A063993 Number of ways of writing n as an unordered sum of exactly 3 nonzero triangular numbers.

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, 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, 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

Offset

0,13

Comments

a(A002097(n)) = 0; a(A111638(n)) = 1; a(A064825(n)) = 2. - Reinhard Zumkeller, Jul 20 2012

Links

T. D. Noe, Table of n, a(n) for n=0..5050

Example

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

Maple

A063993 := proc(n)
    local a, t1idx, t2idx, t1, t2, t3;
    a := 0 ;
    for t1idx from 1 do
        t1 := A000217(t1idx) ;
        if 3*t1 > n then
            break;
        end if;
        for t2idx from t1idx do
            t2 := A000217(t2idx) ;
            if t1+t2 > n then
                break;
            end if;
            t3 :=  n-t1-t2 ;
            if t3 >= t2 then
                if isA000217(t3) then
                    a := a+1 ;
                end if;
            end if ;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Apr 28 2020

Mathematica

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

Prog

(Haskell)
a063993 n = length [() | let ts = takeWhile (< n) $ tail a000217_list,
                    x <- ts, y <- takeWhile (<= x) ts,
                    let z = n - x - y, 0 < z, z <= y, a010054 z == 1]
-- Reinhard Zumkeller, Jul 20 2012
(PARI) trmx(n)=my(k=sqrtint(8*n+1)\2); if(k^2+k>2*n, k-1, k)
trmn(n)=trmx(ceil(n)-1)+1
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

Crossrefs

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

Cf. A000217, A010054.

Column k=3 of A319797.

Sequence in context: A309228 A309778 A143223 * A353445 A115722 A115721

Adjacent sequences: A063990 A063991 A063992 * A063994 A063995 A063996

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Sep 18 2001

Extensions

More terms from Robert G. Wilson v, Sep 20 2001

Status

approved