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Search: id:A008443
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| A008443 |
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Number of ordered ways of writing n as the sum of 3 triangular numbers. |
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12
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| 1, 3, 3, 4, 6, 3, 6, 9, 3, 7, 9, 6, 9, 9, 6, 6, 15, 9, 7, 12, 3, 15, 15, 6, 12, 12, 9, 12, 15, 6, 13, 21, 12, 6, 15, 9, 12, 24, 9, 18, 12, 9, 18, 15, 12, 13, 24, 9, 15, 24, 6, 18, 27, 6, 12, 15, 18, 24, 21, 15, 12, 27, 9, 13, 18, 15, 27, 27, 9, 12, 27, 15, 24, 21, 12, 15, 30, 15, 12
(list; graph; listen)
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OFFSET
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0,2
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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.
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REFERENCES
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Andrews, George E., EYPHKA! num = Delta + Delta + Delta, J. Number Theory 23 (1986), 285-293. [The Y in the title is really the Greek letter Upsilon and Delta is really the Greek letter of that name.]
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
M. Nathanson, Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, Volume 165, Springer-Verlag, 1996. See Chapter 1.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..5050
M. D. Hirschhorn & J. A. Sellers, Partitions Into Three Triangular Numbers
M. D. Hirschhorn & J. A. Sellers, On Representations Of A Number As A Sum Of Three Triangles
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FORMULA
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Expansion of Jacobi theta constant theta_2^3 /8.
Expansion of psi(q)^3 in powers of q where psi() is a Ramanujan theta functions. - Michael Somos Oct 25 2006
Euler transform of period 2 sequence [ 3, -3, ...]. - Michael Somos Oct 25 2006
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EXAMPLE
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5 can be written as 3+1+1, 1+3+1, 1+1+3, so a(5) = 3.
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MAPLE
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s1 := sum(q^(n*(n+1)/2), n=0..30): s2 := series(s1^3, q, 250): for i from 0 to 200 do printf(`%d, `, coeff(s2, q, i)) od:
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PROGRAM
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(PARI) {a(n)=if(n<0, 0, polcoeff( sum(k=0, (sqrtint(8*n+1)-1)\2, x^((k^2+k)/2), x*O(x^n))^3, n))} /* Michael Somos Oct 25 2006 */
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)^2/eta(x+A))^3, n))} /* Michael Somos Oct 25 2006 */
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CROSSREFS
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Cf. A053604, A002636.
Partial sums are in A038835. a(n) = A005869(n)/2 = A005886(n)/4 = A005878(n)/8.
Sequence in context: A100091 A104806 A043551 this_sequence A074883 A011762 A092200
Adjacent sequences: A008440 A008441 A008442 this_sequence A008444 A008445 A008446
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 07 2001
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