OFFSET
0,4
COMMENTS
We have a(n)=C(n;-1), A121449(n)=A(n;-1), A085810(n+1)=-B(n+1;-1), where A(n;d), B(n;d), and C(n;d), n in N, d in C, are so-called quasi-Fibonacci numbers defined and discussed in the comments to A121449 and in Witula-Slota-Warzynski's paper. It follows from formulas (3.47-49) in this paper that the value of A(n;1/3), B(n;1/3) and C(n;1/3) could be obtained from special convolution type identities for sequences a(n), A121449, and A085810.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Ilya Amburg, Krishna Dasaratha, Laure Flapan, Thomas Garrity, Chansoo Lee, Cornelia Mihaila, Nicholas Neumann-Chun, Sarah Peluse, and Matthew Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239 [math.CO], 2015.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-1).
FORMULA
G.f.: x^2/(1-4*x+3*x^2+x^3).
a(n) = (1/7)*((c(2)-c(4))*(1-c(1))^n + (c(4)-c(1))*(1-c(2))^n + (c(1)-c(2))*(1-c(4))^n), where c(j):=2*cos(2*Pi*j/7) - this formula is the Binet formula for a(n) (see the Binet formula (3.17) for the respective quasi-Fibonacci number C(n;d) for value d=-1 in the Witula-Slota-Warzynski paper).
MATHEMATICA
LinearRecurrence[{4, -3, -1}, {0, 0, 1}, 50]
CoefficientList[Series[x^2/(1 - 4 x + 3 x^2 + x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 18 2015 *)
PROG
(PARI) Vec(x^2/(1-4*x+3*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012
(Magma) I:=[0, 0, 1]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-Self(n-3): n in [1..35]]; // Vincenzo Librandi, Sep 18 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roman Witula, Aug 09 2012
STATUS
approved