is equivalent to the system of sums: Sum_{k=1..5} cos(2Pi*k/11) = -1/2 and Sum_{k=1..5} sin(2Pi*k/11) = sqrt(11)/2, and which is equivalent to the system of products: P_{k=1..5} cos(2Pi*k/11) = -1/32 and P_{k=1..5} sin(2Pi*k/11) = sqrt(11)/32 - for details see Witula's book. At last we note that ((-1 + i*sqrt(11))/2)^n + ((-1 - i*sqrt(11))/2)^n = 2*a(n) - A214733(n). - Roman Witula, Jul 27 2012
REFERENCES
Roman Witula, On Some Applications of Formulae for Unimodular Complex Numbers, Jacek Skalmierski's Press, Gliwice 2011.
FORMULA
a(n) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-3)^(j-k)*C(k, j-k).
From Roman Witula, Jul 27 2012: (Start)
a(n+2) + a(n+1) + 3*a(n) = 0.
a(n+1) = (-1)^n*(3*i*sqrt(11)/11)*(((1 + i*sqrt(11))/2)^(n-1) - ((1 - i*sqrt(11))/2)^(n-1)). (End)
From G. C. Greubel, Dec 28 2023: (Start)
a(n) = (-1)^n*3^((n-1)/2)*( sqrt(3)*ChebyshevU(n, 1/(2*sqrt(3))) - ChebyshevU(n-1, 1/(2*sqrt(3))) ).
a(n) = A106852(n) - A106852(n-1).
a(n) = (-1)^n*( A214733(n+1) + A214733(n) ). (End)
MATHEMATICA
CoefficientList[Series[(1+x)/(1+x+3*x^2), {x, 0, 50}], x] (* G. C. Greubel, Aug 30 2017 *)
LinearRecurrence[{-1, -3}, {1, 0}, 40] (* Harvey P. Dale, Jul 02 2022 *)
PROG
(PARI) my(x='x+O('x^50)); Vec((1+x)/(1+x+3*x^2)) \\ G. C. Greubel, Aug 30 2017
(Magma) [n le 2 select 2-n else -(Self(n-1) +3*Self(n-2)): n in [1..50]]; // G. C. Greubel, Dec 28 2023
(SageMath)
@CachedFunction # a = A110523
def a(n): return 1-n if n<2 else -a(n-1) -3*a(n-2)
[a(n) for n in range(41)] # G. C. Greubel, Dec 28 2023
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Jul 24 2005
STATUS
approved