A000401 - OEIS

A000401

Numbers of form x^2 + y^2 + 2*z^2.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

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OFFSET

1,3

COMMENTS

Numbers represented by quadratic form with Gram matrix [ 1, 0, 0; 0, 1, 0; 0, 0, 2 ].

These are the numbers not of the form 4^k*(16*m + 14). [Dickson] - Everett W. Howe, May 18 2008

The asymptotic density of this sequence is 11/12. - Amiram Eldar, Mar 29 2025

REFERENCES

Wacław Sierpiński, Elementary Theory of Numbers, (Ed. A. Schinzel), North-Holland, 1988, see Exercise 4 on p. 395.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33 (1927), 63-70.

Gabriele Nebe and N. J. A. Sloane, Home page for this lattice.

FORMULA

a(n) = 12n/11 + O(log n). - Charles R Greathouse IV, Jan 24 2026

MAPLE

L := [seq(0, i=1..1)]: for x from 0 to 20 do for y from 0 to 20 do for z from 0 to 20 do if member(x^2+y^2+2*z^2, L)=false then L := [op(L), x^2+y^2+2*z^2] fi: od: od: od: L2 := sort(L): for i from 1 to 100 do printf(`%d, `, L2[i]) od:

MATHEMATICA

q=16; imax=q^2; Select[Union[Flatten[Table[x^2+y^2+2*z^2, {z, 0, q}, {y, 0, q}, {x, 0, q}]]], #<=imax&] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)

Select[Range[0, 100], Mod[# / 4^IntegerExponent[#, 4], 16] != 14 &] (* Amiram Eldar, Mar 29 2025 *)

PROG

(Python)

def A000401(n):

def f(x): return n-1+sum(((x>>i)-7>>3)+1 for i in range(1, x.bit_length(), 2))

m, k = n-1, f(n-1)

while m != k: m, k = k, f(k)

return m # Chai Wah Wu, Feb 24 2025

CROSSREFS

Complement of A055039.

Sequence in context: A195099 A195098 A102452 * A023807 A023755 A335522

Adjacent sequences: A000398 A000399 A000400 * A000402 A000403 A000404