A341794 -id:A341794 - cp's OEIS frontend

A341796 Number of ways to write n as an ordered sum of 5 nonzero tetrahedral numbers.

1, 0, 0, 5, 0, 0, 10, 0, 0, 15, 0, 0, 25, 0, 0, 31, 0, 0, 30, 5, 0, 35, 20, 0, 30, 30, 0, 20, 40, 0, 20, 65, 0, 10, 65, 0, 5, 70, 10, 5, 90, 30, 0, 70, 30, 1, 85, 40, 0, 80, 60, 0, 50, 50, 0, 70, 90, 10, 50, 90, 20, 50, 80, 10, 60, 130, 20, 65, 70, 20, 65, 90, 30, 50, 110, 70, 65, 100

Offset: 5

Ilya Gutkovskiy, Feb 19 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 5..1000

Cf. A000292, A023533, A023670, A282172, A282582, A340950, A341776, A341794, A341795, A341797, A341806, A341807, A341808, A341809.

R:=PowerSeriesRing(Integers(), 100);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..20]])^5 )); // G. C. Greubel, Jul 20 2022

nmax = 82; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..20) ) )^m
def A341796_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(5, x) ).list()
a=A341796_list(120); a[5:100] # G. C. Greubel, Jul 20 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^5.

A341797 Number of ways to write n as an ordered sum of 6 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 6, 0, 0, 15, 0, 0, 26, 0, 0, 45, 0, 0, 66, 0, 0, 76, 6, 0, 90, 30, 0, 96, 60, 0, 80, 90, 0, 75, 150, 0, 60, 192, 0, 35, 210, 15, 30, 270, 60, 15, 270, 90, 6, 270, 120, 6, 306, 195, 0, 240, 210, 1, 246, 270, 20, 240, 360, 60, 180, 330, 60, 216, 450, 80, 210, 435, 120, 216, 360

Offset: 6

Ilya Gutkovskiy, Feb 19 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 6..1000

Cf. A000292, A023533, A023670, A282582, A340951, A341777, A341794, A341795, A341796, A341797, A341806, A341807, A341808, A341809.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..20]])^6 )); // G. C. Greubel, Jul 20 2022

nmax = 77; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..20) ) )^m
def A341797_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(6, x) ).list()
a=A341797_list(100); a[6:81] # G. C. Greubel, Jul 20 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^6.

A341795 Number of ways to write n as an ordered sum of 4 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 4, 0, 0, 6, 0, 0, 8, 0, 0, 13, 0, 0, 12, 0, 0, 10, 4, 0, 12, 12, 0, 6, 12, 0, 4, 16, 0, 4, 24, 0, 0, 16, 0, 1, 24, 6, 0, 24, 12, 0, 16, 6, 0, 28, 12, 0, 12, 12, 0, 12, 12, 0, 16, 30, 4, 12, 12, 4, 16, 24, 0, 16, 24, 4, 24, 6, 0, 12, 24, 12, 12, 18, 12, 13, 36, 0, 0, 24, 12

Offset: 4

Ilya Gutkovskiy, Feb 19 2021

nonn

Cf. A000292, A023533, A023670, A282582, A340949, A341775, A341794, A341796, A341797.

nmax = 85; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^4.

A341806 Number of ways to write n as an ordered sum of 7 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 7, 0, 0, 21, 0, 0, 42, 0, 0, 77, 0, 0, 126, 0, 0, 168, 7, 0, 211, 42, 0, 252, 105, 0, 252, 182, 0, 245, 315, 0, 231, 469, 0, 175, 574, 21, 140, 735, 105, 105, 854, 210, 56, 875, 315, 42, 987, 525, 21, 952, 693, 7, 882, 840, 42, 924, 1155, 140, 770, 1260, 211, 749, 1470

Offset: 7

Ilya Gutkovskiy, Feb 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 7..1000

Cf. A000292, A023533, A023670, A282582, A340952, A341778, A341794, A341795, A341796, A341797, A341807, A341808, A341809.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..20]])^7 )); // G. C. Greubel, Jul 19 2022

nmax = 72; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..20) ) )^m
def A341806_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(7, x) ).list()
a=A341806_list(100); a[7:81] # G. C. Greubel, Jul 19 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^7.

A341807 Number of ways to write n as an ordered sum of 8 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 8, 0, 0, 28, 0, 0, 64, 0, 0, 126, 0, 0, 224, 0, 0, 336, 8, 0, 456, 56, 0, 589, 168, 0, 672, 336, 0, 708, 616, 0, 728, 1016, 0, 658, 1400, 28, 560, 1856, 168, 476, 2352, 420, 336, 2632, 728, 238, 2968, 1260, 168, 3192, 1904, 84, 3096, 2464, 112, 3192, 3360, 308, 3024, 4144

Offset: 8

Ilya Gutkovskiy, Feb 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 8..1000

Cf. A000292, A023533, A023670, A282582, A340953, A341791, A341794, A341795, A341796, A341797, A341806, A341808, A341809.

