A063993 -id:A063993 - cp's OEIS frontend

A002636 Number of ways of writing n as an unordered sum of at most 3 nonzero triangular numbers.

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 1, 2, 3, 2, 2, 2, 1, 4, 3, 2, 2, 2, 2, 3, 3, 1, 4, 4, 2, 2, 3, 2, 3, 4, 2, 3, 3, 2, 4, 3, 2, 4, 4, 2, 4, 4, 1, 4, 5, 1, 2, 3, 4, 6, 4, 3, 2, 5, 2, 3, 3, 3, 6, 5, 2, 2, 5, 3, 5, 4, 2, 4, 5, 3, 4, 5, 2, 4, 6, 2, 6, 3, 3, 6, 3, 2, 3, 7, 3, 6, 6, 2, 4, 6, 3, 2

Offset: 0

N. J. A. Sloane, Sep 18 2001

Fermat asserted that every number is the sum of three triangular numbers. This was proved by Gauss, who recorded in his Tagebuch entry for Jul 10 1796 that: EYPHKA! num = DELTA + DELTA + DELTA.
a(n) <= A167618(n). - Reinhard Zumkeller, Nov 07 2009
Equivalently, number of ways of writing n as an unordered sum of exactly 3 triangular numbers. - Jon E. Schoenfield, Mar 28 2021

			0 : empty sum
1 : 1
2 : 1+1
3 : 3 = 1+1+1
4 : 3+1
5 : 3+1+1
6 : 6 = 3+3
7 : 6+1 = 3+3+1
...
13 : 10 + 3 = 6 + 6 + 1, so a(13) = 2.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102, eq. (8).
D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302.
G. Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian) Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15.
Mel Nathanson, Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, Volume 165, Springer-Verlag, 1996. See Chapter 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Gino Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian). Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15. Also D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302. [Annotated scanned copies]
Eric T. Mortenson, A Kronecker-type identity and the representations of a number as a sum of three squares, arXiv:1702.01627 [math.NT], 2017.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

Cf. A007294, A053604, A008443, A063993, A061262.

# reuses code in A000217
A002636 := proc(n)
    local a,i,Ti, j,Tj, Tk ;
    a := 0 ;
    for i from 0 do
        Ti := A000217(i) ;
        if Ti > n then
            break ;
        end if;
        for j from i do
            Tj := A000217(j) ;
            if Ti+Tj > n then
                break ;
            end if;
            Tk := n-Ti-Tj ;
            if Tk >= Tj and isA000217(Tk) then
                a := a+1 ;
            end if;
            if Tk < Tj then
                break ;
            end if;
        end do:
    end do:
    a ;
end proc:
seq(A002636(n),n=0..40) ; # R. J. Mathar, May 26 2025

a = Table[ n(n + 1)/2, {n, 0, 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

first(n)=my(v=vector(n+1),A,B,C); for(a=0,n, A=a*(a+1)/2; if(A>n, break); for(b=0,a, B=A+b*(b+1)/2; if(B>n, break); for(c=0,b, C=B+c*(c+1)/2; if(C>n, break); v[C+1]++))); v \\ Charles R Greathouse IV, Jun 23 2017

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

Entry revised by N. J. A. Sloane, Feb 25 2007

A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1

Offset: 0

Alois P. Heinz, Sep 28 2018

Equals A181506 when the first column is removed. - Georg Fischer, Jul 26 2023

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 1, 0, 1;
  0, 0, 1, 0, 1;
  0, 0, 0, 1, 0, 1;
  0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A010054 (for n>0), A052344, A063993, A319814, A319815, A319816, A319817, A319818, A319819, A319820.
Row sums give A007294.
T(2n,n) gives A319799.
Cf. A000217, A117278, A181506, A243148, A319394.

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..20);

h[n_] := h[n] = If[n < 1, 0, If[IntegerQ @ Sqrt[8*n + 1], n, h[n - 1]]];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, h[i - 1]] + Expand[ x*b[n - i, h[Min[n - i, i]]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, h[n]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A000217(j)).

A307598 Number of partitions of n into 3 distinct positive triangular numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 1, 1, 0, 3, 0, 2, 1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 0, 4, 1, 2, 1, 1, 1, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 1, 3, 2, 2, 2, 2, 1, 3, 2, 0, 4, 1, 1, 5, 1, 3, 2, 2, 3, 2, 2, 1, 4, 1, 2, 4, 2, 2, 3, 2, 1, 3, 2, 4, 3, 3, 2, 2, 3, 1, 6

Offset: 0

Ilya Gutkovskiy, Apr 17 2019

nonn

The greedy inverse starts 0, 10, 19, 37, 52, 82, 109, 136, 241, 226, 217, 247, 364, 427, 457, 541, 532, 577, 637, 961, 721, 787, 1066, 1102, 1381, 1267, 1564, 1192, 1396, 1816, 1501, 1612, 1927, 1942, 2242, 1792, 2842, 2587, 2557, 2422, ... - R. J. Mathar, Apr 28 2020

			a(19) = 2 because we have [15, 3, 1] and [10, 6, 3].

Cf. A000217, A002244, A008443, A024940, A025442, A053604, A063993, A224326, A307597.

a(n) = [x^n y^3] Product_{k>=1} (1 + y*x^(k*(k+1)/2)).

A341774 Number of partitions of n into 3 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0, 2

Offset: 3

Ilya Gutkovskiy, Feb 19 2021

nonn

Cf. A000292, A023533, A024692, A063993, A068980, A341775, A341776, A341777, A341778.

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 *)

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]]

A064181 Smallest number that can be written in n ways as an unordered sum of 3 nonzero triangular numbers.

Original entry on oeis.org

0, 3, 12, 30, 61, 52, 82, 136, 142, 147, 192, 277, 247, 367, 552, 457, 516, 507, 646, 637, 721, 822, 787, 1171, 1066, 1137, 1227, 1402, 1192, 1396, 1696, 1501, 1612, 1876, 1927, 1792, 2551, 2587, 2926, 2761, 2422, 2947, 2887, 3262, 3271, 3412, 3937

Offset: 0

Robert G. Wilson v, Sep 20 2001

nonn

Donovan Johnson, Table of n, a(n) for n = 0..3082

Cf. A063993.

a = Table[ n(n + 1)/2, {n, 1, 100} ]; b = {0}; c = Table[ 0, {5000} ]; Do[ b = Append[ b, a[ [ i ] ] + a[ [ j ] ] + a[ [ k ] ] ], {k, 1, 100}, {j, 1, k}, {i, 1, j} ]; b = Delete[ b, 1 ]; b = Sort[ b ]; l = Length[ b ]; Do[ If[ b[ [ n ] ] < 5000, c[ [ b[ [ n ] ] + 1 ] ]++ ], {n, 1, l} ]; Do[ k = 1; While[ c[ [ k ] ] != n, k++ ]; Print[ k - 1 ], {n, 0, 54} ]

More terms from Don Reble, Sep 21 2001

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

A002636 Number of ways of writing n as an unordered sum of at most 3 nonzero triangular numbers.

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A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A307598 Number of partitions of n into 3 distinct positive triangular numbers.

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A341774 Number of partitions of n into 3 nonzero tetrahedral numbers.

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A002097 Numbers that are not the sum of 3 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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A064181 Smallest number that can be written in n ways as an unordered sum of 3 nonzero triangular numbers.

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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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