A002324 -id:A002324 - cp's OEIS frontend

A035187 Sum over divisors d of n of Kronecker symbol (5|d).

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

Offset: 1

N. J. A. Sloane

Let tau be the golden ratio (1+sqrt(5))/2; let zetaQ(tau)(s)=sum(1/(Z(tau):a)^s) the Dedekind zeta function where a runs through the nonzero ideals of Z(tau) and where (Z(tau):a) is the norm of a; then zetaQ(tau)(s)=sum(n>=1,a(n)/n^s). - Benoit Cloitre, Dec 29 2002
First occurrence of k beginning at zero, or 0 if not yet known: 2, 1, 11, 121, 209, 14641, 2299, 1771561, 6061, 43681, 278179, 0, 66671, 0, 33659659, 5285401, 187891, 0, 1266749, 0, 8067191, 639533521, 0, 0, 2066801, 0, 0, 36735721, 976130111, 0, 153276629, 0, 7703531, 0, 0, 0, 39269219, 0, 0, 0, 250082921, 0, 0, 0, 0, 0, 0, 0, 84738841, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 454508329, ..., .
If k is prime, the 0 above can be replaced by the smallest p^(k-1) with p a prime == {1,4} (mod 5), which is p=11. This follows from the multiplicative formula. - R. J. Mathar, Apr 02 2011
The terms often equal A001157(n) mod 5; the exceptions are at n = 2299, 3509, 3751, 3971, 4961, 6061, 6479, ... - R. J. Mathar, Apr 02 2011
Coefficients of Dedekind zeta function for the quadratic number field of discriminant 5. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			G.f. = x + x^4 + x^5 + x^9 + 2*x^11 + x^16 + 2*x^19 + x^20 + x^25 + 2*x^29 + ...

G. C. Greubel, Table of n, a(n) for n = 1..5000
M. Baake, Algebra, Combinatorics and Number Theory.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.

Cf. A031363 (for indices of nonzero terms), A078428.
Cf. A001622, A086466.
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.

A035187 := proc(n) local f,p; f := ifactors(n)[2] ; if nops(f) = 1 then p := op(1,f) ; if op(1,p) = 5 then 1; elif op(1,p) mod 5 in {1,4} then op(2,p)+1 ; else (1+(-1)^op(2,p))/2 ; end if; else mul(procname(op(1,p)^op(2,p) ),p=f) ; end if;
end proc: # R. J. Mathar, Apr 02 2011

f[n_] := Plus @@ (KroneckerSymbol[5, #] & /@ Divisors@ n); Array[f, 105] (* Robert G. Wilson v *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ 5, #] &]]; (* Michael Somos, Jun 12 2014 *)

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

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p = A[k,1], e = A[k,2]; if( p==5, 1, if((p%5==1) || (p%5==4), e+1, !(e%2))))))}; \\ Michael Somos, Jun 06 2005

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( 5, d) ) )}; \\ Michael Somos, Oct 29 2005

Dirichlet g.f.: Product_p ( (1 - p^(-s)) (1 - Kronecker( 5, p)*p^(-s)) )^(-1).
Sum_{k=1..n} a(k) is asymptotic to c*n where c=2*log(tau)/sqrt(5) (A086466).
Multiplicative with a(5^e) = 1, a(p^e) = e+1 if p == 1, 4 (mod 5), a(p^e) = (1+(-1)^e)/2 if p == 2, 3 (mod 5). - Michael Somos, Jun 06 2005
Moebius transform is period 5 sequence A080891. - Michael Somos, Oct 29 2005
q-series for a(n): Sum_{n >= 1} -(-1)^nq^(n(n+1)/2)(1-q)(1-q^2)...(1-q^(n-1))/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))). - Jeremy Lovejoy, Jun 12 2009

