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A002636 Number of ways of writing n as an unordered sum of at most 3 nonzero triangular numbers.
(Formerly M0076 N0027)
14
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 (list; graph; refs; listen; history; text; internal format)
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: EYPHEKA! num = DELTA + DELTA + DELTA.

a(n) <= A167618(n). - Reinhard Zumkeller, Nov 07 2009

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+2

7 : 6+1 = 3+3+1

...

13 : 10 + 3 + 0 = 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

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

Sequence in context: A199596 A074265 A254688 * A196062 A283682 A087974

Adjacent sequences:  A002633 A002634 A002635 * A002637 A002638 A002639

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

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Last modified December 6 06:34 EST 2019. Contains 329784 sequences. (Running on oeis4.)