A005875 -id:A005875 - cp's OEIS frontend

A071609 Squared radii of the spheres around (0,0,0) that contain record numbers of lattice points.

1, 2, 5, 9, 14, 26, 41, 74, 89, 101, 146, 194, 269, 314, 341, 446, 614, 626, 689, 794, 854, 941, 1106, 1109, 1154, 1286, 1361, 1634, 1781, 1889, 2141, 2609, 2966, 3134, 3401, 3449, 3506, 4241, 4289, 4826, 5381, 5561, 6254, 7229, 7829, 8069, 8126

Offset: 1

Hugo Pfoertner, May 25 2002

nonn

The number of lattice points (i,j,k) on the sphere around (0,0,0) with i^2 + j^2 + k^2 = a(n) is given by A071611(n).

Indices of records in A005875. - Daniel Suteu, Aug 13 2021

Hugo Pfoertner, Table of n, a(n) for n = 1..189

Cf. A005875, A071342, A071343, A071344, A071610, A071611.

A083703 Expansion of eta(q)^4/eta(q^4) in powers of q.

Original entry on oeis.org

1, -4, 2, 8, -4, -8, -8, 16, 6, -12, 8, 8, -8, -24, 0, 16, 12, -16, 10, 24, -8, -16, -24, 16, 8, -28, 8, 32, -16, -8, 0, 32, 6, -32, 16, 16, -12, -40, -24, 16, 24, -16, 16, 40, -8, -40, 0, 32, 24, -36, 10, 16, -24, -24, -32, 48, 0, -32, 24, 24, -16, -40, 0, 48, 12, -16, 16, 56, -16, -32, -48, 16, 30, -64, 8, 40, -24

Offset: 0

Michael Somos, May 04 2003

sign

Euler transform of period 4 sequence [ -4,-4,-4,-3,...].

Seiichi Manyama, Table of n, a(n) for n = 0..10000

A080965(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).
a(2n)=0 iff A005875(n)=0.
Cf. A274327, A285895.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d, 4)=0, -3, -4), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jan 07 2017

CoefficientList[QPochhammer[x]^4/QPochhammer[x^4] + O[x]^80, x] (* Jean-François Alcover, Sep 19 2016 *)

a(n)=if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^4/eta(X^4),n))

G.f.: Product_{n>0} (1-x^n)^4/(1-x^(4n)).

a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A285895(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

A085121 Number of ways of writing n as the sum of three odd squares.

Original entry on oeis.org

0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 56, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 72

Offset: 0

N. J. A. Sloane, Apr 25 2004

nonn

Number of ways of writing n as the sum of the squares of three odd numbers (see example). Equals 8*A008437 because each summand can be the square of either a positive or negative odd number, and there are three summands, thus 2^3 = 8. - Antti Karttunen & Michel Marcus, Jul 23 2018

			a(3) = 8 because 3 = (+1)^2 + (+1)^2 + (+1)^2 = (-1)^2 + (+1)^2 + (+1)^2 = (+1)^2 + (-1)^2 + (+1)^2 = (+1)^2 + (+1)^2 + (-1)^2 = (-1)^2 + (-1)^2 + (+1)^2 = (-1)^2 + (+1)^2 + (-1)^2 = (+1)^2 + (-1)^2 + (-1)^2 = (-1)^2 + (-1)^2 + (-1)^2. - _Antti Karttunen_, Jul 23 2018

Antti Karttunen, Table of n, a(n) for n = 0..65537
J. E. Jones [Lennard-Jones] and A. E. Ingham, On the calculation of certain crystal potential constants and on the cubic crystal of least energy, Proc. Royal Soc., A 107 (1925), 636-653 (see p. 650).

Cf. A005875, A008437. The nonzero coefficients give A005878.

A008442(n) = if( n<1 || n%4!=1, 0, sumdiv(n, d, (d%4==1) - (d%4==3))); \\ From A008442.
A290081(n) = if(n%2,0,A008442(n/2));
A008437(n) = if((n<3)||!(n%2),0,my(s=0, k = sqrtint(n)); k -= ((1+k)%2); while(k>=1, s += A290081(n-(k*k)); k -= 2); (s));
A085121(n) = 8*A008437(n); \\ Antti Karttunen, Jul 22 2018

G.f.: (Sum_{n=-oo..oo} q^((2n+1)^2))^3.

A169784 Number of solutions to a^2 + b^2 + 5*c^2 = n.