A035185 Number of divisors of n == 1 or 7 (mod 8) minus number of divisors of n == 3 or 5 (mod 8).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = 2.
Let zetaQ(sqrt(2))(s) = Sum (1/(Z(sqrt(2)):A)^s), a Dedekind zeta function, where A runs through the nonzero ideals of Z(sqrt(2)) and where (Z(sqrt(2)):A) is the norm of A; then zetaQ(sqrt(2))(s) = Sum_{n>=1}, a(n)/n^s); Sum{k=1..n} a(k) is asymptotic to c*n where c = log(1 + sqrt(2))/sqrt(2). - Benoit Cloitre, Jan 01 2003
Inverse Moebius transform of A091337.
a(n) is the number of solutions to the equation n = x^2 - 2*y^2 in integers where -x < 2*y <= x. [Uspensky and Heaslet] - Michael Somos, Feb 17 2020
Coefficients of Dedekind zeta function for the quadratic number field of discriminant 8. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			G.f. = x + x^2 + x^4 + 2*x^7 + x^8 + x^9 + 2*x^14 + x^16 + 2*x^17 + x^18 + ...
a(7) = 2 because 7 = 3^2 - 2*(+1)^2 = 3^2 - 2*(-1)^2. - _Michael Somos_, Feb 17 2020

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 368.

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

Cf. A035251, A078462, A188169, A188170, A188171, A188172, A259829.
Cf. A002193, A091648.
Moebius transform gives A091337.
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.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ 2, #] &]]; (* Michael Somos, Jul 06 2015 *)
a[ n_] := SeriesCoefficient[ Sum[ x^k (1 - x^(2 k)) / (1 + x^(4 k)), {k, n}], {x, 0, n}]; (* Michael Somos, Jul 06 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # <= 2, 1, Mod[#, 8] > 1 && Mod[#, 8] < 7, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* Michael Somos, Jul 06 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker(2, d)))};

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

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%8>1 && p%8<7, !(e%2), e+1)))}; \\ Michael Somos, Aug 17 2006

{a(n) = if( n<1, 0, polcoeff( sum(k=1, n, x^k * (1 - x^(2*k)) / (1 + x^(4*k)), x * O(x^n)), n))}; \\ Michael Somos, Jul 06 2015

G.f.: Sum_{k>0} x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
-(-1)^(n*(n-1)/2)*a(n) = Sum_{n >= 1} (-1)^n * q^(n*(n+1)/2)*(1-q)*(1-q^2)*...*(1-q^(n-1))/ ((1+q)*(1+q^2)*...*(1+q^n)). - Jeremy Lovejoy, Jun 12 2009
a(n) = (-1)^floor(n/2) * A259829(n). - Michael Somos, Jul 06 2015
a(n) is multiplicative with a(2^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if p == 3, 5 (mod 8), a(p^e) = e + 1 if p == 1, 7 (mod 8). - Jianing Song, Sep 07 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(sqrt(2)+1)/sqrt(2) = A091648/A002193 = 0.623225... . - Amiram Eldar, Oct 11 2022

A035182 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -7.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 5*v^2 + 4*w^2 - 8*v*w - 4*u*v + 2*u*w + v - w. - Michael Somos, Jul 21 2004
Half of the number of integer solutions to x^2 + x*y + 2*y^2 = n. - Michael Somos, Jun 05 2005
Inverse Moebius transform of A175629. - Jianing Song, Sep 07 2018
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -7. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			G.f. = x + 2*x^2 + 3*x^4 + x^7 + 4*x^8 + x^9 + 2*x^11 + 2*x^14 + 5*x^16 + ...

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

Cf. A002652, A326919.
Moebius transform gives A175629.
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.

