A005875 -id:A005875 - cp's OEIS frontend

A004539 Expansion of sqrt(2) in base 2.

1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1

Offset: 1

N. J. A. Sloane

Bailey, Borwein, Crandall, & Pomerance prove a general result that the first n terms contain >> sqrt(n) 1's. Vandehey improves this to sqrt(2*n)(1 + o(1)). - Charles R Greathouse IV, Nov 07 2017

			1.0110101000001001111001...

T. D. Noe, Table of n, a(n) for n = 1..1000
B. Adamczewski and N. Rampersad, On patterns occurring in binary algebraic numbers, Proc. Amer. Math. Soc. 136 (2008), 3105-3109.
David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Carl Pomerance, On the binary expansions of algebraic numbers, J. Théor. Nombres Bordeaux, 16 (2004), 487-518.
R. L. Graham and H. O. Pollak, Note on a nonlinear recurrence related to sqrt(2), Mathematics Magazine, Volume 43, Pages 143-145, 1970. Zbl 201.04705.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Richard Isaac, On the simple normality to base 2 of the square root of s, for s not a perfect square., arXiv:math/0512404 [math.NT], 2005-2006.
Jason Kimberley, Index of expansions of sqrt(d) in base b
Thomas Stoll, On families of nonlinear recurrences related to digits, Journal of Integer Sequences 8 (2005), 05.3.2.
Thomas Stoll, On a problem of Erdős and Graham concerning digits, Acta Arithmetica 125 (2006), pp. 89-100.
Thomas Stoll, A fancy way to obtain the binary digits of 759250125 sqrt{2}, (2009), Amer. Math. Monthly, 117 (2010), 611-617.
Joseph Vandehey, On the binary digits of sqrt(2), arXiv:1711.01722 [math.NT], 2017.
Eric Weisstein's World of Mathematics, Wolfram's Iteration
Eric Weisstein's World of Mathematics, Pythagoras's Constant

Cf. A002193 (decimal version), A233836 (run lengths of 0's and 1's).

a004539 n = a004539_list !! (n-1)
a004539_list = w 2 0 where
   w x r = bit : w (4 * (x - (4 * r + bit) * bit)) (2 * r + bit)
     where bit = head (dropWhile (\b -> (4 * r + b) * b < x) [0..]) - 1
-- Reinhard Zumkeller, Dec 16 2013

N[Sqrt[2], 200]; RealDigits[%, 2]
RealDigits[Sqrt[2],2,120][[1]] (* Harvey P. Dale, Aug 03 2024 *)

binary(sqrt(2)) \\ Michel Marcus, Nov 06 2017

a(n) = floor(quadgen(8)<<(n-1))%2; \\ Chittaranjan Pardeshi, Sep 09 2024

bc
```
obase=2 scale=200 sqrt(2)
```

a(k) = floor(Sum_{n>=1} A005875(n)/exp(Pi*n/(2^((2/3)*k+(1/3))))) mod 2. Will give the k-th binary digit of sqrt(2). A005875 : number of ways to write n as sum of 3 squares. - Simon Plouffe, Dec 30 2023

A066535 Number of ways of writing n as a sum of n squares.

Original entry on oeis.org

1, 2, 4, 8, 24, 112, 544, 2368, 9328, 34802, 129064, 491768, 1938336, 7801744, 31553344, 127083328, 509145568, 2035437440, 8148505828, 32728127192, 131880275664, 532597541344, 2153312518240, 8710505815360, 35250721087168, 142743029326162, 578472382307304

Offset: 0

Peter Bertok (peter(AT)bertok.com), Jan 07 2002

nonn

			There are a(3) = 8 solutions (x,y,z) of 3 = x^2 + y^2 + z^2: (1,1,1), (-1,-1,-1), 3 permutations of (1,1,-1) and 3 permutations of (1,-1,-1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
John Holley-Reid and Jeremy Rouse, The number of representations of n as a growing number of squares, arXiv:1910.01001 [math.NT], 2019.

