OFFSET
1,2
COMMENTS
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
LINKS
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.
Girtrude Hamm, Classification of lattice triangles by their two smallest widths, arXiv:2304.03007 [math.CO], 2023.
David James Sycamore, Triangular array.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
a(n) = ceiling((n^2+1)/2).
From Paul Barry, Apr 12 2008: (Start)
G.f.: x*(1+x-x^2+x^3)/((1+x)(1-x)^3).
a(n) = n*(n+1)/2-floor((n-1)/2). [corrected by R. J. Mathar, Jul 14 2013] (End)
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
E.g.f.: ((2 + x + x^2)*cosh(x) + (1 + x + x^2)*sinh(x) - 2)/2. - Stefano Spezia, Jan 27 2024
MAPLE
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Feb 28 2003
STATUS
approved