R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..70]])^8 )); // G. C. Greubel, Jul 19 2022

nmax = 70; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..8) ) )^m
def A341807_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(8, x) ).list()
a=A341807_list(100); a[8:81] # G. C. Greubel, Jul 19 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^8.

A341808 Number of ways to write n as an ordered sum of 9 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 9, 0, 0, 36, 0, 0, 93, 0, 0, 198, 0, 0, 378, 0, 0, 624, 9, 0, 918, 72, 0, 1269, 252, 0, 1597, 576, 0, 1836, 1134, 0, 2025, 2025, 0, 2058, 3096, 36, 1926, 4356, 252, 1764, 5877, 756, 1470, 7182, 1512, 1134, 8388, 2772, 882, 9576, 4608, 588, 10035, 6552, 462

Offset: 9

Ilya Gutkovskiy, Feb 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 9..1000

Cf. A000292, A023533, A023670, A282582, A340954, A341792, A341794, A341795, A341796, A341797, A341806, A341807, A341809.

R:=PowerSeriesRing(Integers(), 70);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..70]])^9 )); // G. C. Greubel, Jul 18 2022

nmax = 66; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..8) ) )^m
def A341808_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(9, x) ).list()
a=A341808_list(100); a[9:71] # G. C. Greubel, Jul 18 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^9.

A341809 Number of ways to write n as an ordered sum of 10 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 10, 0, 0, 45, 0, 0, 130, 0, 0, 300, 0, 0, 612, 0, 0, 1095, 10, 0, 1740, 90, 0, 2565, 360, 0, 3490, 930, 0, 4351, 1980, 0, 5130, 3790, 0, 5680, 6330, 45, 5820, 9540, 360, 5715, 13620, 1260, 5292, 17950, 2880, 4530, 22140, 5670, 3780, 26490, 10170, 2940, 29770, 15840

Offset: 10

Ilya Gutkovskiy, Feb 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 10..1000

Cf. A000292, A023533, A023670, A282582, A340955, A341793, A341794, A341795, A341796, A341797, A341806, A341807, A341808.

R:=PowerSeriesRing(Integers(), 70);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..70]])^10 )); // G. C. Greubel, Jul 18 2022

nmax = 66; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^10, {x, 0, nmax}], x] // Drop[#, 10] &

def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..8) ) )^m
def A341809_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(10, x) ).list()
a=A341809_list(100); a[10:71]  # G. C. Greubel, Jul 18 2022

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^10.

A343491 Number of representations of n! as a sum of 3 tetrahedral numbers (A000292).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 3, 5, 2, 3, 6, 5, 8, 8, 7, 2, 7, 8, 3, 11, 2, 2

Offset: 1

Altug Alkan, Apr 17 2021

Conjecture I: There are infinitely many n such that a(n) >= 1.
Conjecture II: Natural density of numbers n such that a(n) >= 1 is 1.
Conjecture III: Numbers n such that a(n) = 0 is a finite sequence.
Conjecture IV: a(n) >= 1 for all n.
See Links section for some solutions.

			a(4) = 2 because 4! = 0 + 4 + 20 = 4 + 10 + 10.
a(24) = 2 because 24! = f(11393630) + f(118661018) + f(127041924) = f(81298034) + f(61098204) + f(143537134) where f = A000292.

Altug Alkan, A note on sequence

Cf. A000142, A000292, A102796, A104246, A102799, A267414, A308852, A341794.

Table[Length[Solve[{i*(i + 1)*(i + 2) + j*(j + 1)*(j + 2) + k*(k + 1)*(k + 2) == 6*n!, i >= 0, j >= 0, k >= 0, i <= j, j <= k, k < (6*n!)^(1/3)}, Integers]], {n, 1, 10}] (* Vaclav Kotesovec, Apr 19 2021 *)

cp's OEIS Frontend

A341796 Number of ways to write n as an ordered sum of 5 nonzero tetrahedral numbers.

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A341797 Number of ways to write n as an ordered sum of 6 nonzero tetrahedral numbers.

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A341795 Number of ways to write n as an ordered sum of 4 nonzero tetrahedral numbers.

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A341806 Number of ways to write n as an ordered sum of 7 nonzero tetrahedral numbers.

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A341807 Number of ways to write n as an ordered sum of 8 nonzero tetrahedral numbers.

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A341808 Number of ways to write n as an ordered sum of 9 nonzero tetrahedral numbers.

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A341809 Number of ways to write n as an ordered sum of 10 nonzero tetrahedral numbers.

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A343491 Number of representations of n! as a sum of 3 tetrahedral numbers (A000292).

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