A004144 -id:A004144 - cp's OEIS frontend

A357576 Half area of the convex hull of {(x,y)| x,y integers and x^2 + y^2 < n^2}.

0, 2, 8, 17, 28, 46, 63, 87, 112, 142, 173, 204, 244, 287, 333, 378, 428, 485, 540, 602, 661, 737, 802, 869, 947, 1030, 1118, 1197, 1278, 1378, 1469, 1575, 1670, 1776, 1889, 1990, 2108, 2219, 2353, 2472, 2587, 2723, 2854, 3002, 3135, 3275, 3424, 3563, 3721

Offset: 1

Gerhard Kirchner, Oct 05 2022

nonn

a(n) is odd if there is an edge connecting two vertices (x,y) and (y,x), x > y > 0, such that x-y is odd. Otherwise, a(n) is even. a(n)/n^2 is not monotonous but tends to Pi/2. The convex hull has four symmetry axes: x=0, y=0, y=x, y=-x. Therefore it is sufficient to find the least area of a quarter polygon (multiplied by 2). The half area is an integer because the area of any convex polygon whose vertex coordinates are integers is a multiple of 1/2.

			For n=4: 5+6+6 = 17 square units -> a(4)=17.
     _______
   /|_|_|_|_|\  5
  |_|_|_|_|_|_| 6
  |_|_|_|_|_|_| 6

Gerhard Kirchner, Examples for n <= 13.

Cf. A000328, A004144, A292276, A357575.

block(nmax: 40, a: makelist(0,i,1,nmax), a[1]:0,
for n from 2 thru nmax do
  (x0:0, y0:n, xa:0, ya:n, m1:0, m0:2, ar:0,
    while xa

from math import isqrt
from sympy import convex_hull
def A357576(n): return 0 if n == 1 else int(2*convex_hull(*[(0,0),(n-1,0)]+[(x,isqrt((n-x)*(n+x)-1)) for x in range(n)]).area) # Chai Wah Wu, Oct 23 2022

a(n) = A357575(n) - 2*floor(sqrt(2*n-1)) if n is a nonhypotenuse number (A004144).

A244662 Decimal expansion of 'C' (as designated by D. Shanks), a constant appearing in the second order term of the asymptotic expansion of the number of non-hypotenuse numbers not exceeding a given bound.

Original entry on oeis.org

7, 0, 4, 7, 5, 3, 4, 5, 1, 7, 0, 5, 9, 4, 7, 8, 8, 4, 1, 2, 2, 5, 5, 8, 1, 9, 7, 5, 9, 1, 8, 9, 8, 8, 1, 8, 5, 2, 1, 5, 9, 9, 7, 6, 4, 5, 4, 9, 2, 3, 5, 8, 3, 1, 6, 1, 7, 4, 4, 5, 4, 8, 8, 3, 4, 1, 3, 6, 2, 8, 4, 6, 3, 9, 0, 3, 1, 8, 8, 4, 4, 4, 6, 0, 6, 3, 6, 4, 9, 2, 5, 3, 5, 2, 2, 3, 0, 2, 6, 4

Offset: 0

Jean-François Alcover, Jul 04 2014

			0.70475345170594788412255819759189881852...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.3 Landau-Ramanujan Constant, p. 101.

Daniel Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation, Vol. 18 (1964), pp. 75-86.
Daniel Shanks, Non-hypotenuse Numbers, Fib. Quart., 13:4 (1975), pp. 319-321.
Eric Weisstein's MathWorld, Lemniscate Constant

Cf. A009003, A004144, A062539, A227158, A244659 (first order term).

digits = 100; m0 = 5; dm = 5; beta[x_] := 1/4^x*(Zeta[x, 1/4] - Zeta[x, 3/4]); L = Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2; Clear[f]; f[m_] := f[m] = 1/2*(1 - Log[Pi*E^EulerGamma/(2*L)]) - 1/4*NSum[Zeta'[2^k]/Zeta[2^k] - beta'[2^k]/beta[2^k] + Log[2]/(2^(2^k) - 1), {k, 1, m}, WorkingPrecision -> digits + 10]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], m = m + dm]; c = A227158 = f[m]; c + 1/2 Log[(Pi/L)^2*Exp[EulerGamma]/2] // RealDigits[#, 10, digits] & // First

C = c + 1/2*log((Pi/L)^2*exp(gamma)/2), where c is A227158 and L the Lemniscate constant A062539.

A309228 a(n) is the greatest possible height of a binary tree where all nodes hold positive squares and all interior nodes also equal the sum of their two children and the root node has value n^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 3, 2, 1, 1, 1, 3, 2, 1, 3, 1, 3, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 3, 3, 1, 2, 1, 1, 3, 1, 2, 3, 1, 1, 1, 4, 1, 1, 3, 1, 2, 1, 1, 3, 3, 3, 1, 1, 3, 1, 2, 1, 3, 1, 1, 4, 1, 3

Offset: 1

Rémy Sigrist, Jul 16 2019

nonn

The sequence is unbounded and for any k > 0, A309167(k) is the least n such that a(n) = k.

			a(1) = 1:
              1^2
               |
a(5) = 2:
           3^2    4^2
            \     /
             \   /
              5^2
               |
a(13) = 3:
          3^2    4^2
           \     /
            \   /
             5^2    12^2
              \      /
               \    /
                13^2
                  |

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004144, A309167.

f:= proc(n) option remember; local S,x,y;
   S:= map(t -> subs(t,[x,y]), {isolve(x^2+y^2=n^2)});
   S:= select(t -> type(t,list(posint)) and t[2]>=t[1], S);
   if S = {} then 1 else 1+max(map(procname,map(op,S))) fi
end proc:
map(f, [$1..100]); # Robert Israel, Feb 27 2022

a = Table[1, {m = 100}];
Do[Do[If[IntegerQ@ Sqrt[v2 = n^2-u^2], a[[n]] = Max[a[[n]], 1+Max[a[[u]], a[[Floor@ Sqrt[v2]]]]]], {u, 1, n-1}], {n, 1, m}];
Table[a[[n]], {n, 1, m}] (* Jean-François Alcover, Aug 19 2022, after PARI code *)

a = vector(87,n,1); for (n=1, #a, for (u=1, n-1, if (issquare(v2=n^2-u^2), a[n]=max(a[n],1+max(a[u],a[sqrtint(v2)])))); print1 (a[n]", "))

a(n) = 1 iff n belongs to A004144.
a(A309167(n)) = n.
If n^2 = u^2 + v^2 with u > v > 0, then a(n) >= 1 + max(a(u), a(v)).

cp's OEIS Frontend

A357576 Half area of the convex hull of {(x,y)| x,y integers and x^2 + y^2 < n^2}.

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A244662 Decimal expansion of 'C' (as designated by D. Shanks), a constant appearing in the second order term of the asymptotic expansion of the number of non-hypotenuse numbers not exceeding a given bound.

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A309228 a(n) is the greatest possible height of a binary tree where all nodes hold positive squares and all interior nodes also equal the sum of their two children and the root node has value n^2.

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