A063993 - OEIS

%I #14 Jul 07 2022 14:43:24

%S 0,0,0,1,0,1,0,1,1,1,1,0,2,1,1,1,1,2,1,2,0,2,2,2,1,1,2,2,2,0,3,2,2,2,

%T 2,2,1,3,1,2,3,2,2,2,2,3,2,2,3,3,1,2,5,1,2,1,2,5,3,3,1,4,2,3,2,2,4,4,

%U 2,1,4,3,3,3,2,4,3,3,3,4,2,1,6,1,5,3,3,5,2,2,2,5,2,5,4,2,4,5,3,1

%N Number of ways of writing n as an unordered sum of exactly 3 nonzero triangular numbers.

%C a(A002097(n)) = 0; a(A111638(n)) = 1; a(A064825(n)) = 2. - _Reinhard Zumkeller_, Jul 20 2012

%H T. D. Noe, <a href="/A063993/b063993.txt">Table of n, a(n) for n=0..5050</a>

%e 5 = 3 + 1 + 1, so a(5) = 1.

%p A063993 := proc(n)

%p     local a,t1idx,t2idx,t1,t2,t3;

%p     a := 0 ;

%p     for t1idx from 1 do

%p         t1 := A000217(t1idx) ;

%p         if 3*t1 > n then

%p             break;

%p         end if;

%p         for t2idx from t1idx do

%p             t2 := A000217(t2idx) ;

%p             if t1+t2 > n then

%p                 break;

%p             end if;

%p             t3 :=  n-t1-t2 ;

%p             if t3 >= t2 then

%p                 if isA000217(t3) then

%p                     a := a+1 ;

%p                 end if;

%p             end if ;

%p         end do:

%p     end do:

%p     a ;

%p end proc: # _R. J. Mathar_, Apr 28 2020

%t a = Table[ n(n + 1)/2, {n, 1, 15} ]; b = {0}; c = Table[ 0, {100} ]; Do[ b = Append[ b, a[ [ i ] ] + a[ [ j ] ] + a[ [ k ] ] ], {k, 1, 15}, {j, 1, k}, {i, 1, j} ]; b = Delete[ b, 1 ]; b = Sort[ b ]; l = Length[ b ]; Do[ If[ b[ [ n ] ] < 100, c[ [ b[ [ n ] ] + 1 ] ]++ ], {n, 1, l} ]; c

%o (Haskell)

%o a063993 n = length [() | let ts = takeWhile (< n) $ tail a000217_list,

%o                     x <- ts, y <- takeWhile (<= x) ts,

%o                     let z = n - x - y, 0 < z, z <= y, a010054 z == 1]

%o -- _Reinhard Zumkeller_, Jul 20 2012

%o (PARI) trmx(n)=my(k=sqrtint(8*n+1)\2);if(k^2+k>2*n,k-1,k)

%o trmn(n)=trmx(ceil(n)-1)+1

%o a(n)=if(n<3, return(0)); sum(a=trmn(n/3),trmx(n-2),my(t=n-a*(a+1)/2);sum(b=trmn(t/2),min(trmx(t-1),a), ispolygonal(t-b*(b+1)/2,3))) \\ _Charles R Greathouse IV_, Jul 07 2022

%Y Cf. A053604, A008443, A002636, A064181 (greedy inverse), A307598 (3 distinct positive).

%Y Cf. A000217, A010054.

%Y Column k=3 of A319797.

%K nonn,easy,nice

%O 0,13

%A _N. J. A. Sloane_, Sep 18 2001

%E More terms from _Robert G. Wilson v_, Sep 20 2001