Cf. A004018, A005875, A000118, A066536.
Cf. A122141, A166952. - Paul D. Hanna, Oct 25 2009
a(n^2) gives A361431.

b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1) +2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 16 2014

Join[{1}, Table[SquaresR[n, n], {n, 24}]]

{a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n)); polcoeff(THETA3^n, n)} /* Paul D. Hanna, Oct 25 2009 */

a(n) equals the coefficient of x^n in the n-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Oct 25 2009

a(n) ~ c * d^n / sqrt(n), where d = 4.13273137623493996302796465... (= 1/radius of convergence A166952), c = 0.2820942036723951157919967... . - Vaclav Kotesovec, Sep 12 2014

Edited by Dean Hickerson, Jan 12 2002
a(0) added by Paul D. Hanna, Oct 25 2009
Edited by R. J. Mathar, Oct 29 2009

A117609 Number of lattice points inside the ball x^2 + y^2 + z^2 <= n.

Original entry on oeis.org

1, 7, 19, 27, 33, 57, 81, 81, 93, 123, 147, 171, 179, 203, 251, 251, 257, 305, 341, 365, 389, 437, 461, 461, 485, 515, 587, 619, 619, 691, 739, 739, 751, 799, 847, 895, 925, 949, 1021, 1021, 1045, 1141, 1189, 1213, 1237, 1309, 1357, 1357, 1365, 1419, 1503

Offset: 0

John L. Drost, Apr 06 2006

nonn

			a(2) = 1 + 6 + 12 = 19, since (0,0,0) and (0, 0, +-1) and cyclic permutations (for a total of 6 points), and +-(0, 1, +-1) and cyclic permutations (for a total 12 points) are inside or on x^2 + y^2 + z^2 = 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
S. K. K. Choi, A. V. Kumchev and R. Osburn, On sums of three squares, arXiv:math/0502007 [math.NT], 2005.

Partial sums of A005875.
Cf. A000605 (number of points of norm <= n in cubic lattice).
Cf. A210639, A000092 and references therein.
Cf. A057655.

Table[Sum[SquaresR[3,k], {k,0,n}], {n,0,50}] (* T. D. Noe, Apr 08 2006, revised Sep 27 2011 *)

A117609(n)=sum(x=0,sqrtint(n),(sum(y=1,sqrtint(t=n-x^2),1+2*sqrtint(t-y^2))*2+sqrtint(t)*2+1)*2^(x>0)) \\ M. F. Hasler, Mar 26 2012

q='q+O('q^66); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^3/(1-q)) /* Joerg Arndt, Apr 08 2013 */

# uses Python code for A057655
from math import isqrt
def A117609(n): return A057655(n)+(sum(A057655(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 23 2024

a(n) ~ (4/3)*Pi*n^(3/2) ~ A210639(n).
a(n) = A122510(3,n). - R. J. Mathar, Apr 21 2010
G.f.: T3(q)^3/(1-q) where T3(q) = 1 + 2*Sum_{k>=1} q^(k^2). - Joerg Arndt, Apr 08 2013
a(n^2) = A000605(n). - R. J. Mathar, Aug 03 2025

A085469 Decimal expansion of Madelung constant (negated) for NaCl structure.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 01 2003

This is the electrostatic potential at the origin produced by unit charges of sign (-1)^(i+j+k) at all nonzero lattice points (i,j,k).
The NaCl structure consists of two interpenetrating face-centered cubic lattices of ions with charges +1 and -1, together occupying all the sites of the simple cubic lattice. - Andrey Zabolotskiy, Oct 21 2019
Named after the German physicist Erwin Madelung (1881-1972). - Amiram Eldar, Apr 02 2022

			-1.7475645946331821906362120355443974034851614366247417581528253507...

Richard E. Crandall, Topics in Advanced Scientific Computation, Springer, Telos books, 1996, pages 73-79.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 76.
Sadri Hassani, Mathematical Methods Using Mathematica: For Students of Physics and Related Fields, Springer, NY, page 60.

