A097700 -id:A097700 - cp's OEIS frontend

A002479 Numbers of the form x^2 + 2*y^2.

0, 1, 2, 3, 4, 6, 8, 9, 11, 12, 16, 17, 18, 19, 22, 24, 25, 27, 32, 33, 34, 36, 38, 41, 43, 44, 48, 49, 50, 51, 54, 57, 59, 64, 66, 67, 68, 72, 73, 75, 76, 81, 82, 83, 86, 88, 89, 96, 97, 98, 99, 100, 102, 107, 108, 113, 114, 118, 121, 123, 128, 129, 131

Offset: 1

N. J. A. Sloane

A positive number k belongs to this sequence if and only if every prime p == 5, 7 (mod 8) dividing k occurs to an even power. - Sharon Sela (sharonsela(AT)hotmail.com), Mar 23 2002
Norms of numbers in Z[sqrt(-2)]. - Alonso del Arte, Sep 23 2014
Euler (E256) shows that these numbers are closed under multiplication, according to the Euler Archive. - Charles R Greathouse IV, Jun 16 2016
In addition to the previous comment: The proof was already given 1100 years before Euler by Brahmagupta's identity (a^2 + m*b^2)*(c^2 + m*d^2) = (a*c - m*b*d)^2 + m*(a*d + b*c)^2. - Klaus Purath, Oct 07 2023

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 421.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 59.
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).

N. J. A. Sloane, Table of n, a(n) for n = 1..3148 (first 1000 terms from T. D. Noe)
L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.
L. Euler, (E256) Specimen de usu observationum in mathesi pura, Novi Commentarii academiae scientiarum Petropolitanae 6 (1761), pp. 185-230.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Complement of A097700. Subsequence of A000408. For primes see A033203.

Cf. A035251, A027748, A124010, A003628.

a002479 n = a002479_list !! (n-1)
a002479_list = 0 : filter f [1..] where
   f x = all (even . snd) $ filter ((`elem` [5,7]) . (`mod` 8) . fst) $
                            zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Feb 20 2014

[n: n in [0..131] | NormEquation(2, n) eq true]; // Arkadiusz Wesolowski, May 11 2016

lis:={}; M:=50; M2:=M^2;
for x from 0 to M do for y from 0 to M do
if x^2+2*y^2 <= M2 then lis:={op(lis),x^2+2*y^2}; fi; od: od:
sort(convert(lis,list)); # N. J. A. Sloane, Apr 30 2015

q = 16; imax = q^2; Select[Union[Flatten[Table[x^2 + 2y^2, {y, 0, q/Sqrt[2]}, {x, 0, q}]]], # <= imax &] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)
Union[#[[1]]+2#[[2]]&/@Tuples[Range[0,10]^2,2]] (* Harvey P. Dale, Nov 24 2014 *)

is(n)=my(f=factor(n));for(i=1,#f[,1],if(f[i,1]%8>4 && f[i,2]%2, return(0)));1 \\ Charles R Greathouse IV, Nov 20 2012

list(lim)=my(v=List()); for(a=0,sqrtint(lim\=1), for(b=0,sqrtint((lim-a^2)\2), listput(v,a^2+2*b^2))); Set(v) \\ Charles R Greathouse IV, Jun 16 2016

from itertools import count, islice
from sympy import factorint
def A002479_gen(): # generator of terms
    return filter(lambda n:all(p & 7 < 5 or e & 1 == 0 for p, e in factorint(n).items()),count(0))
A002479_list = list(islice(A002479_gen(),30)) # Chai Wah Wu, Jun 27 2022

A001394 Number of n-step self-avoiding walks on diamond.

Original entry on oeis.org

1, 4, 12, 36, 108, 324, 948, 2796, 8196, 24060, 70188, 205284, 597996, 1744548, 5073900, 14774652, 42922452, 124814484, 362267652, 1052271732, 3051900516, 8857050204, 25671988020, 74449697484, 215677847460, 625096195404, 1810062340812, 5243388472212

Offset: 0

N. J. A. Sloane

Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004

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).

J. W. Essam and M. F. Sykes, The crystal statistics of the diamond lattice, Physica, 29 (1963), 378-388.
A. J. Guttmann, On the critical behavior of self-avoiding walks II, J. Phys. A 22 (1989), 2807-2813.
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
J. L. Martin, The exact enumeration of self-avoiding walks on a lattice, Proc. Camb. Phil. Soc., 58 (1962), 92-101.

Cf. A001395, A001396, A001397, A001398, A097700, A176086, A227715, A227716.