A := Basis( ModularForms( Gamma1(14), 1), 106); B := (-1 + A[1] + 2*A[2] + 4*A[3] + 6*A[5]) / 2; B; // Michael Somos, Jun 10 2015

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -7, d], { d, Divisors[ n]}]]; (* Michael Somos, Jan 23 2014 *)
a[ n_] := If[ n < 1, 0, Length @ FindInstance[ n == x^2 + x y + 2 y^2, {x, y}, Integers, 10^9] / 2]; (* Michael Somos, Jan 23 2014 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -7, #] &]]; (* Michael Somos, Jun 10 2015 *)

{a(n) = my(A, p, e); if( n<0, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; [ !(e%2), 1, e+1] [kronecker( -7, p) + 2]))}; \\ Michael Somos, May 28 2005

{a(n) = if( n<1, 0, qfrep([ 2, 1; 1, 4], n, 1)[n])}; \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -7, p)*X)))[n])}; \\ Michael Somos, Jun 05 2005

a(n) is multiplicative with a(7^e) = 1, a(p^e) = e + 1 if p == 1, 2, 4 (mod 7), a(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7). - Michael Somos, May 28 2005
2 * a(n) = A002652(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(7) =  1.187410... (A326919). - Amiram Eldar, Oct 11 2022

A035170 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = -20.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

Coefficients of Dedekind zeta function for the quadratic number field of discriminant -20. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			q + q^2 + 2*q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + q^10 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 253.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A028586, A033718, A033764, A111949, A124233, A129390.
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.

QP = QPochhammer; s = (1/q) * (QP[q^2]*QP[q^4]*QP[q^5]*(QP[q^10] / (QP[q]* QP[q^20]))-1) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 04 2015 *)
a[n_] := If[n < 0, 0, DivisorSum[ n, KroneckerSymbol[-20, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Dec 12 2017 *)

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

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -20, d)))} \\ Michael Somos, Sep 10 2005

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -20, p) * X) )[n])} \\ Michael Somos, Sep 10 2005

{a(n) = if( n<1, 0, qfrep([1, 0; 0, 5], n)[n] + qfrep([2, 1; 1, 3], n)[n])} \\ Michael Somos, Oct 21 2006

Multiplicative with a(2^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20). - Michael Somos, Sep 10 2005
G.f.: Sum_{k>0} x^k * (1 + x^(2*k)) * (1 + x^(6*k)) / (1 + x^(10*k)). - Michael Somos, Sep 10 2005
a(2*n) = a(5*n) = a(n), a(20*n + 11) = a(20*n + 13) = a(20*n + 17) = a(20*n + 19) = 0.
Moebius transform is period 20 sequence [ 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...]. - Michael Somos, Oct 21 2006
Expansion of -1 + (phi(q) * phi(q^5) + phi(q^2) * phi(q^10) + 4 * q^3 * psi(q^4)* psi(q^20)) / 2 in powers of q where phi(), psi() are Ramanujan theta functions.
2*a(n) = A028586(n) + A033718(n) if n>0. - Michael Somos, Oct 21 2006
a(n) = A124233(n) unless n=0. a(n) = |A111949(n)|. a(2*n + 1) = A129390(n). a(4*n + 3) = 2 * A033764(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(5) = 1.404962... . - Amiram Eldar, Oct 11 2022

A035179 a(n) = Sum_{d|n} Kronecker(-11, d).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661. - Michael Somos, Jun 07 2015
Half of the number of integer solutions to x^2 + x*y + 3*y^2 = n. - Michael Somos, Jun 05 2005
From Jianing Song, Sep 07 2018: (Start)
Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -11.
Inverse Moebius transform of A011582. (End)
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -11. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			G.f. = x + 2*x^3 + x^4 + 2*x^5 + 3*x^9 + x^11 + 2*x^12 + 4*x^15 + x^16 + 2*x^20 + ...

Henry McKean and Victor Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032).

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

Cf. A028609, A065099, A129522, A138661.
Moebius transform gives A011582.
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.

