A171769 Partial sums of A042964 (numbers congruent to 2 or 3 mod 4).

2, 5, 11, 18, 28, 39, 53, 68, 86, 105, 127, 150, 176, 203, 233, 264, 298, 333, 371, 410, 452, 495, 541, 588, 638, 689, 743, 798, 856, 915, 977, 1040, 1106, 1173, 1243, 1314, 1388, 1463, 1541, 1620, 1702, 1785, 1871, 1958, 2048, 2139, 2233, 2328, 2426, 2525

Offset: 1

Author

Jaroslav Krizek, Dec 18 2009

Keywords

• nonn
• easy

Comments

If we insert an initial 0, and alternate the signs: 0,2,-5,11,-18,28,..., we get a sequence where the average of the first n terms is an integer, with no repeats: specifically A001057(n-1). The sum of the first n terms is (-1)^(n-1)*A093353(n-1). - Franklin T. Adams-Watters, May 20 2010

Suppose that n cards have the numbers 1..2n written on them randomly, one number to a side, and are set out on a table randomly. You have the task of maximizing the sum of the visible numbers by flipping cards. If you have no information other than the numbers on the upper faces, and may not flip any particular card more than once, a(n) is the largest sum you can guarantee in the worst case. - Andrew Woods, Jun 06 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A042964, A001057, A093353. - Franklin T. Adams-Watters, May 20 2010

Programs

GAP

a:=[2,5,11,18];; for n in [5..60] do a[n]:=2*a[n-1]-2*a[n-3] + a[n-4]; od; a; # G. C. Greubel, Jul 02 2019

Magma

[Ceiling((2*n+1)*n/2): n in [1..60]]; // Vincenzo Librandi, Jul 02 2019

Mathematica

a[n_]:=Ceiling[((2n+1)n/2)]; Array[a, 60] (* Vincenzo Librandi, Jul 02 2019 *)
LinearRecurrence[{2,0,-2,1}, {2,5,11,18}, 60] (* G. C. Greubel, Jul 02 2019 *)

PARI

Vec(x*(x^2+x+2)/((1-x)^3*(x+1)) + O(x^60)) \\ Colin Barker, Jun 04 2014

Sage

[ceiling(n*(1+2*n)/2) for n in (1..60)] # G. C. Greubel, Jul 02 2019

Formula

a(n) = Sum_{i=1..n} A042964(i).

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - R. H. Hardin, Nov 13 2011

a(n) = ceiling((2*n+1)*n/2). - Andrew Woods, Jun 06 2013

G.f.: x*(2+x+x^2) / ((1-x)^3*(x+1)). - Colin Barker, Jun 04 2014

a(n) = round(n/(1-exp(-1/n))). - Richard R. Forberg, Jan 28 2015