A215404 a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3), with a(0)=0, a(1)=0 and a(2)=1.

0, 0, 1, 4, 13, 39, 113, 322, 910, 2561, 7192, 20175, 56563, 158535, 444276, 1244936, 3488381, 9774440, 27387681, 76739023, 215018609, 602469686, 1688083894, 4729907909, 13252910268, 37133833451, 104046695091, 291532369743, 816855560248, 2288778436672, 6413014696201

Offset: 0

Author

Roman Witula, Aug 09 2012

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A077998, A006054, A121449, A085810, A052975, A094789, A005021, A121442.

Programs

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

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 *)

PARI

Vec(x^2/(1-4*x+3*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012

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).