A := Basis( ModularForms( Gamma1(11), 1), 88); B := (-1 + A[1] + 2*A[2] + 4*A[4] + 2*A[5]) / 2; B; // Michael Somos, Jun 07 2015

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -11, #] &]]; (* Michael Somos, Jun 07 2015 *)

{a(n) = if( n<1, 0, qfrep([2, 1; 1, 6], n, 1)[n])}; \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -11, p)*X))) [n])}; \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -11, d)))};

a(n) is multiplicative with a(11^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = e + 1 if p == 1, 3, 4, 5, 9 (mod 11). - Michael Somos, Jan 29 2007
Moebius transform is period 11 sequence [ 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 0, ...]. - Michael Somos, Jan 29 2007
G.f.: Sum_{k>0} Kronecker(-11, k) * x^k / (1 - x^k). - Michael Somos, Jan 29 2007
A028609(n) = 2 * a(n) unless n = 0. - Michael Somos, Jun 24 2011
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(11) = 0.947225... . - Amiram Eldar, Oct 11 2022

A112604 Number of representations of n as a sum of three times a square and two times a triangular number.

Original entry on oeis.org

1, 0, 1, 2, 0, 2, 1, 0, 0, 2, 0, 0, 3, 0, 2, 2, 0, 0, 2, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 2, 1, 0, 2, 4, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 0, 4, 0, 1, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 1, 0, 0, 4, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 2

Offset: 0

James Sellers, Dec 21 2005

nonn

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Number of representations of 2n as a sum of three times a triangular number and a triangular number.

			a(12) = 3 since we can write 12 = 3(2)^2 + 0 = 3(-2)^2 + 0 = 0 + 2*6.
2*12 = 24 = 3*1+21 = 3*3+15 = 3*6+6 so a(12) = 3.
G.f. = 1 + x^2 + 2*x^3 + 2*x^5 + x^6 + 2*x^9 + 3*x^12 + 2*x^14 + 2*x^15 + ... - _Michael Somos_, Aug 11 2009
G.f. = q + q^9 + 2*q^13 + 2*q^21 + q^25 + 2*q^37 + 3*q^49 + 2*q^57 + 2*q^61 + ... - _Michael Somos_, Aug 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

A112606(n) = a(2*n). 2 * A112607(n) = a(2*n + 1). A123884(n) = a(3*n). A112605(n) = a(3*n + 2). A131961(n) = a(6*n). A112608(n) =a(6*n + 2). 2 * A131963(n) = a(6*n + 3). 2 * A112609(n) = a(6*n + 5). - Michael Somos, Aug 11 2009
Cf. A002324, A033762, A093766, A164272.
Cf. A000700, A000122, A010054, A121373.

a[n_] := DivisorSum[4n+1, Switch[Mod[#, 3], 1, 1, 2, -1, 0, 0]&]; Table[ a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)

{a(n) = if(n<0, 0, n=4*n+1; sumdiv(n, d, (d%3==1) - (d%3==2)))};

{a(n) = my(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^6+A)^5 / eta(x^2+A)*(eta(x^4+A) / eta(x^3+A) / eta(x^12+A))^2, n))}; /* Michael Somos, Feb 14 2006 */

a(n) = A002324(4n+1) = A033762(2n) = d_{1, 3}(4n+1) - d_{2, 3}(4n+1) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
From Michael Somos, Feb 14 2006: (Start)
Expansion of (psi(q)psi(q^3) + psi(-q)psi(-q^3))/2 in powers of q^2 where psi() is a Ramanujan theta function.
G.f.: (Sum_{k} x^k^2)^3*(Sum_{k>0} x^((k^2-k)/2))^2 = Product_{k>0} (1-x^(4k))(1-x^(6k))(1+x^(2k))(1+x^(3k))^2/(1+x^(6k))^2.
Euler transform of period 12 sequence [0, 1, 2, -1, 0, -2, 0, -1, 2, 1, 0, -2, ...]. (End)
From Michael Somos, Aug 11 2009: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A164272.
a(3*n + 1) = 0. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 24 2023

