A128617 -id:A128617 - cp's OEIS frontend

A112609 Number of representations of n as a sum of three times a triangular number and four times a triangular number.

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

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).

			a(30) = 2 since we can write 30 = 3*10 + 4*0 = 3*6 + 4*3
q^7 + q^31 + q^39 + q^63 + q^79 + q^103 + q^111 + q^127 + q^151 + ...

M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

A131962(n) = a(3*n). A112607(n) = a(3*n+1). A128617(n) = a(4*n+3).

A112605(2*n+1) = 2 * a(n). A112607(3*n+1) = a(n). A033762(4*n+3) = 2 * a(n). A112604(6*n+5) = 2 * a(n). A002324(8*n+7) = a(n). A123484(24*n+21) = 2 * a(n).

A112609[n_] := SeriesCoefficient[(QPochhammer[q^6]*QPochhammer[q^8])^2/
(QPochhammer[q^3]*QPochhammer[q^4]), {q,0,n}]; Table[A112609[n], {n, 0, 50}] (* G. C. Greubel, Sep 25 2017 *)

{a(n) = if( n<0, 0, n=8*n+7; sumdiv(n, d, kronecker(-3, d))/2)} /* Michael Somos, Mar 10 2008 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^6 + A) * eta(x^8 + A))^2 / (eta(x^3 + A) * eta(x^4 + A)), n))} /* Michael Somos, Mar 10 2008 */

a(n) = 1/2*( d_{1, 3}(8n+7) - d_{2, 3}(8n+7) ) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of phi(q^3) * psi(q^4) in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Mar 10 2008
Expansion of q^(-7/8) * (eta(q^6) * eta(q^8))^2 / (eta(q^3) * eta(q^4)) in powers of q. - Michael Somos, Mar 10 2008
Euler transform of period 24 sequence [ 0, 0, 1, 1, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 1, -1, 0, -1, 0, 1, 1, 0, 0, -2, ...]. - Michael Somos, Mar 10 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138270.
a(3*n+2) = 0.

A033782 Product t2(q^d); d | 23, where t2 = theta2(q)/(2*q^(1/4)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Also the number of positive odd solutions to equation x^2 + 23y^2 = 8(n + 3). - Seiichi Manyama, May 21 2017

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

Cf. A035162, A128617, A033790.

Expansion of q^(-3) * (eta(q^2) * eta(q^46))^2 / (eta(q) * eta(q^23)) in powers of q. - Seiichi Manyama, May 21 2017

More terms from Seiichi Manyama, May 21 2017

A033790 Product t2(q^d); d | 31, where t2 = theta2(q)/(2*q^(1/4)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Also the number of positive odd solutions to equation x^2 + 31y^2 = 8(n + 4). - Seiichi Manyama, May 21 2017

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

Cf. A035162, A128617, A033782.

Expansion of q^(-4) * (eta(q^2) * eta(q^62))^2 / (eta(q) * eta(q^31)) in powers of q. - Seiichi Manyama, May 21 2017

More terms from Seiichi Manyama, May 21 2017

A082451 Sum over divisors d of n of Kronecker symbol (-60, d).

Original entry on oeis.org

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

Offset: 1

Michael Somos, Apr 25 2003

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

			G.f. =  q + q^2 + q^3 + q^4 + q^5 + q^6 + q^8 + q^9 + q^10 + q^12 + q^15 + q^16 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A128616, A128617.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, 0, DivisorSum[n, KroneckerSymbol[-60, #] &]]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[# < 7, 1, KroneckerSymbol[-60, #] == 1, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger[n])]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3] QPochhammer[ q^5] QPochhammer[ -q, q] QPochhammer[ -q^15, q^15], {q, 0, n}]; (* Michael Somos, Nov 15 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-60, d)))};

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker(-60, p)*X))[n]))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A) / (eta(x + A) * eta(x^15 + A)), n))};

Expansion of q * f(-q^3) * f(-q^5) / (chi(-q) * chi(-q^15)) in powers of q where f(), chi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^3) * eta(q^5) * eta(q^30) / (eta(q) * eta(q^15)) in powers of q.
Euler transform of period 30 sequence [ 1, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, 1, -2, ...].
a(n) is multiplicative with a(2^e) = a(3^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15), a(p^e) = (1 + (-1)^e)/2 if p == 7, 11, 13, 14 (mod 15).
G.f.: x * Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(5*k)) * (1 + x^(15*k)).
a(n) = A128616(n) + A128617(n). - Michael Somos, Nov 15 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(15) = 0.811155... . - Amiram Eldar, Nov 16 2023

A033806 Product t2(q^d); d | 47, where t2 = theta2(q)/(2*q^(1/4)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Also the number of positive odd solutions to equation x^2 + 47y^2 = 8(n + 6). - Seiichi Manyama, May 27 2017

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

Cf. A035162, A128617, A033782, A033790.

Expansion of q^(-6) * (eta(q^2) * eta(q^94))^2 / (eta(q) * eta(q^47)) in powers of q. - Seiichi Manyama, May 27 2017

More terms from Seiichi Manyama, May 27 2017

A287619 Number of positive odd solutions to equation x^2 + 39y^2 = 8*(n + 5).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, May 28 2017

nonn

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

Cf. A035162, A128617.

Number of positive odd solutions to equation x^2 + (8*k - 1)*y^2 = 8*(n + k): A033782 (k=3), A033790 (k=4), this sequence (k=5), A033806 (k=6).

Expansion of q^(-5) * (eta(q^2) * eta(q^78))^2 / (eta(q) * eta(q^39)) in powers of q.

cp's OEIS Frontend

A112609 Number of representations of n as a sum of three times a triangular number and four times a triangular number.

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A033782 Product t2(q^d); d | 23, where t2 = theta2(q)/(2*q^(1/4)).

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A033790 Product t2(q^d); d | 31, where t2 = theta2(q)/(2*q^(1/4)).

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A082451 Sum over divisors d of n of Kronecker symbol (-60, d).

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A033806 Product t2(q^d); d | 47, where t2 = theta2(q)/(2*q^(1/4)).

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A287619 Number of positive odd solutions to equation x^2 + 39y^2 = 8*(n + 5).

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