Edited and extended by Joseph Myers, Jul 21 2013

a(24)-a(27) from Sean A. Irvine, Nov 13 2017

A216282 Number of nonnegative solutions to the equation x^2 + 2*y^2 = n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 1, 2, 0, 1, 1, 0, 0, 0, 1, 1, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 0, 0, 1, 2, 1, 0, 2, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 2, 1, 1, 0, 0, 0, 2, 1, 0, 1, 1, 0, 0, 0, 0, 3, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 1, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0

Offset: 1

V. Raman, Sep 03 2012

nonn

Records occur at 1, 9, 81, 297, 891, 1683, 5049, 15147, 31977, ... - Antti Karttunen, Aug 23 2017

			For n = 9, there are two solutions: 9 = 9^2 + 2*(0^2) = 1^2 + 2*(2^2), thus a(9) = 2.
For n = 81, there are three solutions: 81 = 9^2 + 2*(0^2) = 3^2 + 2*(6^2) = 7^2 + 2*(4^2), thus a(81) = 3.
For n = 65536, there is one solution: 65536 = 256^2 + 2*(0^2) = 65536 + 0, thus a(65536) = 1.
For n = 65537, there is one solution: 65537 = 255^2 + 2*(16^2) = 65205 + 512, thus a(65537) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A092573, A119395, A000161, A025426, A216283.

Cf. A002479 (positions of nonzeros), A097700 (of zeros).

r[n_] := Reduce[x >= 0 && y >= 0 && x^2 + 2 y^2 == n, Integers];
a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === And, 1, Head[rn] === Or, Length[rn], True, -1];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 24 2017 *)

(define (A216282 n) (cond ((< n 2) 1) (else (let loop ((k (- (A000196 n) (modulo (- n (A000196 n)) 2))) (s 0)) (if (< k 0) s (let ((x (/ (- n (* k k)) 2))) (loop (- k 2) (+ s (A010052 x))))))))) ;; Antti Karttunen, Aug 23 2017

Examples from Antti Karttunen, Aug 23 2017

A034030 Imprimitively represented by x^2+2y^2.

Original entry on oeis.org

0, 4, 8, 9, 12, 16, 18, 24, 25, 27, 32, 36, 44, 48, 49, 50, 54, 64, 68, 72, 75, 76, 81, 88, 96, 98, 99, 100, 108, 121, 128, 132, 136, 144, 147, 150, 152, 153, 162, 164, 169, 171, 172, 176, 192, 196, 198, 200, 204, 216, 225, 228

Offset: 1

N. J. A. Sloane

nonn

Cf. A002479, A097700, A057127, A034031-A034034.

# Maple code for A002479, A057127, A034030-A034034 from N. J. A. Sloane, Apr 30 2015
lis:={}; lisP:={}; lisI:={};
M:=50; M2:=M^2;
for x from 0 to M do
   x2:=x^2;
for y from 0 to M do
   N:=x2+2*y^2;
if N <= M2 then
   if gcd(x,y) = 1 then lisP:={op(lisP),N}; else lisI:={op(lisI),N} fi;
   lis:={op(lis),N};
fi;
od: od:
lprint("lis");
Lis:=sort(convert(lis,list));
lprint("lisP");
LisP:=sort(convert(lisP,list));
lprint("lisI");
LisI:=sort(convert(lisI,list));
lprint("lisPnotI");
LisPnotI:=sort(convert(lisP minus lisI, list));
lprint("lisInotP");
LisInotP:=sort(convert(lisI minus lisP,list));
lprint("lisIandP");
LisIandP:=sort(convert(lisI intersect lisP,list));
lprint("liseither");
Liseither:=sort(convert(lis minus (lisI intersect lisP),list));

Corrected by N. J. A. Sloane, Apr 30 2015

A034034 Numbers that are primitively or imprimitively represented by x^2+2y^2, but not both.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 11, 12, 16, 17, 19, 22, 24, 25, 32, 33, 34, 36, 38, 41, 43, 44, 48, 49, 50, 51, 57, 59, 64, 66, 67, 68, 72, 73, 75, 76, 82, 83, 86, 88, 89, 96, 97, 98, 100, 102, 107, 108, 113, 114, 118, 123, 128, 129, 131

Offset: 1

N. J. A. Sloane

nonn

Cf. A002479, A097700, A057127, A034030-A034033.

Corrected by N. J. A. Sloane, Apr 30 2015

A034031 Numbers that are primitively but not imprimitively represented by x^2+2y^2.

