Voozh

A000050

Number of positive integers <= 2^n of form x^2 + y^2.

1, 2, 3, 5, 9, 16, 29, 54, 97, 180, 337, 633, 1197, 2280, 4357, 8363, 16096, 31064, 60108, 116555, 226419, 440616, 858696, 1675603, 3273643, 6402706, 12534812, 24561934, 48168461, 94534626, 185661958, 364869032, 717484560, 1411667114, 2778945873, 5473203125

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Seth A. Troisi, Table of n, a(n) for n = 0..60 (terms 0..35 from N. J. A. Sloane)

P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, arXiv:math/0204332 [math.NT], 2002.

P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, Math. Comp. 73 (2004), no. 245, 451-473.

D. Shanks and L. P. Schmid, Variations on a theorem of Landau. Part I, Math. Comp., 20 (1966), 551-569.

Seth A. Troisi, C++ program

Index entries for sequences related to populations of quadratic forms

FORMULA

a(n) = A102548(2^n). - Andrei Zabolotskii, Jan 30 2026

EXAMPLE

There are 5 integers <= 2^3 of the form x^2 + y^2. The five (x,y) pairs (x <= y) are (0,1), (1,1), (0,2), (1,2), (2,2) and give the integers 1, 2, 4, 5, 8, respectively. So a(3) = 5. - Seth A. Troisi, Apr 27 2022

MATHEMATICA

(* This program is not suitable for a large number of terms *) a[0] = 1; a[n_] := a[n] = (For[cnt = 0; k = 2^(n-1)+1, k <= 2^n, k++, If[SquaresR[2, k] > 0, cnt++]]; cnt + a[n-1]); Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 26}] (* Jean-François Alcover, Mar 20 2014 *)

PROG

(Haskell)

isqrt = a000196

issquare = a010052

a000050 n = foldl f 0 [1..2^n]

where f i j = if a000050' j > 0 then i + 1 else i

a000050' k = foldl f 0 (h k)

where f i y = g y + i

where g y = issquare (k - y^2)

h k = [0..isqrt k]

-- James Spahlinger, Oct 09 2012

CROSSREFS