Jean-François Alcover, Table of n, a(n) for n = 1..2003 [There were errors in the previous b-file, which had 1847 terms contributed by Harry J. Smith.]
David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor and Eric W. Weisstein, Ten Problems in Experimental Mathematics, Amer. Math. Monthly, Vol. 113, No. 6 (2006), pp. 481-509.
R. E. Crandall and J. P. Buhler, Elementary function expansions for Madelung constants, J. Phys. A: Math. Gen., Vol. 20, No. 16 (1987), pp. 5497-5510.
R. E. Crandall and J. P. Buhler, The potential within a crystal lattice, J. Phys. A: Math. Gen., Vol. 20, No. 9 (1987), pp. 2279-2292.
E. R. Fuller, Jr. and E. R. Naimon, Electrostatic Contributions to the Brugger-Type Elastic Constants, Phys. Rev. B, Vol. 6, No. 10 (1971), pp. 3609-3620.
Leslie Glasser, Solid-State Energetics and Electrostatics: Madelung Constants and Madelung Energies, Inorg. Chem., Vol. 51, No. 4 (2012), pp. 2420-2424; DOI: 10.1021/ic2023852.
André Hautot, New applications of Poisson's summation formula, J. of Phys. A, Vol. 8, No. 6 (1975) pp. 853-862.
Simon Plouffe, Madelung constant.
Simon Plouffe, The Levy constant.
Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, and J. A. Berger, Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants, arXiv:2006.01259 [physics.comp-ph], 2020.
Nicolas Tavernier, Gian Luigi Bendazzoli, Véronique Brumas, Stefano Evangelisti, and J. Arjan Berger, Clifford Boundary Conditions for Periodic Systems: the Madelung Constant of Cubic Crystals in 1, 2 and 3 Dimensions, arXiv:2107.04686 [cond-mat.mtrl-sci], 2021.
Sandeep Tyagi, New series representation of the Madelung constant, Prog. Theor. Phys., Vol. 114, No. 3 (2005), pp. 517-521.
Eric Weisstein's World of Mathematics, Benson's Formula.
Eric Weisstein's World of Mathematics, Madelung Constants.
Wikipedia, Madelung constant.
Index entries for sequences related to f.c.c. lattice.

Cf. A004015, A005875, A108778 (continued fraction).

RealDigits[ 12Pi*Sum[ Sech[Pi/2*Sqrt[(2j + 1)^2 + (2k + 1)^2]]^2, {j, 0, 40}, {k, 0, 40}], 10, 111][[1]] (* Robert G. Wilson v, Jul 12 2005 *)
RealDigits[Quiet[12 Pi (Sech[Pi/Sqrt[2]]^2 + NSum[Sum[Sech[Pi Norm[2 v + 1]/2]^2, {v, FrobeniusSolve[{1, 1}, Round[m]]}, Method -> "Procedural"], {m, 1, Infinity}, Compiled -> False, Method -> "WynnEpsilon", NSumTerms -> 33, WorkingPrecision -> 100])]][[1]] (* Jan Mangaldan, Jun 25 2020 *)
digits = 1800; m0 = 800; dm = 10; dd = 10; Clear[f, g];
g[j_, k_] := g[j, k] = 12 Pi Sech[(Pi/2) Sqrt[(2j + 1)^2 + (2k + 1)^2]]^2 // N[#, digits + dd]&;
f[m_] := f[m] =  Sum[g[j, k], {j, 0, m}, {k, 0, m}];
f[m = m0]; f[m += dm];
While[Abs[f[m] - f[m - dm]] > 10^(-digits - dd), Print[m]; m += dm];
A085469 = f[m];
RealDigits[A085469, 10, digits][[1]] (* Jean-François Alcover, May 08 2021, after Robert G. Wilson v *)

Madelung()=my(c=Pi/2,d=asech(2^-default(realbitprecision))\/c+1); sum(j=0,d, sum(k=0,d, sech(c*sqrt((2*j+1)^2+(2*k+1)^2))),0.)*12*Pi \\ Charles R Greathouse IV, Feb 07 2025

Sum_{i, j, k not all 0} (-1)^(i+j+k)/sqrt(i^2+j^2+k^2).

Entry revised by N. J. A. Sloane, Apr 12 2004
Definition corrected by Leslie Glasser, Jan 24 2011
Definition corrected by Andrey Zabolotskiy, Oct 21 2019

A004015 Theta series of face-centered cubic (f.c.c.) lattice.