Original entry on oeis.org

1, 4, 4, 0, 4, 10, 8, 8, 4, 12, 24, 0, 0, 16, 8, 16, 4, 8, 20, 0, 10, 16, 24, 8, 8, 44, 8, 0, 8, 16, 40, 16, 4, 16, 24, 0, 12, 32, 8, 16, 24, 16, 16, 0, 0, 50, 40, 8, 0, 28, 44, 0, 16, 16, 32, 40, 8, 32, 40, 0, 16, 32, 16, 24, 4, 48, 16, 0, 8, 16, 80, 16, 20, 40, 24, 0, 0, 16, 32, 32, 10, 36

Offset: 0

N. J. A. Sloane, May 12 2010

nonn

a(n) = 0 for n of the form 4^i*(8*j+3), A055046. [Zak Seidov, May 12 2010]

Zak Seidov, Table of n,a(n) for n=0..1000

x^2+y^2+k*z^2: A005875, A014455, A034933, A169783.

G.f.: theta_3(q)^2*theta_3(q^5), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A290734 Number of compact partitions of n containing a 1 where each partition is counted with a certain weight.

Original entry on oeis.org

0, -1, 2, -2, 3, -6, 6, -4, 6, -9, 10, -8, 6, -12, 14, -6, 7, -16, 16, -8, 10, -18, 14, -8, 8, -17, 24, -10, 6, -24, 22, -4, 10, -22, 20, -16, 9, -20, 28, -6, 8, -32, 26, -8, 14, -28, 24, -8, 8, -25, 34, -18, 6, -36, 34, -2, 18, -28, 24, -24, 10, -28, 40, -4, 7, -42, 38, -12, 18, -40, 26, -12, 12, -28

Offset: 0

N. J. A. Sloane, Aug 10 2017

sign

See Andrews (2016) for the definition of the particular weight used here.

4*A290733(n) + 2*a(n) = (-1)^n*A005875(n) for n > 0.

George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 2.2.

Cf. A005875, A290733.

M:=101;
B:=proc(a,q,n) local j,t1; global M;
t1:=1;
for j from 0 to M do
t1:=t1*(1-a*q^j)/(1-a*q^(n+j));
od;
t1; end;
# c_1
T1:=add( (-1)^m*q^(m*(m+1)/2)/(B(-q,q,m)*(1+q^m)), m=1..M):
series(T1,q,M); c1seq:=seriestolist(%);

See Maple program for g.f.

A294595 Numbers that are the sum of three squares (square 0 allowed) in exactly six ways.

Original entry on oeis.org

194, 209, 269, 281, 290, 297, 321, 326, 329, 342, 365, 386, 389, 401, 426, 434, 449, 459, 482, 485, 489, 497, 513, 531, 534, 542, 546, 554, 558, 561, 578, 601, 602, 633, 649, 659, 662, 665, 675, 678, 681, 693, 699, 705, 713, 714, 722, 737, 741, 747, 750, 754

Offset: 1

Robert Price, Nov 03 2017

nonn

These are the numbers for which A000164(a(n)) = 6.
a(n) is the n-th largest number which has a representation as a sum of three integer squares (square 0 allowed), in exactly six 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.

Robert Price, Table of n, a(n) for n = 1..1400

Cf. A000164, A005875, A000378, A094942, A224442, A224443, A294577, A294594.

Select[Range[0, 1000], Length[PowersRepresentations[#, 3, 2]] == 6 &]

Updated Mathematica program to Version 11. by Robert Price, Nov 01 2019

A343631 X-coordinate of the points following the 3D spiral defined in A343630.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 28 2021

sign

See the main entry A343630 for details about this 3D generalization of an Ulam type spiral using the Euclidean norm.
Sequences A343632 and A343633 give the y and z coordinates.
The sequence can be seen as a table with row lengths 3*A005875, where A005875(r) is the number of points at distance sqrt(r) from the origin.
Sequence A343641 is the analog for the square spiral variant A343640.

Cf. A343632, A343633 (list of y and z-coordinates).
Cf. A343641 (variant using the sup norm => square spiral).
Cf. A342561 (variant which scans each sphere by increasing z).
Cf. A005875 (number of points on a shell with given radius).
Cf. A004215 (numbers that can't be written as sum of 3 squares => empty shells).

d=1; A343631_vec=concat([[P[1] | P<-S=A343630_row(n,d)]+(#S&&!d*=-1) | n<-[0..8]]) \\ the variable d is necessary to correct the z-scan direction in rows between A004215(2k-1) and A004215(2k).

A343632 Y-coordinate of the points following the 3D spiral defined in A343630.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 28 2021

sign

See the main entry A343630 for details about this 3D generalization of an Ulam type spiral using the Euclidean norm.
Sequences A343631 and A343633 give the x and z coordinates.
The sequence can be seen as a table with row lengths A005875, where A005875(r) is the number of points at distance sqrt(r) from the origin.
Sequence A343642 is the analog for the square spiral variant A343640.

