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.

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A369345 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^3) + 1 + 1/x^(k^3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 5, 17, 31, 61, 139, 309, 701, 1651, 3849, 8929, 22295, 53777, 131025, 335619, 837999, 2107947, 5484373, 14071891, 36275323, 95881995, 250956301, 659257445, 1763642977, 4685724391, 12496708267, 33766814039, 90846586161, 245197523769
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 20 2024

Keywords

Comments

All terms are odd.

Crossrefs

Cf. A000578, A007576, A158118, A350249.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n>(i*(i+1)/2)^2, 0,
     `if`(i=0, 1, b(n, i-1)+b(n+i^3, i-1)+b(abs(n-i^3), i-1)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 21 2024
  • Mathematica
    Table[Coefficient[Product[x^(k^3) + 1 + 1/x^(k^3), {k, 1, n}], x, 0], {n, 0, 33}]

A369358 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 15, 25, 59, 109, 227, 525, 1321, 2917, 7085, 15893, 38759, 90179, 223933, 534867, 1339691, 3246961, 8296441, 20426971, 52715563, 131480623, 342491253, 864759837, 2270860455, 5793103989, 15316065497, 39429185075, 105008858223
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 21 2024

Keywords

Crossrefs

Cf. A000583, A007576, A158465, A350249.

Programs

  • Mathematica
    Table[Coefficient[Product[x^(k^4) + 1 + 1/x^(k^4), {k, 1, n}], x, 0], {n, 0, 25}]

Extensions

a(26)-a(37) from Alois P. Heinz, Jan 21 2024

A007575 Number of stable towers of 2 X 2 LEGO blocks.

Original entry on oeis.org

1, 3, 7, 19, 53, 149, 419, 1191, 3403, 9755, 28077, 81097, 234861, 681697, 1982723, 5777375, 16861521, 49281525, 144222987, 422566835, 1239423303, 3638872529, 10693065215, 31448140529, 92558787745, 272612601065, 803448576111
Offset: 0

Views

Author

Simon Plouffe and N. J. A. Sloane, Robert G. Wilson v

Keywords

References

  • Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • P. J. S. Watson, On "LEGO" towers, J. Rec. Math., 12 (No. 1, 1979-1980), 24-27.

Links

Crossrefs

Cf. A007576.

Programs

  • Maple
    seq(sum(coeff(product(1+x^k+x^(2*k),k=1..n),x,l),l=n*(n+1)/2-n..n*(n+1)/2+n),n=0..20); # Søren Eilers
  • Mathematica
    Array[Sum[SeriesCoefficient[Product[1 + x^k + x^(2 k), {k, #}], {x, 0, j}], {j, # (# + 1)/2 - #, # (# + 1)/2 + #}] &, 27, 0] (* Michael De Vlieger, Feb 24 2020, after Maple *)

Formula

a(n) ~ 3^(n+1) / sqrt(Pi*n). - Vaclav Kotesovec, Jul 11 2018

A369343 a(n) is the constant term in expansion of Product_{k=1..n} (x^(2*k-1) + 1 + 1/x^(2*k-1)).

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 21, 49, 117, 295, 761, 1993, 5261, 14025, 37699, 102151, 278587, 764145, 2106433, 5832863, 16217191, 45255167, 126708863, 355848715, 1002145705, 2829479797, 8007670701, 22711890561, 64547494347, 183790615881, 524239904367, 1497786769295
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 20 2024

Keywords

Comments

All terms are odd.

Links

Crossrefs

Cf. A005408, A007576, A156700, A292476.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n>i^2, 0, `if`(i=0, 1,
      b(n, i-1)+b(n+2*i-1, i-1)+b(abs(n-2*i+1), i-1)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 21 2024
  • Mathematica
    Table[Coefficient[Product[x^(2 k - 1) + 1 + 1/x^(2 k - 1), {k, 1, n}], x, 0], {n, 0, 30}]

Formula

a(n) ~ 3^(n+1) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jan 21 2024

A369734 Number of solutions to 1^2*k_1 + 2^2*k_2 + ... + n^2*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 8, 17, 35, 79, 177, 409, 995, 2475, 6336, 16078, 41401, 107304, 279550, 736032, 1950000, 5199850, 13950852, 37576658, 101670863, 276228026, 753114256, 2060995699, 5658663542, 15583477334, 43039552072, 119179072495, 330836308272, 920537185436
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A007576, A063866, A350249, A369433, A369628, A369735.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^(k^2) + 1 + 1/x^(k^2)), {k, 1, n}], x, 1], {n, 0, 32}]