Original entry on oeis.org

1, 12, 6, 24, 12, 24, 8, 48, 6, 36, 24, 24, 24, 72, 0, 48, 12, 48, 30, 72, 24, 48, 24, 48, 8, 84, 24, 96, 48, 24, 0, 96, 6, 96, 48, 48, 36, 120, 24, 48, 24, 48, 48, 120, 24, 120, 0, 96, 24, 108, 30, 48, 72, 72, 32, 144, 0, 96, 72, 72, 48, 120, 0, 144, 12, 48, 48, 168, 48, 96

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + 12*x + 6*x^2 + 24*x^3 + 12*x^4 + 24*x^5 + 8*x^6 + 48*x^7 + 6*x^8 + ...
G.f. = 1 + 12*q^2 + 6*q^4 + 24*q^6 + 12*q^8 + 24*q^10 + 8*q^12 + 48*q^14 + 6*q^16 + ...
From _Michael Somos_, Jan 05 2012: (Start)
a(2) = 6 since (1, -1, -1) is a solution to x^2 + y^2 + z^2 + x*y + x*z + y*z = 2 and the other 5 solutions are permutations and negations of this one.
a(2) = 6 since (1, 1, -1, -1) is a solution to x + y + z + w = 0 and x^2 + y^2 + z^2 + w^2 = 4 and the other 5 solutions are permutations of this one. (End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 113.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 263.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
L. V. Woodcock, Nature, Jan 09 1997, pp. 141-143.

N. J. A. Sloane, Table of n, a(n) for n = 0..9999
Charles L. Hohn, a(0) to a(15), animated
G. Nebe and N. J. A. Sloane, Home page for this lattice
N. J. A. Sloane, A portion of the f.c.c. lattice packing.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Theta Series
Index entries for sequences related to f.c.c. lattice

Cf. A004013, A005875, A005901, A045828. A055039 gives the positions of the 0's in this sequence.

Cf. A000007, A000122, A004016, A008444, A008445, A008446, A008447, A008448, A008449 (Theta series of lattices A_0, A_1, A_2, A_4, ...)

L := Lattice("A",3); A := ThetaSeries(L, 140); A; /* Michael Somos, Nov 13 2014 */

A := Basis( ModularForms( Gamma1(8), 3/2), 70); A[1] + 12*A[2] + 6*A[3] + 24*A[4]; /* Michael Somos, Sep 08 2018 */

maxd := 201: temp0 := trunc(evalf(sqrt(maxd)))+2: a := 0: for i from -temp0 to temp0 do a := a+q^( (i+1/2)^2): od: th2 := series(a,q,maxd); a := 0: for i from -temp0 to temp0 do a := a+q^(i^2): od: th3 := series(a,q,maxd); th4 := series(subs(q=-q, th3),q,maxd); series((1/2)*(th3^3+th4^3),q,200);

a[n_] := SquaresR[3, 2n]; Table[a[n], {n, 0, 69}] (* Jean-François Alcover, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^3 + EllipticTheta[ 4, 0, q]^3) / 2, {q, 0, 2 n}]; (* Michael Somos, May 24 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^3 + 12 q QPochhammer[ q^4]^3 QPochhammer[ q^8]^2 / QPochhammer[ q^2]^2, {q, 0, n}]; (* Michael Somos, Nov 13 2014 *)
SquaresR[3,2*Range[0,70]] (* Harvey P. Dale, Jun 01 2015 *)

{a(n) = if( n<0, 0, n*=2; polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^3, n))}; /* Michael Somos, Oct 25 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^5 / eta(x^2 + A)^2 / eta(x^8 + A)^2)^3 + 12 * x * eta(x^4 + A)^3 * eta(x^8 + A)^2 / eta(x^2 + A)^2, n))}; /* Michael Somos, May 17 2008 */

{a(n) = if( n<1, n==0, 2 * qfrep( [2, 1, 1; 1, 2, 1; 1, 1, 2], n, 1)[n])}; /* Michael Somos, Jan 02 2012 */

from math import prod, isqrt
from sympy import factorint
def A004018(n): return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in factorint(n).items())<<2 if n else 1
def A004015(n): return A004018(m:=n<<1)+(sum(A004018(m-k**2) for k in range(1,isqrt(m)+1))<<1) # Chai Wah Wu, Feb 24 2025

