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A087811 Numbers k such that ceiling(sqrt(k)) divides k.

%I A087811 #187 Sep 29 2024 13:18:15
%S A087811 1,2,4,6,9,12,16,20,25,30,36,42,49,56,64,72,81,90,100,110,121,132,144,
%T A087811 156,169,182,196,210,225,240,256,272,289,306,324,342,361,380,400,420,
%U A087811 441,462,484,506,529,552,576,600,625,650,676,702,729,756,784,812,841
%N A087811 Numbers k such that ceiling(sqrt(k)) divides k.
%C A087811 Essentially the same as the quarter-squares A002620.
%C A087811 Nonsquare terms of this sequence are given by A002378. - _Max Alekseyev_, Nov 27 2006
%C A087811 This also gives the number of ways to make change for "c" cents using only pennies, nickels and dimes. You must first set n=floor(c/5), to account for the 5-repetitive nature of the task. - _Adam Sasson_, Feb 09 2011
%C A087811 These are the segment boundaries of Oppermann's conjecture (1882): n^2-n < p < n^2 < p < n^2+n. - _Fred Daniel Kline_, Apr 07 2011
%C A087811 a(n) is the number of triples (w,x,y) having all terms in {0..n} and w=2*x+y. - _Clark Kimberling_, Jun 04 2012
%C A087811 a(n+1) is also the number of points with integer coordinates inside a rectangle isosceles triangle with hypotenuse [0,n] (see A115065 for an equilateral triangle). - _Michel Marcus_, Aug 05 2013
%C A087811 a(n) = degree of generating polynomials of Galois numbers in (n+1)-dimensional vector space, defined as total number of subspaces in (n+1) space over GF(n) (see Mathematica procedure), when n is a power of a prime. - _Artur Jasinski_, Aug 31 2016, corrected by _Robert Israel_, Sep 23 2016
%C A087811 Also number of pairs (x,y) with 0 < x <= y <= n, x + y > n. - _Ralf Steiner_, Jan 05 2020
%H A087811 Vincenzo Librandi, <a href="/A087811/b087811.txt">Table of n, a(n) for n = 1..300</a>
%H A087811 Ya-Ping Lu, <a href="/A087811/a087811.pdf">Illustration of the Terms in A087811 on the Square Spiral</a>
%H A087811 Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann%27s_conjecture">Oppermann's conjecture</a>.
%H A087811 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A087811 a(n) = (n + n mod 2)*(n + 2 - n mod 2)/4.
%F A087811 Numbers of the form m^2 or m^2 - m. - _Don Reble_, Oct 17 2003
%F A087811 a(1) = 1, a(2) = 2, a(n) = n + a(n - 2). - _Alonso del Arte_, Jun 18 2005
%F A087811 From _Bruno Berselli_, Feb 09 2011: (Start)
%F A087811 G.f.: x/((1+x)*(1-x)^3).
%F A087811 a(n) = (2*n*(n+2)-(-1)^n+1)/8. (End)
%F A087811 G.f.: G(0)/(2*(1-x^2)*(1-x)), where G(k) = 1 + 1/(1 - x*(2*k+1)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 25 2013
%F A087811 a(n) = (C(n+2,2) - floor((n+2)/2))/2. - _Mircea Merca_, Nov 23 2013
%F A087811 a(n) = ((-1)^n*(-1 + (-1)^n*(1 + 2*n*(2 + n))))/8. - _Fred Daniel Kline_, Jan 06 2015
%F A087811 a(n) = Product_{k=1...n-1} (1 + 2 / (k + k mod 2)), n >= 1. - _Fred Daniel Kline_, Oct 30 2016
%F A087811 E.g.f.: (1/4)*(x*(3 + x)*cosh(x) + (1 + 3*x + x^2)*sinh(x)). - _Stefano Spezia_, Jan 05 2020
%F A087811 a(n) = (n*(n+2)+(n mod 2))/4. - _Chai Wah Wu_, Jul 27 2022
%F A087811 Sum_{n>=1} 1/a(n) = Pi^2/6 + 1. - _Amiram Eldar_, Sep 17 2022
%F A087811 a(n) = A024206(n) + 1. - _Ya-Ping Lu_, Dec 29 2023
%p A087811 f:= gfun:-rectoproc({a(n)=n+a(n-2),a(1)=1,a(2)=2},a(n),remember):
%p A087811 map(f, [$1..100]); # _Robert Israel_, Aug 31 2016
%t A087811 a[1] := 1; a[2] := 2; a[n_] := n + a[n - 2]; Table[a[n], {n, 57}] (* _Alonso del Arte_ *)
%t A087811 GaloisNumber[n_, q_] :=
%t A087811 Sum[QBinomial[n, m, q], {m, 0, n}]; aa = {}; Do[
%t A087811 sub = Table[GaloisNumber[m, n], {n, 0, 200}];
%t A087811 pp = InterpolatingPolynomial[sub, x]; pol = pp /. x -> n + 1;
%t A087811 coef = CoefficientList[pol, n];
%t A087811 AppendTo[aa, Length[coef] - 1], {m, 2, 25}]; aa (* _Artur Jasinski_, Aug 31 2016 *)
%t A087811 Select[Range[900],Divisible[#,Ceiling[Sqrt[#]]]&] (* or *) LinearRecurrence[ {2,0,-2,1},{1,2,4,6},60] (* _Harvey P. Dale_, Nov 06 2016 *)
%o A087811 (Magma) [ n: n in [1..841] | n mod Ceiling(Sqrt(n)) eq 0 ]; // _Bruno Berselli_, Feb 09 2011
%o A087811 (PARI) a(n)=(n+n%2)*(n+2-n%2)/4 \\ _Charles R Greathouse IV_, Apr 03 2012
%o A087811 (PARI) j=0;for(k=1,850,s=sqrtint(4*k+1);if(s>j,j=s;print1(k,", "))) \\ _Hugo Pfoertner_, Sep 17 2018
%o A087811 (Haskell)
%o A087811 a087811 n = (n + n `mod` 2) * (n + 2 - n `mod` 2) `div` 4
%o A087811 -- _Reinhard Zumkeller_, Oct 27 2012
%o A087811 (Python)
%o A087811 def A087811(n): return n*(n+2)+(n&1)>>2 # _Chai Wah Wu_, Jul 27 2022
%Y A087811 Cf. A002378, A002620, A003059, A024206, A110835, A316841, A059260.
%Y A087811 Subsequence of A006446.
%K A087811 nonn,easy
%O A087811 1,2
%A A087811 _Reinhard Zumkeller_, Oct 16 2003