Original entry on oeis.org

1, 2, 3, 6, 11, 17, 19, 22, 33, 34, 38, 41, 43, 51, 57, 59, 66, 67, 73, 82, 83, 86, 89, 97, 102, 107, 113, 114, 118, 123, 129, 131, 134, 137, 139, 146, 163, 166, 177, 178, 179, 187, 193, 194, 201, 209, 211, 214, 219, 226, 227

Offset: 1

N. J. A. Sloane

nonn

Cf. A002479, A097700, A057127, A034030-A034034.

Corrected by N. J. A. Sloane, Apr 30 2015

A034033 Both primitively and imprimitively represented by x^2+2y^2.

Original entry on oeis.org

9, 18, 27, 54, 81, 99, 121, 153, 162, 171, 198, 242, 243, 289, 297, 306, 342, 361, 363, 369, 387, 459, 486, 513, 531, 578, 594, 603, 657, 722, 726, 729, 738, 747, 774, 801, 867, 873, 891, 918, 963, 1017, 1026, 1062, 1083, 1089

Offset: 1

N. J. A. Sloane

nonn

Cf. A002479, A097700, A057127, A034030-A034034.

Corrected by N. J. A. Sloane, Apr 30 2015

A055052 Numbers of the form 4^i(8j+7) or 4^i(8j+5).

Original entry on oeis.org

5, 7, 13, 15, 20, 21, 23, 28, 29, 31, 37, 39, 45, 47, 52, 53, 55, 60, 61, 63, 69, 71, 77, 79, 80, 84, 85, 87, 92, 93, 95, 101, 103, 109, 111, 112, 116, 117, 119, 124, 125, 127, 133, 135, 141, 143, 148, 149, 151, 156, 157, 159, 165, 167, 173, 175

Offset: 1

N. J. A. Sloane, Jun 02 2000

Numbers not of the form x^2+2y^2+8z^2.
The integers that are ratios between the terms constitute the sequence's complement within A003159. - Peter Munn, Feb 07 2024
The asymptotic density of this sequence is 1/3. - Amiram Eldar, Feb 09 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
L. J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).

Disjoint union of A004215 and A055045.

Subsequence of A003159, A097700.

Select[Range[200], MemberQ[{5, 7}, Mod[# / 4^IntegerExponent[#, 4], 8]] &] (* Amiram Eldar, Feb 09 2024 *)

def A055052(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(x.bit_length()>>1):
            m = x>>(i<<1)
            c -= (m-5>>3)+(m-7>>3)+2
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

A034028 Numbers n not of form (x^2+2y^2 with x >= y >= 0).

Original entry on oeis.org

2, 5, 7, 8, 10, 13, 14, 15, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 32, 35, 37, 39, 40, 41, 42, 45, 46, 47, 50, 52, 53, 55, 56, 58, 59, 60, 61, 62, 63, 65, 69, 70, 71, 73, 74, 76, 77, 78, 79, 80, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 97, 98, 101, 103, 104, 105

Offset: 1

N. J. A. Sloane

nonn

Complement of A034027. Cf. A097700.

Definition corrected by N. J. A. Sloane, Apr 30 2015

A034032 Imprimitively but not primitively represented by x^2+2y^2.

Original entry on oeis.org

0, 4, 8, 12, 16, 24, 25, 32, 36, 44, 48, 49, 50, 64, 68, 72, 75, 76, 88, 96, 98, 100, 108, 128, 132, 136, 144, 147, 150, 152, 164, 169, 172, 176, 192, 196, 200, 204, 216, 225, 228, 236, 256, 264, 268, 272, 275, 288, 292, 294

Offset: 1

N. J. A. Sloane

nonn

Cf. A002479, A097700, A057127, A034030-A034034.

Corrected by N. J. A. Sloane, Apr 30 2015

cp's OEIS Frontend

A002479 Numbers of the form x^2 + 2*y^2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

A001394 Number of n-step self-avoiding walks on diamond.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A216282 Number of nonnegative solutions to the equation x^2 + 2*y^2 = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Extensions

A034030 Imprimitively represented by x^2+2y^2.

Views

Author

Keywords

Crossrefs

Programs

Maple

Extensions

A034034 Numbers that are primitively or imprimitively represented by x^2+2y^2, but not both.

Views

Author

Keywords

Crossrefs

Extensions

A034031 Numbers that are primitively but not imprimitively represented by x^2+2y^2.

Views

Author

Keywords

Crossrefs

Extensions

A034033 Both primitively and imprimitively represented by x^2+2y^2.

Views

Author

Keywords

Crossrefs

Extensions

A055052 Numbers of the form 4^i*(8j+7) or 4^i*(8j+5).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

A034028 Numbers n not of form (x^2+2y^2 with x >= y >= 0).

Views

Author

Keywords

Crossrefs

Extensions

A055052 Numbers of the form 4^i(8j+7) or 4^i(8j+5).