A008441 -id:A008441 - cp's OEIS frontend

A112301 Expansion of (eta(q) * eta(q^16))^2 / (eta(q^2) * eta(q^8)) in powers of q.

1, -2, 0, 0, 2, 0, 0, 0, 1, -4, 0, 0, 2, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 3, -4, 0, 0, 2, 0, 0, 0, 0, -4, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, -6, 0, 0, 2, 0, 0, 0, 0, -4, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0, 1, -4, 0, 0, 4, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 2, 0, 0, 0, 0

Offset: 1

Michael Somos, Sep 02 2005, Oct 02 2007

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

			G.f. = q - 2*q^2 + 2*q^5 + q^9 - 4*q^10 + 2*q^13 + 2*q^17 - 2*q^18 + 3*q^25 - ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008441, A053692, A113407, A134013.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q^4] / 2, {q, 0, n}]; (* Michael Somos, Oct 19 2013 *)
QP = QPochhammer; s = (QP[q]*QP[q^16])^2/(QP[q^2]*QP[q^8]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^16 + A))^2 / (eta(x^2 + A) * eta(x^8 + A)), n))};

{a(n) = if( n>0 & (n+1)%4\2, (n%2*3 - 2) * sumdiv( n / gcd(n, 2), d, (-1)^(d\2)))};

Expansion of q * phi(-q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (phi(-q^2)^2 - phi(-q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 16 sequence [ -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -2, ...].
a(n) is ultiplicative with a(2) = -2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).
Moebius transform is period 16 sequence [ 1, -3, -1, 2, 1, 3, -1, 0, 1, -3, -1, -2, 1, 3, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 4 (t/i) f(t) where q = exp(2 Pi i t).
G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^(8*k))^2 * (1 + x^(2*k)) * (1 + x^(4*k)).
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)).
a(4*n) = a(4*n + 3) = a(8*n + 6) = 0. a(8*n + 2) = -2 * A008441(n).
a(n) = -(-1)^n * A134013(n). a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n).

A132003 Expansion of (phi(q^3) / phi(q)) * phi(-q^2) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 2, -2, 2, -4, 2, 0, 2, -2, 4, 0, 2, -4, 0, -4, 2, -4, 2, 0, 4, 0, 0, 0, 2, -6, 4, -2, 0, -4, 4, 0, 2, 0, 4, 0, 2, -4, 0, -4, 4, -4, 0, 0, 0, -4, 0, 0, 2, -2, 6, -4, 4, -4, 2, 0, 0, 0, 4, 0, 4, -4, 0, 0, 2, -8, 0, 0, 4, 0, 0, 0, 2, -4, 4, -6, 0, 0, 4, 0

Offset: 0

Michael Somos, Aug 06 2007

sign

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

			G.f. = 1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4 - 4*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + 4*x^10 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Equation (32.72).

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008441, A122857, A125079, A132004.

a[ n_] := If[ n < 1, Boole[n == 0], -2 DivisorSum[ n, (-1)^(n + #) KroneckerSymbol[ -36, #] &]]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -2 Times @@ (Which[ # < 5, -(-1)^#, Mod[#, 4] == 3, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^6] / EllipticTheta[ 3, 0, q], {q, 0, n}]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q]^2 + EllipticTheta[ 4, 0, q^3]^2) / 2, {q, 0, n}]; (* Michael Somos, Mar 05 2023 *)

{a(n) = if( n<1, n==0, -2 * sumdiv(n, d, (-1)^(n+d) * kronecker(-36, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^3), n))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -2 * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, p==2, -1, p%4==1, e+1, 1-e%2)))};

Expansion of eta(q)^2 * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^3) in powers of q.
a(n) = -2*b(n) where b() is multiplicative with b(2^e) = 2*0^e - 1, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
Euler transform of period 12 sequence [-2, 1, 0, 0, -2, -4, -2, 0, 0, 1, -2, -2, ...].
G.f.: 1 - 2 * Sum_{k>0} Kronecker(-36, k) * x^k / (1 + x^k).
a(n) = - A132004(n) unless n=0.
a(2*n) = A122857(n). a(2*n + 1) = -2 * A125079(n). a(3*n) = a(n). a(3*n + 1) = -2 * A258277(n). a(3*n + 2) = 2 * A258278(n). - Michael Somos, Nov 01 2015
a(12*n + 7) = a(12*n + 11) = 0. a(4*n + 1) = -2 * A008441(n).
a(n) = (-1)^n * A122857(n). Expansion of (phi(-q)^2 + phi(-q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 05 2023

