A065033 1 appears three times, other numbers twice.

1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35

Offset: 0

Author

N. J. A. Sloane, Nov 04 2001

Keywords

• nonn
• easy

Comments

Gives the number of terms in n-th row of many common tables.

Number of partitions of the (n+1)-th Fibonacci number into distinct Fibonacci numbers: a(n) = A000119(A000045(n)), see also A098641. - Reinhard Zumkeller, Apr 24 2005

a(n) = length of run n+1 of consecutive 4s in A254338. - Reinhard Zumkeller, Feb 27 2015

This is the Engel expansion of A070910 + A096789. - Benedict W. J. Irwin, Dec 16 2016

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A004526, A008619.

Cf. A254338.

Programs

Haskell

a065033 n = 0 ^ n + div (n + 1) 2  -- Reinhard Zumkeller, Feb 27 2015

Mathematica

Array[Floor[#/2] &, 61] /. 0 -> 1 (* Michael De Vlieger, Mar 10 2020 *)

PARI

a(n) = { if (n<1, n==0, (n+1)\2) } \\ Harry J. Smith, Oct 03 2009

Formula

From Philippe Deléham, Sep 28 2006: (Start)

a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.

G.f.: (1-x^2+x^3)/(1-x-x^2+x^3). (End)

a(n) = floor((n+1)/2) + 0^n. - Reinhard Zumkeller, Feb 27 2015

E.g.f.: (2 + exp(x)*x + sinh(x))/2. - Stefano Spezia, Aug 05 2025