A035187 -id:A035187 - cp's OEIS frontend

A035215 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 33.

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

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 33. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A038907, A038908.

a[n_] := DivisorSum[n, KroneckerSymbol[33, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)

my(m = 33); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(33, d)); \\ Amiram Eldar, Nov 19 2023

From Amiram Eldar, Nov 19 2023: (Start)
a(n) = Sum_{d|n} Kronecker(33, d).
Multiplicative with a(p^e) = 1 if Kronecker(33, p) = 0 (p = 3 or 11), a(p^e) = (1+(-1)^e)/2 if Kronecker(33, p) = -1 (p is in A038908), and a(p^e) = e+1 if Kronecker(33, p) = 1 (p is in A038907 \ {3, 11}).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(4*sqrt(33)+23)/sqrt(33) = 1.332797188186... . (End)

A341783 Absolute values of norms of prime elements in Z[(1+sqrt(5))/2], the ring of integers of Q(sqrt(5)).

Original entry on oeis.org

4, 5, 9, 11, 19, 29, 31, 41, 49, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 169, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 289, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 509, 521, 529

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(5))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes congruent to 0, 1, 4 modulo 5 and the squares of primes congruent to 2, 3 modulo 5.
For primes p == 1, 4 (mod 5), there are two distinct ideals with norm p in Z[(1+sqrt(5))/2], namely (x + y*(1+sqrt(5))/2) and (x + y*(1-sqrt(5))/2), where (x,y) is a solution to x^2 + x*y - y^2 = p; for p = 5, (sqrt(5)) is the unique ideal with norm p; for p == 2, 3 (mod 5), (p) is the only ideal with norm p^2.

			norm((7 + sqrt(5))/2) = norm((7 - sqrt(5))/2) = 11;
norm((9 + sqrt(5))/2) = norm((9 - sqrt(5))/2) = 19;
norm((11 + sqrt(5))/2) = norm((11 - sqrt(5))/2) = 29;
norm(6 + sqrt(5)) = norm(6 - sqrt(5)) = 31.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A080891, A038872, A045468, A003631.
The number of nonassociative elements with absolute value of norm n (also the number of distinct ideals with norm n) is given by A035187.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), this sequence (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341783(n) = my(disc=5); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A327716 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A003823.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 3, 4, 6, 7, 9, 10, 12, 14, 17, 21, 23, 26, 32, 40, 45, 51, 58, 69, 80, 89, 102, 116, 135, 154, 177, 198, 224, 253, 288, 326, 361, 408, 459, 521, 583, 650, 723, 812, 909, 1009, 1122, 1244, 1393, 1547, 1716, 1898, 2101, 2326, 2575, 2845, 3132, 3456, 3809

Offset: 0

Seiichi Manyama, Sep 23 2019

nonn

a(n) > 0.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

Convolution inverse of A327688.

Cf. A003823, A035187, A327690, A327691, A327694, A327717, A327718, A327719, A327720.

nmax = 60; CoefficientList[Series[Product[QPochhammer[x^(5*j - 3)] * QPochhammer[x^(5*j - 2)]/(QPochhammer[x^(5*j - 4)] * QPochhammer[x^(5*j - 1)]), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-x^k)^sumdiv(k, d, kronecker(5, d))))

G.f.: Product_{i>=1} Product_{j>=1} (1-x^(i*(5*j-2))) * (1-x^(i*(5*j-3))) / ((1-x^(i*(5*j-1))) * (1-x^(i*(5*j-4)))).
G.f.: Product_{k>=1} (1-x^k)^(-A035187(k)).
a(n) ~ c * exp(Pi*sqrt(r*n)) / n^(3/4), where r = 4*log((1+sqrt(5))/2) / (3*sqrt(5)) = 0.2869392939760026925..., c = 0.203427046022096... - Vaclav Kotesovec, Sep 24 2019, updated May 09 2020

A327688 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A007325.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, 0, 0, 0, 0, -2, 2, 1, 0, 1, -1, -1, -1, -1, 2, 1, 0, 1, -1, -3, 1, 2, -1, 0, 4, -6, -2, 3, -1, 1, 4, -1, -2, -1, 2, -4, 4, 0, -3, 1, -3, 4, 2, -1, 3, -1, -3, -1, 2, -3, 1, 2, -6, -3, 12, -7, 3, 11, -7, -4, 7, -10, -1, 7, 2, -16, 11, 2, -10, 14, -4, 3, -3

Offset: 0

Seiichi Manyama, Sep 22 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

Cf. A007325, A035187, A307757, A327686, A327690, A327691, A327716.

