cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A340949 Number of ways to write n as an ordered sum of 4 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 4, 0, 6, 4, 4, 12, 1, 16, 6, 16, 12, 12, 22, 8, 36, 4, 30, 24, 21, 36, 18, 36, 28, 48, 16, 44, 36, 44, 48, 36, 46, 40, 72, 20, 73, 48, 54, 72, 42, 68, 56, 84, 50, 72, 78, 56, 84, 84, 62, 112, 60, 60, 110, 84, 97, 72, 120, 76, 116, 84, 72, 144, 102, 104, 96, 108, 102, 156, 102, 92
Offset: 4

Views

Author

Ilya Gutkovskiy, Jan 31 2021

Keywords

Links

Crossrefs

Cf. A000217, A008438, A010054, A053603, A053604, A319814, A340950, A340951, A340952, A340953, A340954, A340955.

Programs

  • Maple
    b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..75);  # Alois P. Heinz, Jan 31 2021
  • Mathematica
    nmax = 75; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^4, {x, 0, nmax}], x] // Drop[#, 4] &

Formula

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^4, where theta_2() is the Jacobi theta function.

A340950 Number of ways to write n as an ordered sum of 5 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 5, 0, 10, 5, 10, 20, 5, 35, 11, 40, 30, 35, 55, 30, 90, 25, 100, 60, 80, 120, 60, 140, 90, 161, 100, 165, 135, 165, 210, 140, 220, 180, 265, 170, 295, 200, 285, 330, 205, 365, 260, 395, 295, 391, 350, 355, 480, 340, 455, 490, 415, 480, 515, 445, 600, 510, 565, 550, 680, 545, 555
Offset: 5

Views

Author

Ilya Gutkovskiy, Jan 31 2021

Keywords

Links

Crossrefs

Cf. A000217, A008439, A010054, A053603, A053604, A319815, A340949, A340951, A340952, A340953, A340954, A340955.

Programs

  • Maple
    b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..67);  # Alois P. Heinz, Jan 31 2021
  • Mathematica
    nmax = 67; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^5, {x, 0, nmax}], x] // Drop[#, 5] &

Formula

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^5, where theta_2() is the Jacobi theta function.

A340951 Number of ways to write n as an ordered sum of 6 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 6, 0, 15, 6, 20, 30, 15, 66, 21, 90, 61, 90, 126, 86, 210, 90, 270, 156, 261, 320, 210, 450, 261, 516, 375, 542, 495, 570, 727, 540, 870, 650, 966, 816, 1050, 906, 1155, 1266, 1020, 1560, 1090, 1710, 1416, 1698, 1635, 1746, 2120, 1650, 2376, 1980, 2316, 2490, 2368, 2520, 2835
Offset: 6

Views

Author

Ilya Gutkovskiy, Jan 31 2021

Keywords

Links

Crossrefs

Cf. A000217, A008440, A010054, A053603, A053604, A319816, A340949, A340950, A340952, A340953, A340954, A340955.

Programs

  • Maple
    b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..62);  # Alois P. Heinz, Jan 31 2021
  • Mathematica
    nmax = 62; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^6, {x, 0, nmax}], x] // Drop[#, 6] &

Formula

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^6, where theta_2() is the Jacobi theta function.

A340953 Number of ways to write n as an ordered sum of 8 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 8, 0, 28, 8, 56, 56, 70, 176, 84, 336, 196, 448, 492, 504, 953, 616, 1456, 960, 1814, 1792, 1904, 3032, 2100, 4144, 3052, 4768, 4670, 5264, 6720, 5936, 8876, 7112, 10620, 9648, 11718, 12720, 13216, 15960, 15261, 19608, 17164, 23296, 21226, 25424, 26796, 27272, 32844
Offset: 8

Views

Author

Ilya Gutkovskiy, Jan 31 2021

Keywords

Links

Crossrefs

Cf. A000217, A007331, A010054, A053603, A053604, A319818, A340949, A340950, A340951, A340952, A340954, A340955.

Programs

  • Maple
    b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..56);  # Alois P. Heinz, Jan 31 2021
  • Mathematica
    nmax = 56; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^8, {x, 0, nmax}], x] // Drop[#, 8] &

Formula

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^8, where theta_2() is the Jacobi theta function.

A340954 Number of ways to write n as an ordered sum of 9 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 9, 0, 36, 9, 84, 72, 126, 261, 162, 576, 336, 882, 873, 1092, 1845, 1386, 3061, 2160, 4167, 3957, 4860, 6948, 5580, 10287, 7812, 12777, 12276, 14634, 18363, 17136, 25056, 21282, 31266, 28899, 36075, 39654, 41202, 51348, 49383, 63270, 59391, 76059, 73611, 87319, 93582, 96966
Offset: 9

Views

Author

Ilya Gutkovskiy, Jan 31 2021

Keywords

Links

Crossrefs

Cf. A000217, A010054, A053603, A053604, A226253, A319819, A340949, A340950, A340951, A340952, A340953, A340955.

Programs

  • Maple
    b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 9):
seq(a(n), n=9..56);  # Alois P. Heinz, Jan 31 2021
  • Mathematica
    nmax = 56; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^9, {x, 0, nmax}], x] // Drop[#, 9] &

Formula

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^9, where theta_2() is the Jacobi theta function.

A340955 Number of ways to write n as an ordered sum of 10 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 10, 0, 45, 10, 120, 90, 210, 370, 297, 930, 570, 1620, 1480, 2220, 3375, 2940, 6085, 4590, 8981, 8370, 11430, 15100, 13890, 23832, 19155, 31940, 30195, 38520, 46890, 46440, 66550, 59400, 86355, 81532, 104220, 114390, 122410, 153450, 149490, 193440, 188010, 235350, 238840
Offset: 10

Views

Author

Ilya Gutkovskiy, Jan 31 2021

Keywords

Links

Crossrefs

Cf. A000217, A010054, A053603, A053604, A226254, A319820, A340949, A340950, A340951, A340952, A340953, A340954.

Programs

  • Maple
    b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..54);  # Alois P. Heinz, Jan 31 2021
  • Mathematica
    nmax = 54; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^10, {x, 0, nmax}], x] // Drop[#, 10] &

Formula

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^10, where theta_2() is the Jacobi theta function.

A341024 Number of partitions of n into 7 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 1, 2, 0, 0, 2, 1, 1, 0, 2, 1, 1, 3, 0, 1, 1, 1, 3, 2, 2, 2, 2, 1, 2, 4, 0, 2, 4, 0, 3, 4, 3, 4, 3, 1, 4, 3, 3, 4, 4, 4, 2, 6, 3, 6, 6, 1, 6, 3, 3, 5, 9, 4, 4, 8, 2, 6, 9, 3, 9, 7, 4, 9, 6, 6, 10, 8, 5
Offset: 84

Views

Author

Ilya Gutkovskiy, Feb 02 2021

Keywords

Crossrefs

Cf. A000217, A010054, A226252, A307597, A307598, A319817, A340952, A341021, A341022, A341023, A341025, A341026, A341027.

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

Views

Author

Ilya Gutkovskiy, Feb 20 2021

Keywords

Links

Crossrefs

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

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..20]])^7 )); // G. C. Greubel, Jul 19 2022
  • Mathematica
    nmax = 72; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^7, {x, 0, nmax}], x] // Drop[#, 7] &
  • SageMath
    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

Formula

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^7.
Showing 1-8 of 8 results.

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