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 *)
PROG
(PARI) x='x+O('x^50); Vec((1 + x)/(1 + x + 2*x^2)) \\ G. C. Greubel, Aug 29 2017
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Jul 24 2005
STATUS
approved