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A080827 Rounded up staircase on natural numbers.

%I A080827 #76 Jul 19 2025 18:23:58
%S A080827 1,3,5,9,13,19,25,33,41,51,61,73,85,99,113,129,145,163,181,201,221,
%T A080827 243,265,289,313,339,365,393,421,451,481,513,545,579,613,649,685,723,
%U A080827 761,801,841,883,925,969,1013,1059,1105,1153,1201,1251,1301,1353,1405,1459
%N A080827 Rounded up staircase on natural numbers.
%C A080827 Represents the 'rounded up' staircase diagonal on A000027, arranged as a square array. A000982 is the 'rounded down' staircase.
%C A080827 Partial sums of A131055. - _Paul Barry_, Jun 14 2008
%C A080827 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 A080827 Vincenzo Librandi, <a href="/A080827/b080827.txt">Table of n, a(n) for n = 1..1000</a>
%H A080827 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 A080827 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 A080827 David James Sycamore, <a href="/A080827/a080827.jpg">Triangular array</a>.
%H A080827 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A080827 a(n) = ceiling((n^2+1)/2).
%F A080827 a(1) = 1, a(2n) = a(2n-1) + 2n, a(2n+1) = a(2n) + 2n. - _Amarnath Murthy_, May 07 2003
%F A080827 From _Paul Barry_, Apr 12 2008: (Start)
%F A080827 G.f.: x*(1+x-x^2+x^3)/((1+x)(1-x)^3).
%F A080827 a(n) = n*(n+1)/2-floor((n-1)/2). [corrected by _R. J. Mathar_, Jul 14 2013] (End)
%F A080827 From _Wesley Ivan Hurt_, Sep 08 2015: (Start)
%F A080827 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4.
%F A080827 a(n) = (n^2 + 2 - (1 - (-1)^n)/2)/2.
%F A080827 a(n) = floor(n^2/2) + 1 = A007590(n-1) + 1. (End)
%F A080827 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 A080827 E.g.f.: ((2 + x + x^2)*cosh(x) + (1 + x + x^2)*sinh(x) - 2)/2. - _Stefano Spezia_, Jan 27 2024
%p A080827 A080827:=n->(n^2+2-(1-(-1)^n)/2)/2: seq(A080827(n), n=1..100); # _Wesley Ivan Hurt_, Sep 08 2015
%t A080827 CoefficientList[Series[(1 + x - x^2 + x^3) / ((1 + x) (1 - x)^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 05 2013 *)
%t A080827 Ceiling[(Range[100]^2 + 1)/2] (* _Paolo Xausa_, Jul 19 2025 *)
%o A080827 (Magma) [n*(n+1)/2-Floor((n-1)/2) : n in [1..60]]; // _Vincenzo Librandi_, Aug 05 2013
%o A080827 (GAP) List([1..10],n->Int(n^2/2)+1); # _Muniru A Asiru_, Aug 02 2018
%Y A080827 Apart from leading term identical to A099392.
%Y A080827 Cf. A000027, A000982, A007590, A131055.
%K A080827 nonn,easy
%O A080827 1,2
%A A080827 _Paul Barry_, Feb 28 2003