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

Author

N. J. A. Sloane, Sep 18 2001

Keywords

• nonn
• easy
• nice

Comments

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

Examples

			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.
		

References

• 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).

Links

• 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.

Crossrefs

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

Programs

Maple

# 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

Mathematica

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

PARI

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

Extensions

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

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