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 A110512 Expansion of (1 + x)/(1 + x + 2x^2). 6
 1, 0, -2, 2, 2, -6, 2, 10, -14, -6, 34, -22, -46, 90, 2, -182, 178, 186, -542, 170, 914, -1254, -574, 3082, -1934, -4230, 8098, 362, -16558, 15834, 17282, -48950, 14386, 83514, -112286, -54742, 279314, -169830, -388798, 728458 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row sums of number triangle A110511. The sequences A110512 and A001607 are conjugated by one of the relations ((-1 + i*sqrt(7))/2)^n = a(n) + A001607(n)*(-1 + i*sqrt(7))/2 or ((-1 - i*sqrt(7))/2)^n = a(n) + A001607(n)*(-1 - i*sqrt(7))/2. These relations are connected with the Gauss sums; for example, if e := exp(i*2Pi/7) then e + e^2 + e^4 = (-1 + i*sqrt(7))/2 and e^3 + e^5 + e^6 = (-1 - i*sqrt(7))/2 -- for details see Witula's book. We also have a(n+1) = -2*A001607(n), which implies the Binet formula for a(n) (from the respective Binet formula for A001607(n) given in A001607), and A001607(n+1) = a(n) - A001607(n). - Roman Witula, Jul 27 2012 Pisano period lengths: 1, 1, 8, 1, 24, 8, 42, 1, 24, 24, 10, 8, 168, 42, 24, 2, 144, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012 REFERENCES R. Witula, On some applications of formulas for unimodular complex numbers, Jacek Skalmierski's Press, Gliwice 2011 (in Polish). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (-1,-2). FORMULA a(n) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-2)^(j-k)*C(k, j-k). a(n) = (-1)^n*A078020(n). - R. J. Mathar, Feb 04 2009 a(n+2) + a(n+1) + 2*a(n) = 0. - Roman Witula, Jul 27 2012 G.f.: 2 - x + 2*x^2 + 3*x/Q(0), where Q(k)= 1 - 1/(4^k - 2*x*16^k/(2*x*4^k + 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013 MATHEMATICA CoefficientList[Series[(1 + x)/(1 + x + 2*x^2), {x, 0, 50}], x] (* G. C. Greubel, Aug 29 2017 *) PROG (PARI) x='x+O('x^50); Vec((1 + x)/(1 + x + 2*x^2)) \\ G. C. Greubel, Aug 29 2017 CROSSREFS Sequence in context: A278230 A344859 A230940 * A078020 A339091 A097521 Adjacent sequences:  A110509 A110510 A110511 * A110513 A110514 A110515 KEYWORD easy,sign AUTHOR Paul Barry, Jul 24 2005 STATUS approved

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Last modified November 26 07:57 EST 2022. Contains 358354 sequences. (Running on oeis4.)