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 A110523 Expansion of (1 + x)/(1 + x + 3*x^2). 4
 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 (list; graph; refs; listen; history; text; internal format)
 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 R. 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 Index entries for linear recurrences with constant coefficients, signature (-1,-3). 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) = -(3*i*sqrt(11)/11)*(((-1 - i*sqrt(11))/2)^n - ((-1 + i*sqrt(11))/2)^n). (End) MATHEMATICA CoefficientList[Series[(1 + x)/(1 + x + 3 x^2), {x, 0, 50}], x] (* G. C. Greubel, Aug 30 2017 *) PROG (PARI) x='x+O('x^50); Vec((1 + x)/(1 + x + 3 x^2)) \\ G. C. Greubel, Aug 30 2017 CROSSREFS Cf. A214733. Sequence in context: A123289 A096572 A318540 * A145597 A143418 A092370 Adjacent sequences:  A110520 A110521 A110522 * A110524 A110525 A110526 KEYWORD easy,sign AUTHOR Paul Barry, Jul 24 2005 STATUS approved

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Last modified February 21 01:15 EST 2019. Contains 320364 sequences. (Running on oeis4.)