A033192 a(n) = binomial(Fibonacci(n) + 1, 2).

0, 1, 1, 3, 6, 15, 36, 91, 231, 595, 1540, 4005, 10440, 27261, 71253, 186355, 487578, 1276003, 3339820, 8742471, 22885995, 59912931, 156848616, 410626153, 1075018896, 2814412825, 7368190921, 19290113571, 50502074766, 132215989335, 346145696820, 906220783315

Offset: 0

Author

Simon P. Norton

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000045, A000217, A033191, A081667.

Programs

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 *)

PARI

a(n)=binomial(fibonacci(n)+1,2) \\ Charles R Greathouse IV, Jan 26 2012

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

Equals A000217 o A000045. - M. F. Hasler, Jan 27 2012

a(n) = A032441(n) - 1. - Filip Zaludek, Oct 30 2016