A035171 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -19.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 2, 0, 1, 0, 2, 0, 0, 0, 0, 1, 2, 0, 1, 2, 0, 0, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 2, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 1

Offset: 1

N. J. A. Sloane

From Jianing Song, Sep 07 2018: (Start)
Half of the number of integer solutions to x^2 + x*y + 5*y^2 = n.
Inverse Moebius transform of A011585. (End)
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -19. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

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

Cf. A028641.
Moebius transform gives A011585.
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.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-19, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Jul 17 2018 *)

m = -19; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

From Jianing Song, Sep 07 2018: (Start)
a(n) is multiplicative with a(19^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-19, p) = -1, a(p^e) = e + 1 if Kronecker(-19, p) = 1.
G.f.: Sum_{k>0} Kronecker(-19, k) * x^k / (1 - x^k).
A028641(n) = 2 * a(n) unless n = 0.
(End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(19) = 0.720730... . - Amiram Eldar, Oct 11 2022

A035175 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = -15.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant -15. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

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

G. C. Greubel, 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.

QP = QPochhammer; s = (QP[q^3]*QP[q^5])^2/(QP[q]*QP[q^15])/q - 1/q + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-15, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Jul 17 2018 *)

m = -15; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

{a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-15,d)))} \\ Michael Somos, Aug 25 2006

{a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3||p==5, 1, if((p%15)!=2^valuation(p%15,2), (e+1)%2, (e+1))))))} \\ Michael Somos, Aug 25 2006

{a(n)=if(n<1, 0, (qfrep([2, 1;1, 8],n, 1)+qfrep([4, 1;1, 4], n, 1))[n])} \\ Michael Somos, Aug 25 2006

{a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( eta(x^3+A)^2*eta(x^5+A)^2/eta(x+A)/eta(x^15+A), n))} \\ Michael Somos, Aug 25 2006

From Michael Somos, Aug 25 2006: (Start)
Expansion of -1 + (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) in powers of q.
Euler transform of period 15 sequence [ 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -2, ...]. if a(0)=1.
Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...].
Given g.f. A(x), then B(x) = 1 + A(x) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = -v^3 + 4*u*v*w - 2*u*w^2 - u^2*w.
G.f.: -1 + x * Product_{k>0} ((1 - x^(3*k)) * (1 - x^(5*k)))^2 / ((1 - x^k) * (1 - x^(15*k))).
G.f.: -1 + (1/2) * (Sum_{n,m} x^(n^2 + n*m + 4*m^2) + x^(2*n^2 + n*m + 2*m^2)).
a(n) is multiplicative with a(3^e) = a(5^e) = 1, a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15).
a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0.
a(3*n) = a(n). a(n) = |A106406(n)| unless n=0. a(n) = A123864(n) unless n=0. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(15) = 1.622311... . - Amiram Eldar, Oct 11 2022

A035203 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 21.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 21. 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. A038893, A038894.

a[n_] := DivisorSum[n, KroneckerSymbol[21, #] &]; Array[a, 100] (* Amiram Eldar, Oct 11 2022 *)

my(m = 21); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(21, d)); \\ Amiram Eldar, Nov 19 2023

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log((5+sqrt(21))/2)/sqrt(21) = 0.683807... . - Amiram Eldar, Oct 11 2022
From Amiram Eldar, Nov 19 2023: (Start)
a(n) = Sum_{d|n} Kronecker(21, d).
Multiplicative with a(p^e) = 1 if Kronecker(21, p) = 0 (p = 3 or 7), a(p^e) = (1+(-1)^e)/2 if Kronecker(21, p) = -1 (p is in A038894), and a(p^e) = e+1 if Kronecker(21, p) = 1 (p is in A038893 \ {3, 7}). (End)

