Search: a226748 -id:a226748
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A003108
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Number of partitions of n into cubes.
(Formerly M0209)
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+10
55
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1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 17, 17, 17, 18, 18, 18, 19, 19, 21, 21, 21, 22, 22, 22, 23, 23, 25, 26, 26, 27, 27, 27, 28
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OFFSET
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0,9
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COMMENTS
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The g.f. 1/(z+1)/(z**2+1)/(z**4+1)/(z-1)**2 conjectured by Simon Plouffe in his 1992 dissertation is wrong.
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REFERENCES
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H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
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LINKS
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Eric Weisstein's World of Mathematics, Partition
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FORMULA
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G.f.: Sum_{n>=0} x^(n^3) / Product_{k=1..n} (1 - x^(k^3)). - Paul D. Hanna, Mar 09 2012
a(n) ~ exp(4 * (Gamma(1/3)*Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3)*Zeta(4/3))^(3/4) / (24*Pi^2*n^(5/4)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016
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EXAMPLE
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a(16) = 3 because we have [8,8], [8,1,1,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].
G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 +...
such that the g.f. A(x) satisfies the identity [Paul D. Hanna]:
A(x) = 1/((1-x)*(1-x^8)*(1-x^27)*(1-x^64)*(1-x^125)*...)
A(x) = 1 + x/(1-x) + x^8/((1-x)*(1-x^8)) + x^27/((1-x)*(1-x^8)*(1-x^27)) + x^64/((1-x)*(1-x^8)*(1-x^27)*(1-x^64)) +...
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MAPLE
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g:=1/product(1-x^(j^3), j=1..30): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=0..65); # Emeric Deutsch, Mar 30 2006
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Product[1/(1 - x^(k^3)), {k, 1, nmax^(1/3)}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)
nmax = 60; cmax = nmax^(1/3);
s = Table[n^3, {n, cmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)
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PROG
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(PARI) {a(n)=polcoeff(1/prod(k=1, ceil(n^(1/3)), 1-x^(k^3)+x*O(x^n)), n)} /* Paul D. Hanna, Mar 09 2012 */
(PARI) {a(n)=polcoeff(1+sum(m=1, ceil(n^(1/3)), x^(m^3)/prod(k=1, m, 1-x^(k^3)+x*O(x^n))), n)} /* Paul D. Hanna, Mar 09 2012 */
(Haskell)
a003108 = p $ tail a000578_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
(Magma) [#RestrictedPartitions(n, {d^3:d in [1..n]}): n in [0..150]]; // Marius A. Burtea, Jan 02 2019
(Python)
from functools import lru_cache
from sympy import integer_nthroot, divisors
@lru_cache(maxsize=None)
@lru_cache(maxsize=None)
def a(n): return integer_nthroot(n, 3)[1]
@lru_cache(maxsize=None)
def c(n): return sum(d for d in divisors(n, generator=True) if a(d))
return (c(n)+sum(c(k)*A003108(n-k) for k in range(1, n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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A068980
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Number of partitions of n into nonzero tetrahedral numbers.
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+10
32
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1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 11, 11, 12, 12, 15, 15, 16, 16, 19, 19, 22, 22, 25, 25, 28, 29, 32, 32, 35, 36, 42, 42, 45, 46, 52, 53, 56, 57, 63, 64, 70, 71, 77, 78, 84, 87, 94, 95, 101, 104, 115, 116, 122, 125, 136, 139, 146, 149, 160, 163, 175
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OFFSET
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0,5
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LINKS
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FORMULA
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G.f.: 1 / Product_(k>=3} (1 - z^binomial(k, 3)).
G.f.: Sum_{i>=0} x^(i*(i+1)*(i+2)/6) / Product_{j=1..i} (1 - x^(j*(j+1)*(j+2)/6)). - Ilya Gutkovskiy, Jun 08 2017
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EXAMPLE
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a(10) = 4 because we can write 10 = 10 = 4 + 4 + 1 + 1 = 4 + 1 + 1 + 1 + 1 + 1 = 1 + ... + 1.
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Product[1/(1-x^(k*(k+1)*(k+2)/6)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)
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CROSSREFS
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See also A007294 (partitions into triangular numbers), A000292 (tetrahedral numbers).
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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1, 4, 6, 8, 10, 12, 19, 20, 27, 35, 44, 48, 56, 64, 84, 85, 120, 124, 125, 146, 165, 216, 220, 231, 255, 286, 343, 344, 364, 455, 456, 489, 512, 560, 670, 680, 729, 742, 816, 891, 969, 1000, 1128, 1140, 1156, 1330, 1331, 1469, 1540, 1629, 1728, 1771, 1834
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OFFSET
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1,2
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COMMENTS
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LINKS
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MATHEMATICA
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nn = 25; t1 = Table[n (n + 1) (n + 2)/6, {n, nn}]; t2 = Table[n^3, {n, nn}]; t3 = Table[(2*n^3 + n)/3, {n, nn}]; t4 = Table[n (3*n - 1) (3*n - 2)/2, {n, nn}]; t5 = Table[n (5*n^2 - 5*n + 2)/2, {n, nn}]; Select[Union[t1, t2, t3, t4, t5], # <= t1[[-1]] &] (* T. D. Noe, Oct 13 2012 *)
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PROG
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(Haskell)
a053012 n = a053012_list !! (n-1)
a053012_list = tail $ f
[a000292_list, a000578_list, a005900_list, a006566_list, a006564_list]
where f pss = m : f (map (dropWhile (<= m)) pss)
where m = minimum (map head pss)
(PARI) listpoly(lim, poly[..])=my(v=List()); for(i=1, #poly, my(P=poly[i], x=variable(P), f=k->subst(P, x, k), n, t); while((t=f(n++))<=lim, listput(v, t))); Set(v)
list(lim)=my(n='n); listpoly(lim, n*(n+1)*(n+2)/6, n^3, (2*n^3+n)/3, n*(3*n-1)*(3*n-2)/2, n*(5*n^2-5*n+2)/2) \\ Charles R Greathouse IV, Oct 11 2016
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 22 2000
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STATUS
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approved
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A226749
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Number of partitions of n into distinct Platonic numbers, cf. A053012.
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+10
3
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1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 7, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 9, 9, 9, 11, 11, 11, 12, 13, 13, 12, 12, 13, 15, 15, 16, 17, 17, 16, 18, 18, 19, 19, 21, 21, 23, 24, 25, 24, 24, 24, 26, 26, 29, 32
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OFFSET
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0,11
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LINKS
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EXAMPLE
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First Platonic numbers: 1, 4, 6, 8, 10, 12, 19, 20, 27, ...
a(10) = #{10, 6+4} = 2;
a(11) = #{10+1, 6+4+1} = 2;
a(12) = #{12, 8+4} = 2;
a(13) = #{12+1, 8+4+1} = 2;
a(14) = #{10+4, 8+6} = 2;
a(15) = #{10+4+1, 8+6+1} = 2;
a(16) = #{12+4, 10+6} = 2;
a(17) = #{12+4+1, 10+6+1} = 2;
a(18) = #{12+6, 10+8, 8+6+4} = 3;
a(19) = #{19, 12+6+1, 10+8+1, 8+6+4+1} = 4;
a(20) = #{20, 19+1, 12+8, 10+6+4} = 4.
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PROG
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(Haskell)
a226749 = p a053012_list where
p _ 0 = 1
p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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