Expansion of phi(q^2)^3 + 12 * q * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 25 2006
Expansion of (phi(q)^3 + phi(-q)^3) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006
Expansion of b(q) * phi(q^18) + c(q^3) * phi(q^2) in powers of q^3 where b(), c() are cubic AGM theta functions and phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006
Expansion of (theta_3(q)^3 + theta_4(q)^3) / 2 in powers of q^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(7/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A004013.
a(n) = A005875(2*n).
G.f.: Sum_{i, j, k in Z} x^( i*i + j*j + k*k + i*j + i*k + j*k ). - Michael Somos, Jan 02 2012
From Michael Somos, Jan 05 2012: (Start)
Number of integer solutions to x^2 + y^2 + z^2 + x*y + x*z + y*z = n.
Number of integer solutions to x + y + z even and x^2 + y^2 + z^2 = 2 * n.
Number of integer solutions to x + y + z + w = 0 and x^2 + y^2 + z^2 + w^2 = 2 * n. (End)
a(2*n) = A005875(n). a(2*n+1) = 12 * A045828(n). - Michael Somos, Dec 28 2017

A094942 Numbers having a unique partition into three squares.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 16, 19, 20, 21, 22, 24, 30, 32, 35, 37, 40, 42, 43, 44, 46, 48, 52, 56, 58, 64, 67, 70, 76, 78, 80, 84, 88, 91, 93, 96, 115, 120, 128, 133, 140, 142, 148, 160, 163, 168, 172, 176, 184, 190, 192, 208, 224, 232, 235

Offset: 1

T. D. Noe, May 24 2004

nonn

Note that squares are allowed to be zero.
From Wolfdieter Lang, Apr 09 2013: (Start)
These are the numbers for which A000164(a(n)) = 1.
a(n) is the n-th largest number which has a representation as a sum of three squares (square 0 allowed), in exactly one way, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity with order and signs taken into account are A005875(a(n)).
These numbers are a proper subset of A000378.
(End)
Note that all these numbers are 4^k * A094739(n) for some k >= 0. - T. D. Noe, Nov 08 2013

			From _Wolfdieter Lang_, Apr 09 2013 (Start)
a(1) = 0 because 0 = 0^2 + 0^2 + 0^2 and 0 is the first number m with A000164(m)=1.
a(8) = 8 = 0^2 + 2^2 + 2^2, the 8th largest number m for which A000164(m) is 1.
(End)

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

Cf. A025321 (numbers having a unique partition into three positive squares), A094739 (primitive n having a unique partition into three squares).

Cf. A000164, A005875, A000378, A224442 (two ways), A224443 (three ways).

lim=25; nLst=Table[0, {lim^2}]; Do[n=a^2+b^2+c^2; If[n>0 && nRay Chandler, Oct 31 2019 *)

The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 1, m >= 0}.

0 added by T. D. Noe, Apr 09 2013

A319574 A(n, k) = [x^k] JacobiTheta3(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, 0, 4, 6, 1, 0, 2, 0, 12, 8, 1, 0, 0, 4, 8, 24, 10, 1, 0, 0, 8, 6, 32, 40, 12, 1, 0, 0, 0, 24, 24, 80, 60, 14, 1, 0, 0, 0, 24, 48, 90, 160, 84, 16, 1, 0, 2, 4, 0, 96, 112, 252, 280, 112, 18, 1, 0, 0, 4, 12, 64, 240, 312, 574, 448, 144, 20, 1

Offset: 0

Peter Luschny, Oct 01 2018

Number of ways of writing k as a sum of n squares.

			[ 0] 1,  0,    0,    0,     0,     0,     0      0,     0,     0, ... A000007
[ 1] 1,  2,    0,    0,     2,     0,     0,     0,     0,     2, ... A000122
[ 2] 1,  4,    4,    0,     4,     8,     0,     0,     4,     4, ... A004018
[ 3] 1,  6,   12,    8,     6,    24,    24,     0,    12,    30, ... A005875
[ 4] 1,  8,   24,   32,    24,    48,    96,    64,    24,   104, ... A000118
[ 5] 1, 10,   40,   80,    90,   112,   240,   320,   200,   250, ... A000132
[ 6] 1, 12,   60,  160,   252,   312,   544,   960,  1020,   876, ... A000141
[ 7] 1, 14,   84,  280,   574,   840,  1288,  2368,  3444,  3542, ... A008451
[ 8] 1, 16,  112,  448,  1136,  2016,  3136,  5504,  9328, 12112, ... A000143
[ 9] 1, 18,  144,  672,  2034,  4320,  7392, 12672, 22608, 34802, ... A008452
[10] 1, 20,  180,  960,  3380,  8424, 16320, 28800, 52020, 88660, ... A000144
   A005843,   v, A130809,  v,  A319576,  v ,   ...      diagonal: A066535
           A046092,    A319575,       A319577,     ...

