A110512 Expansion of (1 + x)/(1 + x + 2x^2).

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

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

Tejo V. Madhavarapu, The Most Malicious Maître D', arXiv:2407.09000 [math.CO], 2024. See p. 3.

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 *)
LinearRecurrence[{-1, -2}, {1, 0}, 40] (* Harvey P. Dale, Dec 30 2024 *)

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