A002097 -id:A002097 - 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

A051611 Numbers that are not the sum of 2 nonzero triangular numbers.

Original entry on oeis.org

1, 3, 5, 8, 10, 14, 15, 17, 19, 23, 26, 28, 32, 33, 35, 40, 41, 44, 45, 47, 50, 52, 53, 54, 59, 62, 63, 68, 71, 74, 75, 77, 78, 80, 82, 85, 86, 89, 95, 96, 98, 103, 104, 105, 107, 109, 113, 116, 117, 118, 122, 124, 125, 128, 129, 131, 134, 138, 140, 143, 145, 147

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

A053603(a(n)) = 0. - Reinhard Zumkeller, Jun 28 2013

T. D. Noe, Table of n, a(n) for n=1..1000

Integers not in the sequence A051533. Cf. A002097, A000217, A007294, A051611, A053603.

a051611 n = a051611_list !! (n-1)
a051611_list = filter ((== 0) . a053603) [1..]
-- Reinhard Zumkeller, Jun 28 2013

notSumQ[n_] := Reduce[ 0 < x <= y && n == x*(x + 1)/2 + y*(y + 1)/2, {x, y}, Integers] === False; Select[ Range[150], notSumQ] (* Jean-François Alcover, Jun 27 2012 *)
With[{trnos=Accumulate[Range[100]]},Complement[Range[150],Total/@ Tuples[ trnos,2]]] (* Harvey P. Dale, Jun 01 2016 *)

A064825 Numbers which are the sums of three positive triangular numbers in exactly two different ways.

Original entry on oeis.org

12, 17, 19, 21, 22, 23, 26, 27, 28, 31, 32, 33, 34, 35, 39, 41, 42, 43, 44, 46, 47, 51, 54, 56, 62, 64, 65, 68, 74, 80, 88, 89, 90, 92, 95, 106, 113, 123, 128, 137, 141, 146, 153, 164, 179, 194, 200, 218, 245, 281, 326, 335

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 23 2001

nonn

Checked up to 1000000 but haven't found any other values.
A063993(a(n)) = 2. - Reinhard Zumkeller, Jul 20 2012
Without the qualifier "positive" in the Name, sequence A071530 would result. - Jon E. Schoenfield, Jan 01 2020
No further terms <= 3*10^7. - Michael S. Branicky, Dec 17 2021

			19 = 10 + 6 + 3 = 15 + 3 + 1.

Cf. A063993, A064816, A002097, A071530, A111638.

[k:k in [1..350]|#RestrictedPartitions(k, 3, {m*(m+1) div 2:m in [1..200]}) eq 2 ]; // Marius A. Burtea, Jan 01 2020

from collections import Counter
from itertools import count, takewhile, combinations_with_replacement as mc
def aupto(N):
    tris = takewhile(lambda x: x <= N, (i*(i+1)//2 for i in count(1)))
    sum3 = filter(lambda x: x <= N, (sum(c) for c in mc(tris, 3)))
    sum3counts = Counter(sum3)
    return sorted(k for k in sum3counts if sum3counts[k] == 2)
print(aupto(1000)) # Michael S. Branicky, Dec 17 2021

Offset changed to 1 by Michel Marcus, Jan 14 2014

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.

cp's OEIS Frontend

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

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A051611 Numbers that are not the sum of 2 nonzero triangular numbers.

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A064825 Numbers which are the sums of three positive triangular numbers in exactly two different ways.

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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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