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 1-5 of 5 results.

A350249 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 7, 19, 43, 95, 189, 429, 1003, 2457, 6319, 16165, 41601, 107969, 280253, 737065, 1950865, 5201941, 13954313, 37593679, 101695957, 276296549, 753191093, 2061201397, 5658850121, 15583938539, 43040609115, 119182143639, 330841253283, 920550527585
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 28 2022

Keywords

Links

Crossrefs

Cf. A007576, A158092, A350495, A350880.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1,
      expand((x^(n^2)+1+1/x^(n^2))*b(n-1)))
    end:
a:= n-> coeff(b(n),x,0):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 28 2022
  • Mathematica
    Table[Coefficient[Product[x^(k^2) + 1 + 1/x^(k^2), {k, 1, n}], x, 0], {n, 0, 30}] (* Vaclav Kotesovec, Feb 05 2022 *)

Formula

Conjecture: a(n) ~ sqrt(5) * 3^(n + 1/2) / (2*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Feb 04 2022

A369390 a(n) = [x^prime(n)] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).

Original entry on oeis.org

1, 1, 2, 4, 6, 13, 31, 77, 188, 449, 1191, 3014, 7920, 21498, 57833, 154073, 412733, 1141274, 3106771, 8576977, 24015471, 66489615, 185886699, 517837152, 1435964205, 4034697191, 11438332340, 32395341851, 92396549863, 263233759500, 736127855014, 2093027604453
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 22 2024

Keywords

Links

Crossrefs

Cf. A063890, A316706, A350880.

Programs

  • Maple
    s:= proc(n) s(n):= `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      b(n, i-1)+b(n+ithprime(i), i-1)+b(abs(n-ithprime(i)), i-1)))
    end:
a:= n-> b(ithprime(n), n):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 22 2024
  • Mathematica
    Table[Coefficient[Product[(x^Prime[k] + 1 + 1/x^Prime[k]), {k, 1, n}], x, Prime[n]], {n, 1, 32}]

A369560 a(n) = [x^n] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 16, 38, 91, 225, 547, 1407, 3570, 9250, 24578, 65740, 175626, 470084, 1279101, 3482419, 9547953, 26445796, 73251187, 203818706, 567543095, 1577629707, 4408095456, 12400615844, 34995570604, 99241500366, 282037360250, 795846583187
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 25 2024

Keywords

Comments

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

Crossrefs

Cf. A000040, A022894, A113040, A316706, A350404, A350695, A350880, A369390.

Programs

  • Maple
    s:= proc(n) s(n):= `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      b(n, i-1)+b(n+ithprime(i), i-1)+b(abs(n-ithprime(i)), i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Jan 25 2024
  • Mathematica
    Table[Coefficient[Product[x^Prime[k] + 1 + 1/x^Prime[k], {k, 1, n}], x, n], {n, 0, 31}]

A369388 a(n) is the constant term in expansion of Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k))^2.

Original entry on oeis.org

1, 3, 9, 45, 249, 1373, 9177, 62257, 453179, 3320531, 24087877, 183643865, 1394580343, 10794949627, 85730722969, 686171829489, 5487361175591, 43981108061647, 358362244544957, 2922625435214613, 24006575088945973, 199229783030494775, 1653790732247194785
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2024

Keywords

Crossrefs

Cf. A047653, A156181, A350880, A350881.

Programs

  • Mathematica
    Table[Coefficient[Product[(x^Prime[k] + 1 + 1/x^Prime[k])^2, {k, 1, n}], x, 0], {n, 0, 22}]

A369733 Number of solutions to 2*k_1 + 3*k_2 + ... + prime(n)*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

Original entry on oeis.org

0, 0, 1, 1, 3, 8, 18, 39, 95, 233, 565, 1440, 3640, 9409, 24957, 66699, 177931, 475584, 1292985, 3517163, 9636135, 26675682, 73847316, 205379443, 571621138, 1588186858, 4435778209, 12474517743, 35194193531, 99781874834, 283513309423, 799779819641
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2024

Keywords

Crossrefs

Cf. A007576, A063866, A113040, A350880, A369560, A369628.

Programs

  • Maple
    s:= proc(n) s(n):= `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      b(n, i-1)+b(n+ithprime(i), i-1)+b(abs(n-ithprime(i)), i-1)))
    end:
a:= n-> b(1, n):
seq(a(n), n=0..32);  # Alois P. Heinz, Jan 30 2024
  • Mathematica
    Table[Coefficient[Product[(x^Prime[k] + 1 + 1/x^Prime[k]), {k, 1, n}], x, 1], {n, 0, 31}]

Formula

a(n) = [x^1] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).
Showing 1-5 of 5 results.

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