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A080827
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Rounded up staircase on natural numbers.
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13
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1, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405, 1459
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OFFSET
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1,2
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COMMENTS
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Represents the 'rounded up' staircase diagonal on A000027, arranged as a square array. A000982 is the 'rounded down' staircase.
a(1)= 1, a(2n) = a(2n-1) + 2n, a(2n+1) = a(2n) +2n. - Amarnath Murthy, May 07 2003
Partial sums of A131055. - Paul Barry, Jun 14 2008
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
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. C. F. de Winter, Using the Student's t-test with extremely small sample sizes, Practical Assessment, Research & Evaluation, 18(10), 2013.
David James Sycamore, Triangular array.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
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FORMULA
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a(n) = ceiling((n^2+1)/2).
G.f.: x*(1+x-x^2+x^3)/((1+x)(1-x)^3); a(n) = n*(n+1)/2-floor((n-1)/2). - Paul Barry, Apr 12 2008 [corrected by R. J. Mathar, Jul 14 2013]
From Wesley Ivan Hurt, Sep 08 2015: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4.
a(n) = (n^2 + 2 - (1 - (-1)^n)/2)/2.
a(n) = floor(n^2/2) + 1 = A007590(n-1) + 1. (End)
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
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MAPLE
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A080827:=n->(n^2+2-(1-(-1)^n)/2)/2: seq(A080827(n), n=1..100); # Wesley Ivan Hurt, Sep 08 2015
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MATHEMATICA
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s1=0; lst={}; Do[s1+=n; If[EvenQ[s1], s1-=1]; AppendTo[lst, s1], {n, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 06 2009 *)
CoefficientList[Series[(1 + x - x^2 + x^3) / ((1 + x) (1 - x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 05 2013 *)
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PROG
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(Magma) [n*(n+1)/2-Floor((n-1)/2) : n in [1..60]]; // Vincenzo Librandi, Aug 05 2013
(GAP) List([1..10], n->Int(n^2/2)+1); # Muniru A Asiru, Aug 02 2018
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CROSSREFS
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Apart from leading term identical to A099392.
Cf. A000027, A000982, A007590, A131055.
Sequence in context: A118028 A209974 A099392 * A200919 A213207 A102378
Adjacent sequences: A080824 A080825 A080826 * A080828 A080829 A080830
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KEYWORD
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nonn,easy
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AUTHOR
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Paul Barry, Feb 28 2003
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STATUS
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approved
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