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954.
J. Carlos Moreno and Samuel S. Wagstaff Jr., Sums Of Squares Of Integers, Chapman & Hall/CRC, (2006).

Seiichi Manyama, Descending antidiagonals n = 0..139, flattened
L. Carlitz, Note on sums of four and six squares, Proc. Amer. Math. Soc. 8 (1957), 120-124.
S. H. Chan, An elementary proof of Jacobi's six squares theorem, Amer. Math. Monthly, 111 (2004), 806-811.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
Index entries for sequences related to sums of squares

Variant starting with row 1 is A122141, transpose of A286815.

A319574row := proc(n, len) series(JacobiTheta3(0, x)^n, x, len+1);
[seq(coeff(%, x, j), j=0..len-1)] end:
seq(print([n], A319574row(n, 10)), n=0..10);
# Alternative, uses function PMatrix from A357368.
PMatrix(10, n -> A000122(n-1)); # Peter Luschny, Oct 19 2022

A[n_, k_] := If[n == k == 0, 1, SquaresR[n, k]];
Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

for n in (0..10):
    Q = DiagonalQuadraticForm(ZZ, [1]*n)
    print(Q.theta_series(10).list())

A063730 Number of solutions to w^2 + x^2 + y^2 + z^2 = n in positive integers.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 6, 0, 4, 4, 0, 12, 1, 0, 12, 4, 6, 4, 12, 12, 0, 12, 6, 12, 12, 0, 24, 16, 0, 12, 18, 12, 13, 16, 12, 28, 6, 0, 36, 16, 12, 24, 24, 24, 4, 16, 30, 24, 18, 12, 36, 36, 0, 28, 42, 12, 36, 16, 24, 52, 1, 24, 48, 28, 18, 24, 60, 36, 12

Offset: 0

N. J. A. Sloane, Aug 23 2001

nonn

Cf. A063691, A063725.

Column k=4 of A337165.

r[n_] := Reduce[ w > 0 && x > 0 && y > 0 && z > 0 && w^2 + x^2 + y^2 + z^2 == n, {w, x, y, z}, Integers]; a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === Or, Length[rn], True, 1]; Table[a[n], {n, 0, 72}] (* Jean-François Alcover, Jul 22 2013 *)
a[n_ ] := Length[FindInstance[{n == w^2 + x^2 + y^2 + z^2, w > 0, x > 0, y > 0, z > 0}, {w, x, y, z}, Integers, 10^18]]; (* Michael Somos, Jun 23 2023 *)

seq(n)=Vec((sum(k=1, sqrtint(n), x^(k^2)) + O(x*x^n))^4 + O(x*x^n), -(n+1)) \\ Andrew Howroyd, Aug 08 2018

G.f.: (Sum_{m>=1} x^(m^2))^4.
a(n) = ( A000118(n) - 4*A005875(n) + 6*A004018(n) - 4*A000122(n) + A000007(n) )/16. - Max Alekseyev, Sep 29 2012
G.f.: ((theta_3(q) - 1)/2)^4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018

A224443 Numbers that are the sum of three squares (square 0 allowed) in exactly three ways.

