A064825 -id:A064825 - cp's OEIS frontend

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

N. J. A. Sloane, Sep 18 2001

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

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

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

Cf. A053604, A008443, A002636, A064181 (greedy inverse), A307598 (3 distinct positive).
Cf. A000217, A010054.
Column k=3 of A319797.

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

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

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

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

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

A002097 Numbers that are not the sum of 3 nonzero triangular numbers.

Original entry on oeis.org

1, 2, 4, 6, 11, 20, 29

Offset: 1

N. J. A. Sloane, Dan Hoey

A063993(a(n)) = 0. - Reinhard Zumkeller, Jul 20 2012

Cf. A111638, A064825.

Complement[Range[30],Union[Total/@Tuples[Accumulate[Range[8]],3]]] (* Harvey P. Dale, Oct 06 2017 *)

A111638 Numbers having a unique partition into three positive triangular numbers.

Original entry on oeis.org

3, 5, 7, 8, 9, 10, 13, 14, 15, 16, 18, 24, 25, 36, 38, 50, 53, 55, 60, 69, 81, 83, 99, 110, 119

Offset: 1

T. D. Noe, Aug 10 2005

A063993(a(n)) = 1. - Reinhard Zumkeller, Jul 20 2012

			Example: 119=55+36+28

Cf. A060773 (n having a unique partition into three nonnegative triangular numbers).

Cf. A002097, A064825.

trig[n_]:=n(n+1)/2; trigInv[x_]:=Ceiling[Sqrt[Max[0, 2x]]]; lim=100; nLst=Table[0, {trig[lim]}]; Do[n=trig[a]+trig[b]+trig[c]; If[n>0 && n<=trig[lim], nLst[[n]]++ ], {a, 1, lim}, {b, a, trigInv[trig[lim]-trig[a]]}, {c, b, trigInv[trig[lim]-trig[a]-trig[b]]}]; Flatten[Position[nLst, 1]]

A071530 Numbers that are the sum of 3 triangular numbers in exactly 2 ways.

Original entry on oeis.org

3, 6, 7, 9, 10, 13, 15, 17, 18, 19, 23, 24, 25, 26, 32, 33, 35, 38, 41, 44, 47, 54, 60, 62, 68, 69, 74, 80, 83, 89, 95, 99, 110, 113, 119, 128, 179, 194

Offset: 1

Benoit Cloitre, Jun 02 2002

nonn

If it is required that the triangular numbers be positive, sequence A064825 results. - Jon E. Schoenfield, Jan 01 2020

			From _Jon E. Schoenfield_, Jan 01 2020: (Start)
15 is a term of the sequence because there are exactly 2 ways to express 15 as the sum of 3 triangular numbers: 15 = 6 + 6 + 3 = 15 + 0 + 0.
60 is a term because there are exactly 2 ways to express 60 as the sum of 3 triangular numbers: 60 = 36 + 21 + 3 = 45 + 15 + 0.
12 can be expressed as the sum of 3 triangular numbers in 3 ways, so it is not a term: 12 = 10 + 1 + 1 = 6 + 6 + 0 = 6 + 3 + 3. (End)

Cf. A000217, A060773, A061262, A002636.

Cf. A064825, A002097, A008443, A064816.

With[{max = 20}, t = Accumulate[Range[0, max]]; Select[Range[t[[-1]]], Length[IntegerPartitions[#, {3}, t]] == 2 &]] (* Amiram Eldar, May 14 2025 *)

for(n=1,150,if(sum(i=0,n,sum(j=0,i,sum(k=0,j,if(i*(i+1)/2+j*(j+1)/2+k*(k+1)/2-n,0,1))))==2,print1(n,",")))

{n: A002636(n) =2}. - R. J. Mathar, May 26 2025

More terms from Vladeta Jovovic, Jun 07 2002

Removed keyword "more" because this is probably finite. - R. J. Mathar, May 26 2025

A330810 a(n) is the largest number that can be expressed as the sum of three triangular numbers in exactly n ways.

Original entry on oeis.org

53, 194, 470, 788, 1730, 2000, 2693, 4310, 6053, 6845, 10688, 11348, 13970, 12923, 20768, 17135, 27830, 26480, 36245, 31688, 37073, 39983, 57860, 46940, 49148, 68258, 62810, 66515, 76985, 73868, 82850, 123878, 87890, 119810, 111053, 118490, 118880, 119183

Offset: 1

Jon E. Schoenfield, Jan 01 2020

nonn

One or more of the three triangular numbers may be zeros. If it were required that the triangular numbers be positive, sequence A330811 would result.

Cf. A000217, A002097, A060773, A061262, A063993, A064825, A071530, A111638, A258257, A258258, A330811.

A330811 a(n) is the largest number that can be expressed as the sum of three positive triangular numbers in exactly n ways.

Original entry on oeis.org

29, 119, 335, 713, 1730, 1328, 3413, 3485, 4565, 6053, 6950, 10688, 11348, 13970, 16778, 20768, 18173, 36245, 26480, 27203, 37073, 35033, 39983, 57860, 46940, 49148, 68258, 62810, 66515, 76985, 73868, 123878, 103403, 87890, 119810, 111053, 118490, 118880

Offset: 0

Jon E. Schoenfield, Jan 01 2020

nonn

If the triangular numbers were not required to be positive, sequence A330810 would result.

Cf. A000217, A002097, A060773, A061262, A063993, A064825, A071530, A111638, A258257, A258258, A330810.

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

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A002097 Numbers that are not the sum of 3 nonzero triangular numbers.

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A111638 Numbers having a unique partition into three positive triangular numbers.

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A071530 Numbers that are the sum of 3 triangular numbers in exactly 2 ways.

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A330810 a(n) is the largest number that can be expressed as the sum of three triangular numbers in exactly n ways.

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A330811 a(n) is the largest number that can be expressed as the sum of three positive triangular numbers in exactly n ways.

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