Cf. A343631, A343633 (list of x and z-coordinates).
Cf. A343642 (variant using the sup norm => square spiral).
Cf. A342562 (variant which scans each sphere by increasing z).
Cf. A005875 (number of points on a shell with given radius).
Cf. A004215 (numbers that can't be written as sum of 3 squares => empty shells).

d=1; A343632_vec=concat([[P[2] | P<-S=A343630_row(n,d)]+(#S&&!d*=-1) | n<-[0..9]]) \\ the variable d is necessary to correct the z-scan direction in rows between A004215(2k-1) and A004215(2k).

A066536 Number of ways of writing n as a sum of n+1 squares.

Original entry on oeis.org

1, 4, 12, 32, 90, 312, 1288, 5504, 22608, 88660, 339064, 1297056, 5043376, 19975256, 80027280, 321692928, 1291650786, 5177295432, 20748447108, 83279292960, 335056780464, 1351064867328, 5456890474248, 22063059606912

Offset: 0

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

nonn

			There are a(2)=12 solutions (x,y,z) of 2=x^2+y^2+z^2: 3 permutations of (1,1,0), 3 permutations of (-1,-1,0) and 6 permutations of (1, -1,0).

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

Cf. A004018, A005875, A000118, A066535.

Cf. A122141, A166952. - Paul D. Hanna, Oct 26 2009

Join[{1}, Table[SquaresR[n+1, n], {n, 24}]]
(* Calculation of constants {d,c}: *) {1/r, Sqrt[s/(2*Pi*r^2*Derivative[0, 0, 2][EllipticTheta][3, 0, r*s])]} /. FindRoot[{s == EllipticTheta[3, 0, r*s], r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] == 1}, {r, 1/4}, {s, 5/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 16 2023 *)

{a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n));polcoeff(THETA3^(n+1), n)} /* Paul D. Hanna, Oct 26 2009*/

a(n) equals the coefficient of x^n in the (n+1)-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Oct 26 2009
a(n) is divisible by n+1: a(n)/(n+1) = A166952(n) for n>=0. - Paul D. Hanna, Oct 26 2009
a(n) ~ c * d^n / sqrt(n), where d = 4.13273137623493996302796465... (= 1/radius of convergence A166952), c = 0.70710538549959357505200... . - Vaclav Kotesovec, Sep 10 2014

Edited by Dean Hickerson, Jan 12 2002
a(0) added by Paul D. Hanna, Oct 26 2009
Edited by R. J. Mathar, Oct 29 2009

A080965 Expansion of eta(q^2)^12/(eta(q)^4eta(q^4)^5) in powers of q.

Original entry on oeis.org

1, 4, 2, -8, -4, 8, -8, -16, 6, 12, 8, -8, -8, 24, 0, -16, 12, 16, 10, -24, -8, 16, -24, -16, 8, 28, 8, -32, -16, 8, 0, -32, 6, 32, 16, -16, -12, 40, -24, -16, 24, 16, 16, -40, -8, 40, 0, -32, 24, 36, 10, -16, -24, 24, -32, -48, 0, 32, 24, -24, -16, 40, 0, -48, 12, 16, 16

Offset: 0

Michael Somos, Feb 28 2003

sign

Euler transform of period 4 sequence [4,-8,4,-3,...].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

a(n)=A080964(4n)=2*A072071(4n)-A072070(4n).
A083703(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).
a(2n)=0 iff A005875(n)=0.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add([-3, 4, -8, 4]
      [1+irem(d, 4)]*d, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 05 2015

a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, {-3, 4, -8, 4}[[1 + Mod[#, 4]]]*#&]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 25 2015, after Alois P. Heinz *)

a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^-4*eta(X^2)^12*eta(X^4)^-5,n))

G.f.: Product_{n>0} (1-x^(2n))^12/((1-x^n)^4(1-x^(4n))^5).

cp's OEIS Frontend

A071609 Squared radii of the spheres around (0,0,0) that contain record numbers of lattice points.

Views

Author

Keywords

Comments

Links

Crossrefs

A083703 Expansion of eta(q)^4/eta(q^4) in powers of q.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A085121 Number of ways of writing n as the sum of three odd squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A169784 Number of solutions to a^2 + b^2 + 5*c^2 = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A290734 Number of compact partitions of n containing a 1 where each partition is counted with a certain weight.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A294595 Numbers that are the sum of three squares (square 0 allowed) in exactly six ways.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A343631 X-coordinate of the points following the 3D spiral defined in A343630.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A343632 Y-coordinate of the points following the 3D spiral defined in A343630.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A066536 Number of ways of writing n as a sum of n+1 squares.

Views

Author

Keywords

Examples