This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A093005 #105 Jul 07 2024 07:58:22
%S A093005 1,2,6,8,15,18,28,32,45,50,66,72,91,98,120,128,153,162,190,200,231,
%T A093005 242,276,288,325,338,378,392,435,450,496,512,561,578,630,648,703,722,
%U A093005 780,800,861,882,946,968,1035,1058,1128,1152,1225,1250,1326,1352,1431,1458
%N A093005 a(n) = n * ceiling(n/2).
%C A093005 Old name was: The lone multiple of n among the next n numbers.
%C A093005 Another old name: a(n) = n*floor((n+1)/2).
%C A093005 Consider the triangle
%C A093005 1
%C A093005 2 3
%C A093005 4 5 6
%C A093005 7 8 9 10
%C A093005 11 12 13 14 15
%C A093005 16 17 18 19 20 21
%C A093005 22 23 24 25 26 27 28
%C A093005 ... Then sequence contains the multiple of n in the n-th row.
%C A093005 Interleaves A000384 and A001105. - _Paul Barry_, Jun 29 2006
%C A093005 Termwise products of the natural numbers and the natural numbers repeated.
%C A093005 Number of pairs (x,y) having the same parity, with x in {0,...,n} and y in {0,...,2n}. - _Clark Kimberling_, Jul 02 2012
%C A093005 Similar to generalized hexagonal numbers A000217. Other members of this family are A210977, A006578, A210978, A181995, A210981, A210982. - _Omar E. Pol_, Aug 09 2012
%C A093005 For even n, a(n) gives the sum of all the parts in the partitions of n into exactly two parts. For odd n>1, a(n) gives n plus the sum of all the parts in the partitions of n into exactly two parts. - _Wesley Ivan Hurt_, Nov 14 2013
%C A093005 Number of regions of the plane that do not contain the origin, in the arrangement of lines with polar equations rho = 1/cos(theta-k*2*Pi/n), k=0..n-1; or, by inversion, number of bounded regions in the arrangement of circles with radius 1 and centers the n-th roots of unity. - _Luc Rousseau_, Feb 08 2019
%C A093005 Numbers k such that floor(sqrt(2k)+1/2) | k. - _Wesley Ivan Hurt_, Dec 01 2020
%H A093005 Harvey P. Dale, <a href="/A093005/b093005.txt">Table of n, a(n) for n = 1..1000</a>
%H A093005 Torleiv Kløve, <a href="https://doi.org/10.7146/math.scand.a-11464">Linear recurring sequences in boolean rings</a>, Math. Scand., 33 (1973), 5-12.
%H A093005 Torleiv Kløve, <a href="/A005984/a005984.pdf">Linear recurring sequences in boolean rings</a>, Math. Scand., 33 (1973), 5-12. (Annotated scanned copy)
%H A093005 Luc Rousseau, <a href="/A093005/a093005.png">Illustration, a(n) viewed as a number of regions in an arrangement of lines / of circles</a>.
%H A093005 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A093005 a(n) = n*floor((n+1)/2).
%F A093005 a(n) = n*A008619(n).
%F A093005 a(2*n-1) = n*(2*n-1), a(2*n) = 2*n^2.
%F A093005 From _Paul Barry_, Jun 29 2006: (Start)
%F A093005 G.f.: x*(1+x+2*x^2)/((1+x)^2*(1-x)^3).
%F A093005 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F A093005 a(n) = n*((2*n+1) - (-1)^n)/4. (End)
%F A093005 a(n) = n * ceiling(n/2). - _Wesley Ivan Hurt_, Nov 14 2013
%F A093005 E.g.f.: (1/2)*x*( (x+2)*cosh(x) + (x+1)*sinh(x) ). - _G. C. Greubel_, Mar 14 2024
%F A093005 Sum_{n>=1} 1/a(n) = Pi^2/12 + 2*log(2). - _Amiram Eldar_, Mar 15 2024
%F A093005 a(n) = A183207(n) - A370980(n) + 1, by Euler's formula. - _Scott R. Shannon_, Jul 07 2024
%p A093005 A093005:=n->n*ceil(n/2); seq(A093005(n), n=1..100); # _Wesley Ivan Hurt_, Nov 14 2013
%t A093005 a[n_Integer] := n*Floor[(n + 1)/2] (* _Olivier Gérard_, Jun 21 2007 *)
%t A093005 Table[n*Ceiling[n/2],{n,60}] (* or *) LinearRecurrence[{1,2,-2,-1,1},{1,2,6,8,15},60] (* _Harvey P. Dale_, May 08 2014 *)
%o A093005 (Python)
%o A093005 for n in range(1,55):
%o A093005 print(n*((n+1)//2), end=",") # _Alex Ratushnyak_, Apr 26 2012
%o A093005 (PARI) a(n)=(n+1)\2*n \\ _Charles R Greathouse IV_, Jun 11 2015
%o A093005 (Magma) [n*(n+(n mod 2))/2: n in [1..70]]; // _G. C. Greubel_, Mar 14 2024
%o A093005 (SageMath) [n*(n +(n%2))/2 for n in range(1,71)] # _G. C. Greubel_, Mar 14 2024
%Y A093005 Cf. A000217, A000384, A001105, A006578, A008619 (a(n)/n), A181995.
%Y A093005 Cf. A183207, A210977, A210978, A210981, A210982, A370980.
%K A093005 nonn,easy
%O A093005 1,2
%A A093005 _Amarnath Murthy_, Mar 29 2004
%E A093005 Corrected and extended by _Joshua Zucker_, May 08 2006
%E A093005 New name from _Alex Ratushnyak_, Apr 26 2012
%E A093005 New name from _Wesley Ivan Hurt_, Nov 14 2013