OFFSET
0,4
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
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.
EXAMPLE
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.
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
PROG
(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
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Sep 18 2001
EXTENSIONS
More terms from Robert G. Wilson v, Sep 20 2001
Entry revised by N. J. A. Sloane, Feb 25 2007
STATUS
approved