Original entry on oeis.org

41, 50, 54, 65, 66, 74, 86, 90, 98, 99, 110, 113, 114, 117, 121, 122, 126, 131, 137, 145, 150, 164, 166, 169, 174, 178, 179, 181, 182, 186, 197, 200, 205, 216, 218, 219, 222, 226, 227, 229, 237, 258, 260, 264, 265, 275, 286, 291, 296, 302

Offset: 1

Wolfdieter Lang, Apr 08 2013

nonn

These are the numbers for which A000164(a(n)) = 3.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly three ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity of a(n) with order and signs taken into account is A005875(a(n)).
This sequence is a proper subsequence of A000378.

			a(1) = 41  = 0^2 + 4^2 + 5^2  = 1^2 + 2^2 + 6^2 = 3^3 + 4^2 + 4^2, and 41 is the first number m with A000164(m) = 3.
The representations [a,b,c] for n = 1, ..., 10, are:
n=1,  41: [0, 4, 5], [1, 2, 6], [3, 4, 4],
n=2,  50: [0, 1, 7], [0, 5, 5], [3, 4, 5],
n=3,  54: [1, 2, 7], [2, 5, 5], [3, 3, 6],
n=4,  65: [0, 1, 8], [0, 4, 7], [2, 5, 6],
n=5,  66: [1, 1, 8], [1, 4, 7], [4, 5, 5],
n=6,  74: [0, 5, 7], [1, 3, 8], [3, 4, 7],
n=7,  86: [1, 2, 9], [1, 6, 7], [5, 5, 6],
n=8,  90: [0, 3, 9], [1, 5, 8], [4, 5, 7],
n=9,  98: [0, 7, 7], [1, 4, 9], [3, 5, 8],
n=10, 99: [1, 7, 7], [3, 3, 9], [5, 5, 7].

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000164, A005875, A000378, A094942 (one way), A224442 (two ways).

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i^23, 4, min(4, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))))
    end:
a:= proc(n) option remember; local k;
      for k from 1 +`if`(n=1, 0, a(n-1))
      while b(k, isqrt(k), 3)<>3 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 09 2013

Select[ Range[0, 400], Length[ PowersRepresentations[#, 3, 2]] == 3 &] (* Jean-François Alcover, Apr 09 2013 *)

This sequence gives the increasingly ordered numbers of the set {m integer | m = a^2 + b^2 + c^2, a, b and c integers with 0 <= a <= b <= c, and m has exactly three such representations}.

The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 3, m >= 0}.

A253663 Number of positive solutions to x^2+y^2+z^2 <= n^2.

Original entry on oeis.org

0, 0, 1, 7, 17, 38, 78, 127, 196, 296, 410, 564, 738, 958, 1220, 1514, 1848, 2235, 2686, 3175, 3719, 4365, 5007, 5758, 6568, 7442, 8415, 9477, 10597, 11779, 13100, 14459, 15954, 17566, 19231, 21029, 22916, 24930, 27030, 29293, 31616, 34103, 36732, 39459

Offset: 0

R. J. Mathar, Jan 07 2015

nonn

Whereas A000604 counts solutions where x>=0, y>=0, z>=0, this sequence counts solutions where x>0, y>0, z>0.

			a(4)=17 counts the following solutions (x,y,z): (1,1,1), (2,2,2), three permutations of (1,1,2), three permutations of (1,1,3), three permutations of (1,2,2), and six permutations of (1,2,3).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000604.

[len([(x,y,z) for x in [1..n] for y in [1..n] for z in [1..n] if x^2+y^2+z^2<=n^2]) for n in [0..43]] # Tom Edgar, Jan 07 2015

a(n) = A211639(n^2).
a(n) = [x^(n^2)] (theta_3(x) - 1)^3/(8*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018
Comment from N. J. A. Sloane, Jun 02 2024 (Start)
The one-dimensional lattice {n: n an integer} , which graphically looks like
   ...o   o   o   o   o   o   ...
has theta series 1 + 2 q + 2 q^4 + 2 q^9 + 2 q^16 + ... = sum {n=-oo..oo} q^(n^2),
and that power series is called theta_3(q), A000122.
  Raising it to the power 3 counts points with x^2+y^2+z^2 = k, A005875.
Dividing it by 1-x gives the partial sums, which basically is what this sequence is.
So a first approximation to a theta series for the sequence is theta_3(q)^8/(1-q).
Subtracting 1 and dividing by 8 is because here we only want positive solutions.
(End)

cp's OEIS Frontend

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A004015 Theta series of face-centered cubic (f.c.c.) lattice.

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A094942 Numbers having a unique partition into three squares.

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