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A104246
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Minimal number of tetrahedral numbers (A000292(k) = k(k+1)(k+2)/6) needed to sum to n.
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35
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1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 4, 5, 3, 1, 2, 3, 4, 2, 2, 3, 4, 3, 3, 2, 3, 4, 4, 3, 3, 4, 5, 4, 4, 2, 1, 2, 3, 3, 2, 3, 4, 4, 3, 3, 2, 3, 4, 4, 2, 3, 4, 5, 3, 3, 2, 3, 4, 4, 3, 4, 5, 5, 1, 2, 3, 4, 2, 3, 3, 2, 3, 4, 2, 3, 3, 4, 3, 4, 4, 3, 4
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OFFSET
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1,2
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COMMENTS
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According to Dickson, Pollock conjectures that a(n) <= 5 for all n. Watson shows that a(n) <= 8 for all n, and Salzer and Levine show that a(n) <= 5 for n <= 452479659. - N. J. A. Sloane, Jul 15 2011
Could be extended with a(0) = 0, in analogy to A061336. Kim (2003, first row of table "d = 3" on p. 73) gives max {a(n)} = 5 as a "numerical result", but the value has no "* denoting exact values" (see Remark at end of paper), which means this could be incorrect. - M. F. Hasler, Mar 06 2017, edited Sep 22 2022
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REFERENCES
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Dickson, L. E., History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 1952, see p. 13.
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LINKS
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MAPLE
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tet:=[seq((n^3-n)/6, n=1..20)];
LAGRANGE(tet, 8, 120); # the LAGRANGE transform of a sequence is defined in A193101. - N. J. A. Sloane, Jul 15 2011
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PROG
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(PARI)
seq(N) = {
my(a = vector(N, k, 8), T = k->(k*(k+1)*(k+2))\6);
for (n = 1, N,
my (k1 = sqrtnint((6*n)\8, 3), k2 = sqrtnint(6*n, 3));
while(n < T(k2), k2--); if (n == T(k2), a[n] = 1; next());
for (k = k1, k2, a[n] = min(a[n], a[n - T(k)] + 1))); a;
};
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CROSSREFS
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Cf. A000292 (tetrahedral numbers), A000797 (numbers that need 5 tetrahedral numbers).
Cf. A061336 (analog for triangular numbers).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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