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^sumdiv(k, d, kronecker(5, d))))

G.f.: Product_{i>=1} Product_{j>=1} (1-x^(i*(5*j-1))) * (1-x^(i*(5*j-4))) / ((1-x^(i*(5*j-2))) * (1-x^(i*(5*j-3)))).

G.f.: Product_{k>=1} (1-x^k)^A035187(k).

A035184 a(n) = Sum_{d|n} Kronecker(-1, d).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 0, 4, 1, 4, 0, 0, 2, 0, 0, 5, 2, 2, 0, 6, 0, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 6, 0, 4, 0, 3, 2, 0, 0, 8, 2, 0, 0, 0, 2, 0, 0, 0, 1, 6, 0, 6, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 0, 0, 0, 0, 0, 10, 1, 4, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 2, 2, 0, 9, 2, 0, 0, 8, 0

Offset: 1

N. J. A. Sloane

			G.f. = x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^8 + x^9 + 4*x^10 + 2*x^13 + 5*x^16 + 2*x^17 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002175, A008441, A019669, A053692, A113407, A121444.
Inverse Moebius transform of A034947.
Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), this sequence (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).

a[n_] := DivisorSum[n, KroneckerSymbol[-1, #] &]; Array[a, 105] (* Jean-François Alcover, Dec 02 2015 *)

{a(n) = if( n<1, 0, direuler( p=2, n, 1/((1 - X) * (1 - kronecker( -1, p) * X))) [n])}; /* Michael Somos, Jan 05 2012 */

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -1, d)))}; /* Michael Somos, Jan 05 2012 */

a(n) is multiplicative with a(2^e) = e + 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4). - Michael Somos, Jan 05 2012
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n).
Dirichlet g.f.: zeta(s)*beta(s)/(1 - 2^(-s)), where beta is the Dirichlet beta function. - Ralf Stephan, Mar 27 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 = 1.570796... (A019669). - Amiram Eldar, Oct 17 2022

A035282 Expansion of zeta function of icosian ring (nonzero terms).

Original entry on oeis.org

1, 5, 6, 10, 24, 21, 40, 30, 31, 60, 64, 50, 84, 120, 60, 50, 144, 120, 124, 85, 144, 200, 160, 126, 91, 180, 240, 240, 155, 204, 220, 300, 410, 320, 156, 264, 280, 210, 360, 300, 304, 384, 420, 170, 400, 504, 360, 300, 364, 384, 250, 400, 504, 960, 424, 720, 310

Offset: 1

N. J. A. Sloane

Let zetaI(s) be the zeta function of icosian ring: zetaI(s) = zetaQ(tau)(2s)*zetaQ(tau)(2s-1) where zetaQ(tau)(s) is defined in A035187; then zetaI(s) = Sum_{n>=1} a(n)/n^(2s).

Nonzero terms of A078473. - Michel Marcus, Mar 03 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. (1999), 51 1258-1276.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999.

Cf. A031363, A035187, A078472, A078473.

f[p_, e_] := Which[p == 5, (5^(e + 1) - 1)/4, (m = Mod[p, 5]) == 2 || m == 3, If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 0], m == 1 || m == 4, Sum[(k + 1)*(e - k + 1)*p^k, {k, 0, e}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 0 &] (* Amiram Eldar, May 13 2022 *)

A035181 a(n) = Sum_{d|n} Kronecker(-9, d).

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 0, 4, 1, 4, 0, 3, 2, 0, 2, 5, 2, 2, 0, 6, 0, 0, 0, 4, 3, 4, 1, 0, 2, 4, 0, 6, 0, 4, 0, 3, 2, 0, 2, 8, 2, 0, 0, 0, 2, 0, 0, 5, 1, 6, 2, 6, 2, 2, 0, 0, 0, 4, 0, 6, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 3, 0, 0, 4, 0, 10, 1, 4, 0, 0, 4, 0, 2, 0, 2, 4, 0, 0, 0, 0, 0, 6, 2, 2, 0, 9, 2, 4, 0, 8, 0

Offset: 1

N. J. A. Sloane

			x + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 4*x^8 + x^9 + 4*x^10 + 3*x^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A008441, A019693, A125079.

Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), A035184 (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -9, d], { d, Divisors[ n]}]] (* Michael Somos, Jun 24 2011 *)

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -9, d)))} \\ Michael Somos, Jun 24 2011

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -9, p) * X))) [n])} \\ Michael Somos, Jun 24 2011

{a(n) = local(A, p, e); if( n<0, 0, A = factor(n); prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if( p==2, e+1, if( p==3, 1, if( p%4==1, e+1, (1 + (-1)^e)/2))))))} \\ Michael Somos, Jun 24 2011

A035181(n)=sumdivmult(n,d,kronecker(-9,d)) \\ M. F. Hasler, May 08 2018

From Michael Somos, Jun 24 2011: (Start)
a(n) is multiplicative with a(2^e) = e + 1, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4) and p > 3.
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker(-9, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker(-9, p) * p^-s)). (End)
a(3*n) = a(n). a(2*n + 1) = A125079(n). a(4*n + 1) = A008441(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/3 = 2.094395... (A019693). - Amiram Eldar, Oct 17 2022

A078473 Expansion of zeta function of icosian ring.

Original entry on oeis.org

1, 0, 0, 5, 6, 0, 0, 0, 10, 0, 24, 0, 0, 0, 0, 21, 0, 0, 40, 30, 0, 0, 0, 0, 31, 0, 0, 0, 60, 0, 64, 0, 0, 0, 0, 50, 0, 0, 0, 0, 84, 0, 0, 120, 60, 0, 0, 0, 50, 0, 0, 0, 0, 0, 144, 0, 0, 0, 120, 0, 124, 0, 0, 85, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 0, 200, 0, 0, 160, 126, 91, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Benoit Cloitre, Dec 31 2002

Let zetaI(s) be the zeta function of icosian ring: zetaI(s) = zetaQ(tau)(2s)*zetaQ(tau)(2s-1) where zetaQ(tau)(s) is defined in A035187. Then zetaI(s) = Sum_{n>=1} a(n)/n^(2s).

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276; arXiv preprint, arXiv:math/9904028 [math.MG], 1999.

Cf. A035187, A035282 (nonzero terms of the sequence), A031363 (n for which a(n) is not zero), A078471.

f[p_, e_] := Which[p == 5, (5^(e + 1) - 1)/4, (m = Mod[p, 5]) == 2 || m == 3, If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 0], m == 1 || m == 4, Sum[(k + 1)*(e - k + 1)*p^k, {k, 0, e}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 13 2022 *)

{a(n)=local(A); if(n<1, 0, A=direuler(p=2,n,1/(1-X)/(1-kronecker(5,p)*X)); sumdiv(n,d,A[d]*d*A[n/d]))} /* Michael Somos, Jun 06 2005 */

pf(p, r) = {if (p==5, (5^(r+1) -1)/4, if (((p % 5) == 2) || ((p % 5) == 3), if (!(r % 2), (p^(r+2) - 1)/(p^2-1), 0), if (((p % 5) == 1) || ((p % 5) == 4), sum(k=0, r, (k+1)*(r-k+1)*p^k))););}
a(n) = {my(f = factor(n)); prod(i=1, #f~, pf(f[i, 1], f[i, 2]));} \\ Michel Marcus, Mar 03 2014

Multiplicative with a(p^e) = (5^(e + 1) - 1)/4 if p = 5, (p^(e + 2) - 1)/(p^2 - 1) or 0 if p == 2 or 3 (mod 5) and e is even or odd, respectively, and Sum_{k=0..e} (k + 1)*(e - k + 1)*p^k if p == 1 or 4 (mod 5). - Amiram Eldar, May 13 2022

A263849 Let R = Z[(1+sqrt(5))/2] denote the ring of integers in the real quadratic number field of discriminant 5. Then a(n) is the largest integer k such that every totally positive element nu in R of norm m = A031363(n) can be written as a sum of three squares in R in at least k ways.

