A008586 - OEIS

%I #153 Mar 14 2023 13:52:34

%S 0,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,

%T 96,100,104,108,112,116,120,124,128,132,136,140,144,148,152,156,160,

%U 164,168,172,176,180,184,188,192,196,200,204,208,212,216,220,224,228

%N Multiples of 4.

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

%C A000466(n), a(n) and A053755(n) are Pythagorean triples. - _Zak Seidov_, Jan 16 2007

%C 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

%C 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

%C For n > 0: numbers having more even than odd divisors: A048272(a(n)) < 0. - _Reinhard Zumkeller_, Jan 21 2012

%C A214546(a(n)) < 0 for n > 0. - _Reinhard Zumkeller_, Jul 20 2012

%C A090418(a(n)) = 0 for n > 0. - _Reinhard Zumkeller_, Aug 06 2012

%C Terms are the differences of consecutive centered square numbers (A001844). - _Mihir Mathur_, Apr 02 2013

%C 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

%C 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

%C 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

%C First differences of A001844. - _Robert Price_, May 13 2016

%C Numbers k such that Fibonacci(k) is a multiple of 3 (A033888). - _Bruno Berselli_, Oct 17 2017

%H T. D. Noe, <a href="/A008586/b008586.txt">Table of n, a(n) for n = 0..1000</a>

%H Tom M. Apostol, <a href="https://archive.org/details/introductiontoan00apos_0/page/2/mode/2up">Introduction to Analytic Number Theory</a>, Springer-Verlag, 1976, page 3.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=316">Encyclopedia of Combinatorial Structures 316</a> [Broken link]

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014, 2015.

%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DoublyEvenNumber.html">Doubly Even Number</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F a(n) = A008574(n), n>0. - _R. J. Mathar_, Oct 28 2008

%F a(n) = Sum_{k>=0} A030308(n,k)*2^(k+2). - _Philippe Deléham_, Oct 17 2011

%F a(n+1) = A000290(n+2) - A000290(n). - _Philippe Deléham_, Mar 31 2013

%F G.f.: 4*x/(1-x)^2. - _David Wilding_, Jun 21 2014

%F E.g.f.: 4*x*exp(x). - _Stefano Spezia_, May 18 2021

%p A008586:=n->4*n; seq(A008586(n), n=0..100); # _Wesley Ivan Hurt_, Feb 24 2014

%t Range[0, 500, 4] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)

%o (PARI) a(n)=n<<2 \\ _Charles R Greathouse IV_, Oct 17 2011

%o (Haskell)

%o a008586 = (* 4)

%o a008586_list = [0, 4 ..]  -- _Reinhard Zumkeller_, May 13 2014

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

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_