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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A301565 Expansion of Product_{k>=0} (1 + x^(5*k+3))*(1 + x^(5*k+4)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 2, 3, 2, 1, 2, 4, 4, 3, 3, 4, 6, 6, 4, 4, 7, 9, 7, 6, 8, 11, 12, 10, 9, 12, 16, 16, 14, 14, 19, 23, 22, 19, 21, 27, 31, 29, 26, 31, 40, 42, 38, 38, 45, 53, 55, 51, 52, 63, 73, 73, 69, 73, 87, 97, 95, 91, 100, 118, 128
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 3 or 4 mod 5.

Examples

			a(17) = 3 because we have [14, 3], [13, 4] and [9, 8].

Links

Crossrefs

Cf. A035374, A047204, A107236, A107237, A203776, A219607, A281243, A281271, A301562, A301563, A301564, A301567, A301568, A301569, A301570.

Programs

  • Mathematica
    nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k + 3)) (1 + x^(5 k + 4)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[QPochhammer[-x^3, x^5] QPochhammer[-x^4, x^5], {x, 0, nmax}], x]
nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{3, 4}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A047204(k)).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(33/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A268044 The odd numbers congruent to {3, 4} mod 5.

Original entry on oeis.org

3, 9, 13, 19, 23, 29, 33, 39, 43, 49, 53, 59, 63, 69, 73, 79, 83, 89, 93, 99, 103, 109, 113, 119, 123, 129, 133, 139, 143, 149, 153, 159, 163, 169, 173, 179, 183, 189, 193, 199, 203, 209, 213, 219, 223, 229, 233, 239, 243, 249, 253, 259, 263, 269, 273, 279, 283, 289, 293, 299
Offset: 1

Views

Author

Mikk Heidemaa, Jan 25 2016

Keywords

Comments

The odd numbers with terminal digit 3 or 9.

Links

Crossrefs

Cf. A001622, A005408, A045435, A047204, A131229.
Second bisection of A045572.

Programs

  • Magma
    [5*n-(3-(-1)^n)/2: n in [1..60]]; // Vincenzo Librandi, Jan 25 2016
  • Mathematica
    Table[5 n - (3 - (-1)^n)/2, {n, 1000}] (* or *) Select[ Range [1000], OddQ[#] && MemberQ[{3, 4}, Mod[#, 5]] &]
LinearRecurrence[{1,1,-1},{3,9,13},60] (* Harvey P. Dale, Feb 12 2023 *)

Formula

G.f.: x*(3 + 6*x + x^2)/((1 + x)*(1 - x)^2).
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = 5*n - (3 - (-1)^n)/2.
a(n) = -A131229(-n+1) with A131229(0) = -3.
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt((5+sqrt(5))/2)*Pi/10 - 3*log(phi)/(2*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

Extensions

Edited by Bruno Berselli, Jan 25 2016
Previous Showing 21-22 of 22 results.

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

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