A080827 - OEIS

%I #68 Jan 29 2024 09:02:10

%S 1,3,5,9,13,19,25,33,41,51,61,73,85,99,113,129,145,163,181,201,221,

%T 243,265,289,313,339,365,393,421,451,481,513,545,579,613,649,685,723,

%U 761,801,841,883,925,969,1013,1059,1105,1153,1201,1251,1301,1353,1405,1459

%N Rounded up staircase on natural numbers.

%C Represents the 'rounded up' staircase diagonal on A000027, arranged as a square array. A000982 is the 'rounded down' staircase.

%C a(1)= 1, a(2n) = a(2n-1) + 2n, a(2n+1) = a(2n) +2n. - _Amarnath Murthy_, May 07 2003

%C Partial sums of A131055. - _Paul Barry_, Jun 14 2008

%C The same sequence arises in the triangular array of integers >= 1 according to a simple "zig-zag" rule for selection of terms. a(n-1) lies in the (n-1)-th row of the array and the second row of that subarray (with apex a(n-1)) contains just two numbers, one odd one even. The one with the same (odd) parity as a(n-1) is a(n). - _David James Sycamore_, Jul 29 2018

%H Vincenzo Librandi, <a href="/A080827/b080827.txt">Table of n, a(n) for n = 1..1000</a>

%H J. C. F. de Winter, <a href="http://pareonline.net/getvn.asp?v=18&amp;n=10">Using the Student's t-test with extremely small sample sizes</a>, Practical Assessment, Research & Evaluation, 18(10), 2013.

%H Girtrude Hamm, <a href="https://arxiv.org/abs/2304.03007">Classification of lattice triangles by their two smallest widths</a>, arXiv:2304.03007 [math.CO], 2023.

%H David James Sycamore, <a href="/A080827/a080827.jpg">Triangular array</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%F a(n) = ceiling((n^2+1)/2).

%F From _Paul Barry_, Apr 12 2008: (Start)

%F G.f.: x*(1+x-x^2+x^3)/((1+x)(1-x)^3).

%F a(n) = n*(n+1)/2-floor((n-1)/2). [corrected by _R. J. Mathar_, Jul 14 2013] (End)

%F From _Wesley Ivan Hurt_, Sep 08 2015: (Start)

%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4.

%F a(n) = (n^2 + 2 - (1 - (-1)^n)/2)/2.

%F a(n) = floor(n^2/2) + 1 = A007590(n-1) + 1. (End)

%F Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/2 + coth(Pi/sqrt(2))*Pi/(2*sqrt(2)) - 1/2. - _Amiram Eldar_, Sep 15 2022

%F E.g.f.: ((2 + x + x^2)*cosh(x) + (1 + x + x^2)*sinh(x) - 2)/2. - _Stefano Spezia_, Jan 27 2024

%p A080827:=n->(n^2+2-(1-(-1)^n)/2)/2: seq(A080827(n), n=1..100); # _Wesley Ivan Hurt_, Sep 08 2015

%t s1=0;lst={};Do[s1+=n;If[EvenQ[s1],s1-=1];AppendTo[lst,s1],{n,6!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jun 06 2009 *)

%t CoefficientList[Series[(1 + x - x^2 + x^3) / ((1 + x) (1 - x)^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 05 2013 *)

%o (Magma) [n*(n+1)/2-Floor((n-1)/2) : n in [1..60]]; // _Vincenzo Librandi_, Aug 05 2013

%o (GAP) List([1..10],n->Int(n^2/2)+1); # _Muniru A Asiru_, Aug 02 2018

%Y Apart from leading term identical to A099392.

%Y Cf. A000027, A000982, A007590, A131055.

%K nonn,easy

%O 1,2

%A _Paul Barry_, Feb 28 2003