A003051 Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are equivalent if they are related by a rotation or reflection preserving the hexagonal lattice.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 3, 5, 4, 4, 3, 8, 4, 5, 6, 9, 4, 8, 5, 10, 8, 7, 5, 15, 7, 8, 9, 13, 6, 14, 7, 15, 10, 10, 10, 20, 8, 11, 12, 20, 8, 18, 9, 17, 16, 13, 9, 28, 12, 17, 14, 20, 10, 22, 14, 25, 16, 16, 11, 34, 12, 17, 21, 27, 16, 26, 13, 24, 18, 26, 13, 40, 14

Offset: 1

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
From Andrey Zabolotskiy, Mar 10 2018: (Start)
If only primitive sublattices are considered, we get A003050.
Here only rotations and reflections preserving the parent hexagonal lattice are allowed. If reflections are not allowed, we get A145394. If any rotations and reflections are allowed, we get A300651.
In other words, the parent lattice of the sublattices under consideration has Patterson symmetry group p6mm, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145393 (p4mm), A145394 (p6).
Rutherford says at p. 161 that his sequence for p6mm differs from this sequence, but it seems that with the current definition and terms of this sequence, this actually is his p6mm sequence, and the sequence he thought to be this one is actually A300651. Also, he says that a(n) != A300651(n) only when A002324(n) > 2 (first time happens at n = 49), but actually these two sequences differ at other terms, too, for example, at n = 42 (see illustration). (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217.
A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973), 201-217. [Annotated and corrected scanned copy]
M. Bernstein, N. J. A. Sloane and P. E. Wright, On Sublattices of the Hexagonal Lattice, Discrete Math. 170 (1997) 29-39 (Abstract, pdf, ps).
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] (see Table 3).
Daejun Kim, Seok Hyeong Lee, and Seungjai Lee, Zeta functions enumerating subforms of quadratic forms, arXiv:2409.05625 [math.NT], 2024.
W. Kurth, Enumeration of Platonic maps on the torus, Discrete Math. 61 (1986), 71-83.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2].
Andrey Zabolotskiy, Sublattices of the hexagonal lattice (illustrations for n = 1..7, 14)
Andrey Zabolotskiy, Coweight lattice A^*_n and lattice simplices, arXiv:2003.10251 [math.CO], 2020.
Index entries for sequences related to sublattices
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003050, A054384, A001615, A006984, A054345, A054346, A000203, A069734, A145391, A145392, A145393, A145394, A112689, A159842.

max = 73; A145390 = Drop[ CoefficientList[ Series[ Sum[(1 + Cos[n*Pi/2])*x^n/(1 - x^n), {n, 1, max}], {x, 0, max}], x], 1]; A002324[n_] := (dn = Divisors[n]; Count[dn, ?(Mod[#, 3] == 1 & )] - Count[dn, ?(Mod[#, 3] == 2 & )]); a[n_] := (DivisorSigma[1, n] + 2 A002324[n] + 3*A145390[[n]])/6; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Oct 11 2011, after given formula *)

a(n) = Sum_{ m^2 | n } A003050(n/m^2).
a(n) = (A000203(n) + 2*A002324(n) + 3*A145390(n))/6. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ d|n } A112689(d+1). - Andrey Zabolotskiy, Aug 29 2019
a(n) = Sum_{ d|n } floor(d/6) + 1 - 1*[d == 2 or 6 (mod 12)] + 1*[d == 4 (mod 12)]. [Kurth] - Brahadeesh Sankarnarayanan, Feb 24 2023

cp's OEIS Frontend

A035187 Sum over divisors d of n of Kronecker symbol (5|d).

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A035185 Number of divisors of n == 1 or 7 (mod 8) minus number of divisors of n == 3 or 5 (mod 8).

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A035182 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -7.

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A035170 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -20.

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A035179 a(n) = Sum_{d|n} Kronecker(-11, d).

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A112604 Number of representations of n as a sum of three times a square and two times a triangular number.

A035182 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -7.

A035170 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = -20.

A035171 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -19.

A035175 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = -15.

A035203 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 21.