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A064323 a(n) = a(n-1)+ceiling(a(n-2)/2) with a(0)=0, a(1)=1.

%I A064323 #24 Jan 26 2023 22:06:12
%S A064323 0,1,1,2,3,4,6,8,11,15,21,29,40,55,75,103,141,193,264,361,493,674,921,
%T A064323 1258,1719,2348,3208,4382,5986,8177,11170,15259,20844,28474,38896,
%U A064323 53133,72581,99148,135439,185013,252733,345240,471607,644227,880031,1202145
%N A064323 a(n) = a(n-1)+ceiling(a(n-2)/2) with a(0)=0, a(1)=1.
%C A064323 a(n)/a(n-1) approaches (1+sqrt(3))/2 = 1.3660254... = A332133 for large n.
%H A064323 Harry J. Smith, <a href="/A064323/b064323.txt">Table of n, a(n) for n = 0..400</a>
%F A064323 a(n) = A064324(n)-1.
%e A064323 a(5) = a(4)+ceiling(a(3)/2) = 3+ceiling(2/2) = 4.
%p A064323 a:= proc(n) option remember;
%p A064323       `if`(n<2, n, a(n-1)+ceil(a(n-2)/2))
%p A064323     end:
%p A064323 seq(a(n), n=0..48);  # _Alois P. Heinz_, Jan 26 2023
%t A064323 RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-1]+Ceiling[a[n-2]/2]},a,{n,50}] (* _Harvey P. Dale_, Nov 06 2013 *)
%o A064323 (PARI)  for (n=0, 400, if (n>1, a=a1 + ceil(a2/2); a2=a1; a1=a, if (n, a=a1=1, a=a2=0)); write("b064323.txt", n, " ", a) )  \\ _Harry J. Smith_, Sep 11 2009
%o A064323 (PARI) first(n)=if(n<2, return([0,1][1..n+1])); my(v=vector(n+1)); v[2]=1; for(k=3,n+1, v[k]=v[k-1]+(v[k-2]+1)\2); v \\ _Charles R Greathouse IV_, Jan 26 2023
%o A064323 (Magma) [n le 2 select n-1 else Self(n-1)+Ceiling(Self(n-2)/2): n in [1..45]]; // _Bruno Berselli_, Apr 20 2012
%Y A064323 Cf. A064324, A332133.
%K A064323 nonn
%O A064323 0,4
%A A064323 _Henry Bottomley_, Sep 11 2001