A110523 Expansion of (1 + x)/(1 + x + 3*x^2).

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

Author

Paul Barry, Jul 24 2005

Keywords

• easy
• sign

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. Lastly 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae, Vol. 38, No. 2 (2024), pp. 284-313. See p. 298.
• Index entries for linear recurrences with constant coefficients, signature (-1,-3).

Crossrefs

Cf. A001607, A106852, A110512, A110522, A214733.

Programs

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

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 *)

PARI

my(x='x+O('x^50)); Vec((1+x)/(1+x+3*x^2)) \\ G. C. Greubel, Aug 30 2017

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

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)

E.g.f.: exp(-x/2)*(sqrt(11)*cos(sqrt(11)*x/2) + sin(sqrt(11)*x/2))/sqrt(11). - Stefano Spezia, Jul 27 2025