Formula

a(n) = [x^1] Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

A369735 Number of solutions to 1^3*k_1 + 2^3*k_2 + ... + n^3*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 3, 4, 6, 15, 28, 56, 125, 287, 646, 1540, 3625, 8484, 21167, 51458, 126342, 323126, 811538, 2052501, 5339265, 13751212, 35589866, 94032931, 246791641, 650227636, 1739032299, 4630165425, 12373805281, 33429284691, 90073865814, 243460560324
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A007576, A063866, A369345, A369437, A369628, A369734.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n>(i*(i+1)/2)^2, 0,
     `if`(i=0, 1, b(n, i-1)+b(n+i^3, i-1)+b(abs(n-i^3), i-1)))
    end:
a:= n-> b(1, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 30 2024
  • Mathematica
    Table[Coefficient[Product[(x^(k^3) + 1 + 1/x^(k^3)), {k, 1, n}], x, 1], {n, 0, 33}]

Formula

a(n) = [x^1] Product_{k=1..n} (x^(k^3) + 1 + 1/x^(k^3)).

A369874 a(n) is the constant term in the expansion of Product_{d|n} (x^d + 1 + 1/x^d).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 13, 1, 13, 1, 1, 1, 103, 1, 1, 1, 7, 1, 77, 1, 1, 1, 1, 1, 175, 1, 1, 1, 63, 1, 49, 1, 1, 5, 1, 1, 463, 1, 1, 1, 1, 1, 41, 1, 39, 1, 1, 1, 2975, 1, 1, 3, 1, 1, 33, 1, 1, 1, 25, 1, 2363, 1, 1, 1, 1, 1, 25, 1, 261
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 03 2024

Keywords

Comments

a(n) is the number of solutions to 0 = Sum_{d|n} c_i * d with c_i in {-1,0,1}, i=1..tau(n), tau = A000005.

Links

Crossrefs

Cf. A000005, A007576, A033630, A083206, A369873.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^d + 1 + 1/x^d), {d, Divisors[n]}], x, 0], {n, 1, 80}]
  • Python
    from collections import Counter
from sympy import divisors
def A369874(n):
    c = {0:1}
    for d in divisors(n,generator=True):
        b = Counter(c)
        for j in c:
            a = c[j]
            b[j+d] += a
            b[j-d] += a
        c = b
    return c[0] # Chai Wah Wu, Feb 05 2024

A369372 a(n) is the constant term in expansion of Product_{k=1..n} (x^k + 1 + 1/x^k)^3.

Original entry on oeis.org

1, 7, 85, 1437, 26707, 534513, 11255951, 245612031, 5503639327, 125900330437, 2928092906281, 69026845135479, 1645689594867257, 39611576627651927, 961279033420170871, 23494000801494204647, 577777092945262623161, 14287061769367391787065, 355010279665452190629001
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 21 2024

Keywords

Crossrefs

Cf. A007576, A124995, A156181, A369373.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^k + 1 + 1/x^k)^3, {k, 1, n}], x, 0], {n, 0, 18}]
  • PARI
    a(n) = polcoef(prod(k=1, n, (x^k + 1 + 1/x^k)^3), 0); \\ Michel Marcus, Jan 22 2024

A369373 a(n) is the constant term in expansion of Product_{k=1..n} (x^k + 1 + 1/x^k)^4.

Original entry on oeis.org

1, 19, 701, 33873, 1884211, 113091013, 7138569079, 466998324373, 31378587089717, 2152644125539205, 150149036955370989, 10616242785424087153, 759159709650751045807, 54809160248598728775119, 3989668904561505824038609, 292488794939698331845055779, 21576667915867159070829849217
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 21 2024

Keywords

Crossrefs

Cf. A007576, A124996, A156181, A369372.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^k + 1 + 1/x^k)^4, {k, 1, n}], x, 0], {n, 0, 16}]

A369714 Number of solutions to 1^4*k_1 + 2^4*k_2 + ... + n^4*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 13, 25, 56, 110, 218, 494, 1216, 2702, 6477, 14752, 35758, 83730, 208107, 499459, 1250815, 3048590, 7787399, 19260830, 49686365, 124430675, 324018684, 820906005, 2155194085, 5514650519, 14578030389, 37630395887, 100201473164
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A007576, A063866, A368478, A369358, A369628, A369734, A369735.

Programs

  • Maple
    b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1, b(n, i-1)+
      b(abs(n-i^4), i-1)+b(n+i^4, i-1))))(i*(i+1)*(2*i+1)*(3*i^2+3*i-1)/30)
    end:
a:= n-> b(1, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 30 2024
  • Mathematica
    b[n_, i_] := b[n, i] = Function[m, If[n > m, 0, If[n == m, 1, b[n, i-1] + b[Abs[n-i^4], i-1] + b[n+i^4, i-1]]]][i*(i+1)*(2*i+1)*(3*i^2 + 3*i-1)/30];
a[n_] := b[1, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jan 27 2025, after Alois P. Heinz *)

Formula

a(n) = [x^1] Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).

Extensions

a(34)-a(37) from Alois P. Heinz, Jan 30 2024
Previous Showing 11-20 of 21 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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