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

0, 1, 1, 2, 3, 4, 6, 8, 11, 15, 21, 29, 40, 55, 75, 103, 141, 193, 264, 361, 493, 674, 921, 1258, 1719, 2348, 3208, 4382, 5986, 8177, 11170, 15259, 20844, 28474, 38896, 53133, 72581, 99148, 135439, 185013, 252733, 345240, 471607, 644227, 880031, 1202145

Offset: 0

Author

Henry Bottomley, Sep 11 2001

Keywords

• nonn

Comments

a(n)/a(n-1) approaches (1+sqrt(3))/2 = 1.3660254... = A332133 for large n.

Examples

			a(5) = a(4)+ceiling(a(3)/2) = 3+ceiling(2/2) = 4.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A064324, A332133.

Programs

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

Maple

a:= proc(n) option remember;
      `if`(n<2, n, a(n-1)+ceil(a(n-2)/2))
    end:
seq(a(n), n=0..48);  # Alois P. Heinz, Jan 26 2023

Mathematica

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 *)

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

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

Formula

a(n) = A064324(n)-1.