Original entry on oeis.org

1, 6, 12, 24, 32, 24, 54, 24, 24, 30, 24, 48, 48, 96, 24, 48, 96, 48, 24, 120, 108, 48, 72, 48, 120, 54, 48, 48, 48, 84, 72, 120, 72, 78, 48, 144, 72, 72, 128, 192, 120, 96, 48, 48, 96, 96, 216, 72, 48, 120, 96, 96, 48, 96, 48, 120, 96, 224, 72, 120, 48, 288, 72, 48, 72, 246, 240, 120, 144

Offset: 0

N. J. A. Sloane, Nov 15 2015

nonn

Let R = Z[(1+sqrt{5})/2] denote the ring of integers in the real quadratic number field of discriminant 5. The main result of Maass (1941) is that every totally positive nu in R is a sum of 3 squares x^2+y^2+z^2 with x,y,z in R. The number N_{nu} of such representations is given by the formula in the theorem on page 191. The norms of the totally positive elements nu are rational integers m belonging to A031363, so we can order the terms of the sequence according to the values m = A031363(n). [Comment based on remarks from Gabriele Nebe.]
The terms were computed with the aid of Magma by David Durstoff, Nov 11 2015.
The attached file from David Durstoff gives list of pairs m=A031363(n), a(n), and also the initial terms of Maass's series theta(tau).  David Durstoff says: "I expressed theta(tau) in terms of two variables q1 and q2. The coefficient of q1^k q2^m is a(nu) with k = trace(nu/delta) and m = trace(nu), where delta = (5+sqrt{5})/2 is a generator of the different ideal. I computed the terms for q1^0 to q1^10 and all possible powers of q2."
Note that there are examples of totally positive elements x and y in R which have the same norm, but for which the number of ways of writing x as a sum of three squares in R is different to the number of ways of writing y as a sum of three squares in R.  E.g. there are 78 ways of writing (27 + 7*sqrt(5))/2 as a sum of three squares in R, and there are 192 ways of writing 11 as a sum of three squares in R, yet both elements have norm 121.  The sequence of possible norms for which this can occur is 121, 209, 341, 361, 451, 484, 551, 589, 605, 649, ... - Robin Visser, Mar 28 2025

			From _Robin Visser_, Mar 30 2025: (Start)
a(1) = 6, as every totally positive element of norm A031363(1)=1 in R can be written as a sum of three squares in R in exactly 6 ways.  E.g. the element 1 in R has norm 1 and can be written as a sum of three squares in R as:
  1 = 1^2 + 0^2 + 0^2 = (-1)^2 + 0^2 + 0^2 = 0^2 + 1^2 + 0^2 = 0^2 + (-1)^2 + 0^2 = 0^2 + 0^2 + 1^2 = 0^2 + 0^2 + (-1)^2.
a(2) = 12, as every totally positive element of norm A031363(2)=4 in R can be written as a sum of three squares in R in exactly 12 ways. E.g. the element 2 in R has norm 4 and can be written as a sum of three squares in R as:
  2 = 1^2 + 1^2 + 0^2 = 1^2 + 0^2 + 1^2 = 0^2 + 1^2 + 1^2 = 1^2 + (-1)^2 + 0^2 = 1^2 + 0^2 + (-1)^2 = 0^2 + 1^2 + (-1)^2 = (-1)^2 + 1^2 + 0^2 = (-1)^2 + 0^2 + 1^2 = 0^2 + (-1)^2 + 1^2 = (-1)^2 + (-1)^2 + 0^2 = (-1)^2 + 0^2 + (-1)^2 = 0^2 + (-1)^2 + (-1)^2.
a(3) = 24, as every totally positive element of norm A031363(3)=5 in R can be written as a sum of three squares in R in exactly 24 ways. E.g. the element (5+sqrt(5))/2 in R has norm 5 and can be written as a sum of three squares in R as:
  (5+sqrt(5))/2 = w^2 + 1^2 + 0^2 = w^2 + 0^2 + 1^2 = 0^2 + w^2 + 1^2 = 1^2 + w^2 + 0^2 = 0^2 + 1^2 + w^2 = 1^2 + 0^2 + w^2 = (-w)^2 + 1^2 + 0^2 = (-w)^2 + 0^2 + 1^2 = 0^2 + (-w)^2 + 1^2 = 1^2 + (-w)^2 + 0^2 = 0^2 + 1^2 + (-w)^2 = 1^2 + 0^2 + (-w)^2 = w^2 + (-1)^2 + 0^2 = w^2 + 0^2 + (-1)^2 = 0^2 + w^2 + (-1)^2 = (-1)^2 + w^2 + 0^2 = 0^2 + (-1)^2 + w^2 = (-1)^2 + 0^2 + w^2 = (-w)^2 + (-1)^2 + 0^2 = (-w)^2 + 0^2 + (-1)^2 = 0^2 + (-w)^2 + (-1)^2 = (-1)^2 + (-w)^2 + 0^2 = 0^2 + (-1)^2 + (-w)^2 = (-1)^2 + 0^2 + (-w)^2, where w = (1+sqrt(5))/2. (End)

