OFFSET
0,4
COMMENTS
a(n) is the sum of n-th row in Wythoff array A003603. [Reinhard Zumkeller, Jan 26 2012]
A subsequence of the triangular numbers A000217. In fact, binomial(F(n)+1,2) = A000217(F(n)). - M. F. Hasler, Jan 27 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2394
James P. Jones and Péter Kiss, Representation of integers as terms of a linear recurrence with maximal index, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 25. (1998) pp. 21-37. See Theorem 3.7 p. 33.
Kálmán Liptai and László Szalay, Random inhomogeneous binary recurrences, Annales Univ. Sci. Budapest, Sect. Comp. 54 (2023) 253-263. See p. 262.
Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).
FORMULA
G.f.: x(x^3-x^2-2x+1)/[(1+x)(1-3x+x^2)(1-x-x^2)].
a(n) = ((Fibonacci(n)+Fibonacci(n)^2)/2). - Gary Detlefs, Dec 24 2010
a(n) = A032441(n) - 1. - Filip Zaludek, Oct 30 2016
MAPLE
a:= n-> (f-> f*(f+1)/2)((<<0|1>, <1|1>>^n)[1, 2]):
seq(a(n), n=0..35); # Alois P. Heinz, Sep 06 2008
MATHEMATICA
Table[Binomial[Fibonacci[n] + 1, 2], {n, 0, 50}] (* Alonso del Arte, Jan 26 2012 *)
LinearRecurrence[{3, 1, -5, -1, 1}, {0, 1, 1, 3, 6}, 40] (* Harvey P. Dale, Apr 04 2020 *)
PROG
(PARI) a(n)=binomial(fibonacci(n)+1, 2) \\ Charles R Greathouse IV, Jan 26 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Simon P. Norton
STATUS
approved