A008586 Multiples of 4.

0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 14 ).

A000466(n), a(n) and A053755(n) are Pythagorean triples. - Zak Seidov, Jan 16 2007

If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-3) is equal to the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007

Number of n-permutations (n>=1) of 5 objects u, v, z, x, y with repetition allowed, containing n-1 u's. Example: if n=1 then n-1 = zero (0) u, a(1)=4 because we have v, z, x, y. If n=2 then n-1 = one (1) u, a(2)=8 because we have vu, zu, xu, yu, uv, uz, ux, uy. A038231 formatted as a triangular array: diagonal: 4, 8, 12, 16, 20, 24, 28, 32, ... - Zerinvary Lajos, Aug 06 2008

For n > 0: numbers having more even than odd divisors: A048272(a(n)) < 0. - Reinhard Zumkeller, Jan 21 2012

A214546(a(n)) < 0 for n > 0. - Reinhard Zumkeller, Jul 20 2012

A090418(a(n)) = 0 for n > 0. - Reinhard Zumkeller, Aug 06 2012

Terms are the differences of consecutive centered square numbers (A001844). - Mihir Mathur, Apr 02 2013

a(n)*Pi = nonnegative zeros of the cycloid generated by a circle of radius 2 rolling along the positive x-axis from zero. - Wesley Ivan Hurt, Jul 01 2013

Apart from the initial term, number of vertices of minimal path on an n-dimensional cubic lattice (n>1) of side length 2, until a self-avoiding walk gets stuck. A004767 + 1. - Matthew Lehman, Dec 23 2013

The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 2688. - Philippe A.J.G. Chevalier, Dec 29 2015

First differences of A001844. - Robert Price, May 13 2016

Numbers k such that Fibonacci(k) is a multiple of 3 (A033888). - Bruno Berselli, Oct 17 2017

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 316 [Broken link]
• Milan Janjic, Two Enumerative Functions
• Tanya Khovanova, Recursive Sequences
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014, 2015.
• William A. Stein, The modular forms database
• Eric Weisstein's World of Mathematics, Doubly Even Number
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A000290, A000466, A001844, A004767, A008574, A030308, A033888, A035008, A038231, A048272, A053755, A090418, A214546.

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008585, A005843, A001477, A000217.

Programs

Haskell

a008586 = (* 4)
a008586_list = [0, 4 ..]  -- Reinhard Zumkeller, May 13 2014

Maple

A008586:=n->4*n; seq(A008586(n), n=0..100); # Wesley Ivan Hurt, Feb 24 2014

Mathematica

Range[0, 500, 4] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)

PARI

a(n)=n<<2 \\ Charles R Greathouse IV, Oct 17 2011

Formula

a(n) = A008574(n), n>0. - R. J. Mathar, Oct 28 2008

a(n) = Sum_{k>=0} A030308(n,k)*2^(k+2). - Philippe Deléham, Oct 17 2011

a(n+1) = A000290(n+2) - A000290(n). - Philippe Deléham, Mar 31 2013

G.f.: 4*x/(1-x)^2. - David Wilding, Jun 21 2014

E.g.f.: 4*x*exp(x). - Stefano Spezia, May 18 2021