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Search: id:A002636
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| A002636 |
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Number of ways of writing n as an unordered sum of at most 3 nonzero triangular numbers.
(Formerly M0076 N0027)
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+0
9
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| 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; listen)
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OFFSET
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0,4
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COMMENT
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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). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 07 2009]
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..10000
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
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EXAMPLE
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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.
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MATHEMATICA
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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
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CROSSREFS
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Cf. A007294, A053604, A008443, A063993, A061262.
Sequence in context: A080236 A025142 A074265 this_sequence A087974 A008679 A029435
Adjacent sequences: A002633 A002634 A002635 this_sequence A002637 A002638 A002639
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KEYWORD
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nonn,easy,nice,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Sep 18 2001
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 20 2001
Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Feb 25 2007
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