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A209083 Largest number of the form C(n,x) + C(n,y) + C(n,z) where x + y + z = n. 1
3, 3, 5, 9, 14, 25, 45, 77, 141, 261, 505, 935, 1849, 3445, 6865, 12885, 25741, 48637, 97241, 184775, 369513, 705453, 1410865, 2704179, 5408313, 10400625, 20801201, 40116627, 80233201, 155117549, 310235041, 601080421, 1202160781, 2333606253, 4667212441 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

lim{n->infinity} a(n+1)/a(n)=2. Subset of A034703. From an idea of Michael B. Porter.

For n > 6, it appears that the solution is always x = n mod 2, y = z = floor(n/2). - T. D. Noe, Mar 05 2012

LINKS

Paolo P. Lava, Table of n, a(n) for n = 0..360

EXAMPLE

For n=5 [x,y,z] can be [0,0,5], [0,1,4], [0,2,3], [1,1,3] and [1,2,2].

C(5,0) + C(5,0) + C(5,5) = 1+1+1 = 3.

C(5,0) + C(5,1) + C(5,4) = 1+5+5 = 11.

C(5,0) + C(5,2) + C(5,3) = 1+10+10 =21.

C(5,1) + C(5,1) + C(5,3) = 5+5+10 = 20.

C(5,1) + C(5,2) + C(5,2) = 5+10+10 = 25.

Therefore 25 is in the sequence.

MAPLE

with(numtheory);

P:=proc(i)

local c, m, n, s, v;

v:=array[1..3];

for n from 3 to i do

  s:=0; v[1]:=0; v[2]:=0; v[3]:=n;

  while v[1]<=floor(n/3) do

    while v[2]<=floor((n-v[1])/2) do

      c:=0;

      for m from 1 to 3 do c:=c+binomial(n, v[m]); od;

      if c>s then s:=c; fi;

      v[2]:=v[2]+1; v[3]:=v[3]-1;

    od;

    v[1]:=v[1]+1; v[2]:=v[1]; v[3]:=n-v[1]-v[2];

  od;

  print(s);

od;

end:

P(1000);

MATHEMATICA

Table[Maximize[{Binomial[n, a] + Binomial[n, b] + Binomial[n, c], a + b + c == n, a >= 0, b >= 0, c >= 0, a <= n, b <= n, c <= n}, {a, b, c}, Integers][[1]], {n, 0, 30}] (* T. D. Noe, Mar 05 2012 *)

PROG

(PARI) A209083(n)={local(a, b, c, s); s=-1; for(a=0, n, for(b=0, n-a, c=n-a-b; s=max(s, binomial(n, a)+binomial(n, b)+binomial(n, c)))); s}

CROSSREFS

Cf. A034703.

Sequence in context: A104220 A321986 A325187 * A137202 A146926 A000198

Adjacent sequences:  A209080 A209081 A209082 * A209084 A209085 A209086

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Mar 05 2012

STATUS

approved

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Last modified September 24 23:43 EDT 2022. Contains 356951 sequences. (Running on oeis4.)