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 A035106 #109 Jan 03 2025 02:19:59
%S A035106 1,2,3,6,8,12,15,20,24,30,35,42,48,56,63,72,80,90,99,110,120,132,143,
%T A035106 156,168,182,195,210,224,240,255,272,288,306,323,342,360,380,399,420,
%U A035106 440,462,483,506,528,552,575,600,624,650,675,702,728,756,783,812,840
%N A035106 1, together with numbers of the form k*(k+1) or k*(k+2), k > 0.
%C A035106 Largest integer m such that every permutation (p_1, ..., p_n) of (1, ..., n) satisfies p_i * p_{i+1} >= m for some i, 1 <= i <= n-1. Equivalently, smallest integer m such that there exists a permutation (p_1, ..., p_n) of (1, ..., n) satisfying p_i * p_{i+1} <= m for every i, 1 <= i <= n-1.
%C A035106 Also, nonsquare positive integers m such that floor(sqrt(m)) divides m. - _Max Alekseyev_, Nov 27 2006
%C A035106 Also, for n>1, a(n) is the number of non-isomorphic simple connected undirected graphs having n+1 edges and a longest path of length n. - _Nathaniel Gregg_, Nov 02 2021
%H A035106 Reinhard Zumkeller, <a href="/A035106/b035106.txt">Table of n, a(n) for n = 1..10000</a>
%H A035106 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A035106 For n > 1, a(n) = n*(n+2)/4 if n is even and (n-1)*(n+3)/4 if n is odd. - _Jud McCranie_, Oct 25 2001
%F A035106 a(n) = a(n-1) + a(n-2) - a(n-3) + 1 = A002620(n+2) + A004526(n+2). - _Henry Bottomley_, Mar 08 2000
%F A035106 a(n+2) = (2*n^2 + 12*n + 3*(-1)^n + 13)/8, with a(1)=1, i.e., a(n+2) = (n+2)*(n+4)/4 if n is even and (n+1)*(n+5)/4 if n is odd. - _Vladeta Jovovic_, Oct 23 2001
%F A035106 From Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004: (Start)
%F A035106 a(n) = a(n-2) + (n-1), where a(1) = 0, a(2) = 0.
%F A035106 a(n) = (2*(n+1)^2 + 3*(-1)^n - 5)/8, n>=2, with a(1)=1. (End)
%F A035106 For n > 1, a(n) = floor((n+1)^4/(4*(n+1)^2+1)). - _Gary Detlefs_, Feb 11 2010
%F A035106 For n > 1, a(n) = n + ceiling((1/4)*(n-1)^2) - 1. - _Clark Kimberling_, Jan 07 2011; corrected by _Arkadiusz Wesolowski_, Sep 25 2012
%F A035106 a(1)=1, a(2)=2, a(3)=3, a(4)=6, a(5)=8; for n > 5, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - _Harvey P. Dale_, May 03 2012
%F A035106 G.f.: x + x^2*(2-x) / ( (1+x)*(1-x)^3 ) = x*(x^4 - 2*x^3 + x^2 - 1)/((x-1)^3*(x+1)). - _Vladeta Jovovic_, Oct 23 2001; _Harvey P. Dale_, May 03 2012
%F A035106 a(n) = floor(n/2)*(1 + ceiling(n/2)), a(1) = 1. - _Arkadiusz Wesolowski_, Sep 25 2012
%F A035106 a(n) = ceiling((n-1)*(n+3)/4), n > 1. - _Wesley Ivan Hurt_, Jun 14 2013
%F A035106 a(n+1) - a(n) = A052938(n-2) for n > 1. - _Reinhard Zumkeller_, Oct 06 2015
%F A035106 E.g.f.: (8*x + 3*exp(-x) - (3-6*x-2*x^2)*exp(x))/8. - _G. C. Greubel_, Jun 10 2019
%F A035106 Sum_{n>=1} 1/a(n) = 11/4. - _Amiram Eldar_, Sep 24 2022
%e A035106 n=5: we must arrange the numbers 1..5 so that the max of the products of pairs of adjacent terms is minimized. The answer is 51324, with max product = 8, so a(5) = 8.
%t A035106 Join[{1},LinearRecurrence[{2,0,-2,1},{2,3,6,8},60]] (* or *) Join[{1}, Table[ If[EvenQ[n],(n(n+2))/4,((n-1)(n+3))/4],{n,2,60}]] (* _Harvey P. Dale_, May 03 2012 *)
%o A035106 (Haskell)
%o A035106 import Data.List.Ordered (union)
%o A035106 a035106 n = a035106_list !! (n-1)
%o A035106 a035106_list = 1 : tail (union a002378_list a005563_list)
%o A035106 -- _Reinhard Zumkeller_, Oct 05 2015
%o A035106 (PARI) my(x='x+O('x^60)); Vec(x*(x^4-2*x^3+x^2-1)/((x-1)^3*(x+1))) \\ _Altug Alkan_, Oct 23 2015
%o A035106 (PARI) A035106(n)=!(n-1)+floor((n^2)/4+n/2); \\ _R. J. Cano_, Jul 24 2023
%o A035106 (Magma) [1] cat [(2*n*(n+2) +3*((-1)^n -1))/8: n in [2..60]]; // _G. C. Greubel_, Jun 10 2019
%o A035106 (Sage) [1]+[(2*n*(n+2) +3*((-1)^n -1))/8 for n in (2..60)] # _G. C. Greubel_, Jun 10 2019
%o A035106 (GAP) Concatenation([1], List([2..60], n-> (2*n*(n+2) +3*((-1)^n -1))/8)); # _G. C. Greubel_, Jun 10 2019
%Y A035106 Cf. A006446, A052938, A064764, A064796, A064797, A064817, A004652, A035104, A035107.
%Y A035106 First differences give (essentially) A028242.
%Y A035106 Bisections: A002378 (pronic numbers) and A005563.
%K A035106 nonn,nice,easy
%O A035106 1,2
%A A035106 _N. J. A. Sloane_, revised Oct 30 2001
%E A035106 Edited by _Max Alekseyev_, Oct 09 2015
%E A035106 Definition modified to allow for the initial 1. - _N. J. A. Sloane_, May 17 2016