|
%I #40 Dec 25 2023 13:46:59
%S 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,
%T 15,15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,
%U 26,27,27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35
%N 1 appears three times, other numbers twice.
%C Gives the number of terms in n-th row of many common tables.
%C 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
%C a(n) = length of run n+1 of consecutive 4s in A254338. - _Reinhard Zumkeller_, Feb 27 2015
%C This is the Engel expansion of A070910 + A096789. - _Benedict W. J. Irwin_, Dec 16 2016
%H Harry J. Smith, <a href="/A065033/b065033.txt">Table of n, a(n) for n = 0..1000</a>
%H Andrei Asinowski, Cyril Banderier, Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(0)=a(1)=a(2)=1, a(3)=2, 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). - _Philippe Deléham_, Sep 28 2006
%F a(n) = floor((n+1)/2) + 0^n. - _Reinhard Zumkeller_, Feb 27 2015
%t Array[Floor[#/2] &, 61] /. 0 -> 1 (* _Michael De Vlieger_, Mar 10 2020 *)
%o (PARI) { for (n=0, 1000, if (n<3, a=1, if (n%2, a++)); write("b065033.txt", n, " ", a) ) } \\ _Harry J. Smith_, Oct 03 2009
%o (Haskell)
%o a065033 n = 0 ^ n + div (n + 1) 2 -- _Reinhard Zumkeller_, Feb 27 2015
%Y Cf. A004526, A008619.
%Y Cf. A254338.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, Nov 04 2001
|