A134013 Expansion of q * phi(q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Oct 02 2007

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

			G.f. = q + 2*q^2 + 2*q^5 + q^9 + 4*q^10 + 2*q^13 + 2*q^17 + 2*q^18 + 3*q^25 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A003881, A008441, A053692, A112301, A113407, A134014.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := SeriesCoefficient[ (1/2) EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^4], {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)

{a(n) = if( n>0 && (n+1)%4\2, (n%4) * sumdiv( n/gcd(n,2), d, (-1)^(d\2)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^8 + A)), n))};

Expansion of eta(q^2)^5 * eta(q^16)^2 / ( eta(q)^2 * eta(q^4)^2 * eta(q^8) ) in powers of q.
Euler transform of period 16 sequence [ 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -2, ...].
Moebius transform is period 16 sequence [ 1, 1, -1, -2, 1, -1, -1, 0, 1, 1, -1, 2, 1, -1, -1, 0, ...].
a(n) is multiplicative with a(2) = 2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134014.
a(4*n) = a(4*n + 3) = a(8*n + 6) = 0. a(8*n + 2) = 2 * a(4*n + 1).
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 + x^k)^2 / (1 - x^(4*k)).
a(n) = -(-1)^n * A112301(n). a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n = 5) = 2 * A053692(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/4 (A003881). - Amiram Eldar, Nov 24 2023

A132004 Expansion of (1 - phi(q^3) / phi(q) * phi(-q^2) * phi(-q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Aug 06 2007

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

			G.f. = x - x^2 + x^3 - x^4 + 2*x^5 - x^6 - x^8 + x^9 - 2*x^10 - x^12 + 2*x^13 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Equation (32.72).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008441, A035154, A053692, A122856, A122865, A125079, A132003, A258277, A258278.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^(n + #) KroneckerSymbol[ -36, #] &]]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 5, -(-1)^#, Mod[#, 4] == 3, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* Michael Somos, Nov 01 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(n+d) * kronecker( -36, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^3)) / 2, 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==3, 1, p==2, -1, p%4==1, e+1, 1-e%2)))};

Expansion of (1 - eta(q)^2 * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^3)) / 2 in powers of q.
a(n) is multiplicative with a(2^e) = 2*0^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).
G.f.: Sum_{k>0} x^k / (1 + x^k) * Kronecker(-36, k).
a(3*n) = a(n). -2 * a(n) = A132003(n) unless n = 0. a(2*n) = - A035154(n). a(2*n + 1) = A125079(n).
a(n) = (-1)^n * A035154(n). a(12*n + 7) = a(12*n + 11) = 0. - Michael Somos, Nov 01 2015
a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(4*n + 1) = A008441(n). a(4*n + 2) = - A125079(n). - Michael Somos, Nov 01 2015
a(6*n) = - A035154(n). a(6*n + 2) = - A122865(n). a(6*n + 4) = - A122856(n). - Michael Somos, Nov 01 2015
a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). - Michael Somos, Nov 01 2015

A134015 Expansion of (1 - phi(-q) * phi(q^4)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Oct 02 2007

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

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

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. A002654, A008441, A113406, A134014.

a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^4]) / 2, {x, 0, n}]; (* Michael Somos, Oct 28 2015 *)
a[ n_] := If[ n < 1 || Mod[n, 4] > 1, 0, (Mod[n, 2] 3 - 2) DivisorSum[ n, KroneckerSymbol[ -4, #]&]]; (* Michael Somos, Oct 28 2015 *)

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

{a(n) = -(-1)^n * if( n<1, 0, qfrep([1, 0; 0, 4], n)[n])};

Moebius transform is period 16 sequence [ 1, -1, -1, -2, 1, 1, -1, 0, 1, -1, -1, 2, 1, 1, -1, 0, ...].
a(n) is multiplicative with a(2) = 0, a(2^e) = -2 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).
a(4*n+2) = a(4*n+3) = 0.
G.f.: x / (1 + x^2) + x^3 / (1 + x^6) - 2 * x^4 / (1 + x^8) + ...
a(n) = -(-1)^n * A113406(n). -2 * a(n) = A134014(n) unless n=0. a(4*n) = -2 * A002654(n). a(4*n + 1) = A008441(n).