Maass, Hans. Über die Darstellung total positiver Zahlen des Körpers R (sqrt(5)) als Summe von drei Quadraten, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Vol. 14. No. 1, pp. 185-191, 1941.

Robin Visser, Table of n, a(n) for n = 0..6000
Harvey Cohn, A computation of some bi-quadratic class numbers, Math. Tables Aids Comput. 12 (1958), 213-217. See pages 215-217.
David Durstoff, Table showing list of pairs m=A031363(n), a(n)

Cf. A031363 (the norms), A035187 (number of ideals with that norm).
See A263850 for another version of this sequence.
Cf. A005875 (sum of 3 squares in Z), A000118 (sum of 4 squares in Z).

More terms from Robin Visser, Mar 28 2025

A263850 Let R = Z[(1+sqrt(5))/2] denote the ring of integers in the real quadratic number field of discriminant 5. Then a(n) is the largest integer k such that every totally positive element of norm n in R can be written as a sum of three squares in R in at least k ways, or 0 if there is no totally positive element of norm n.

Original entry on oeis.org

1, 6, 0, 0, 12, 24, 0, 0, 0, 32, 0, 24, 0, 0, 0, 0, 54, 0, 0, 24, 24, 0, 0, 0, 0, 30, 0, 0, 0, 24, 0, 48, 0, 0, 0, 0, 48, 0, 0, 0, 0, 96, 0, 0, 24, 48, 0, 0, 0, 96, 0, 0, 0, 0, 0, 48, 0, 0, 0, 24, 0, 120, 0, 0, 108, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 72, 0, 0, 48, 120, 54, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0

Offset: 0

N. J. A. Sloane, Nov 18 2015

nonn

The terms were computed with the aid of Magma by David Durstoff, Nov 11 2015. See A263849 for further information.

Note that there are examples of totally positive elements x and y in R which have the same norm, but for which the number of ways of writing x as a sum of three squares in R is different to the number of ways of writing y as a sum of three squares in R. See A263849 for explicit examples. - Robin Visser, Mar 28 2025

Maass, Hans. Über die Darstellung total positiver Zahlen des Körpers R (sqrt(5)) als Summe von drei Quadraten, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Vol. 14. No. 1, pp. 185-191, 1941.

Robin Visser, Table of n, a(n) for n = 0..10000
Harvey Cohn, A computation of some bi-quadratic class numbers, Math. Tables Aids Comput. 12 (1958), 213-217. See pages 215-217.

Cf. A263849 (another version of this sequence), A031363, A035187.

Cf. A005875 (sum of 3 squares in Z), A000118 (sum of 4 squares in Z).

import itertools
def a(n):
    if n==0: return 1
    if any([((r[0]%5 in [2,3]) and (r[1]%2==1)) for r in factor(n)]): return 0
    K. = NumberField(x^2-x-1); cw = w.coordinates_in_terms_of_powers(); ans = []
    for idl in K.ideals_of_bdd_norm(n)[n]:
        for u in [1,-1,w,-w]:
            X,Y = cw(u*idl.gens_reduced()[0]); num = 0
            if (X < 0): continue
            for b in range(-isqrt(X), isqrt(X)+1):
                for d in range(-isqrt(X-b^2), isqrt(X-b^2)+1):
                    for f in range(-isqrt(X-b^2-d^2), isqrt(X-b^2-d^2)+1):
                        S = b^2+d^2+f^2; M = isqrt(X-S);
                        for (a,c,e) in itertools.product(range(-M, M+1), repeat=3):
                            if (a^2+c^2+e^2+S==X) and (2*(a*b+c*d+e*f)+S==Y): num += 1
            if (num > 0): ans.append(num)
    return min(ans)  # Robin Visser, Mar 28 2025

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