A246862 Expansion of phi(x) * f(x^3, x^5) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 05 2014

nonn

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

			G.f. = 1 + 2*x + x^3 + 4*x^4 + x^5 + 2*x^6 + 2*x^7 + 4*x^9 + 2*x^12 + ...
G.f. = q + 2*q^17 + q^49 + 4*q^65 + q^81 + 2*q^97 + 2*q^113 + 4*q^145 + ...

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

Cf. A000122, A008441, A113407, A116604, A125079, A129447, A134343, A138741, A214264, A226192, A246863.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ x^8], {x, 0, n}];

{a(n) = if( n<0, 0, issquare(16 * n + 1) + 2 * sum(i=1, sqrtint(n), issquare(16 * (n - i^2) + 1)))};

Euler transform of period 16 sequence [ 2, -3, 3, -1, 3, -4, 2, -2, 2, -4, 3, -1, 3, -3, 2, -2, ...].
Convolution of A000122 and A214264.
a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = A246863(n).
a(n) = A113407(2*n) = A226192(2*n) = A008441(4*n) = A134343(4*n) = A116604(8*n) = A125079(8*n) = A129447(8*n) = A138741(8*n).

A258291 Expansion of q^(-1/4) * eta(q) * eta(q^2) * eta(q^6) / eta(q^3) in powers of q.

Original entry on oeis.org

1, -1, -2, 2, -1, 0, 3, -1, 0, 2, -1, -4, 1, -1, 0, 2, -2, 0, 2, 0, -2, 4, -1, 0, 2, -1, 0, 2, -1, -4, 1, -2, 0, 0, -1, 0, 4, -1, -4, 2, 0, 0, 3, -1, 0, 2, -2, 0, 2, -1, 0, 4, 0, 0, 0, -2, -6, 2, -1, 0, 2, -1, 0, 0, -1, -4, 4, -1, 0, 2, -1, 0, 3, -1, 0, 0, -2

Offset: 0

Michael Somos, May 25 2015

sign

			G.f. = 1 - x - 2*x^2 + 2*x^3 - x^4 + 3*x^6 - x^7 + 2*x^9 - x^10 - 4*x^11 + ...
G.f. = q - q^5 - 2*q^9 + 2*q^13 - q^17 + 3*q^25 - q^29 + 2*q^37 - q^41 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A002175, A008441, A121444, A258277.

QP := QPochhammer; CoefficientList[Series[QP[q]*QP[q^2]*QP[q^6]/QP[q^3], {q, 0, 50}], q] (* G. C. Greubel, Aug 04 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^6 + A) / eta(x^3 + A), n))};

Euler transform of period 6 sequence [ -1, -2, 0, -2, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 9 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258277.
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 + x^(3*k)).
a(3*n) = A002175(n). a(3*n + 1) = - A121444(n). a(9*n + 2) = -2 * A008441(n). a(9*n + 5) = a(9*n + 8) = 0.

A113414 Expansion of Sum_{k>0} x^k/(1-(-x^2)^k).

Original entry on oeis.org

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

Offset: 1

Michael Somos, Oct 29 2005

nonn

A001227(n) = a(2*n), A008441(n) = a(4*n+1), A099774(n) = a(4*n+2).

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

{a(n)=if(n<1, 0, if(n%4==3, 0, if(n%4==2, numdiv(n/2), if(n%4==0, sumdiv(n,d,d%2), sumdiv(n,d,(-1)^(d\2))))))}

{a(n)=if(n<1, 0, polcoeff( sum(k=1,sqrtint(8*n+1)\2, (-1)^(k%4==2)*x^((k^2+k)/2)/(1-(-1)^(k\2)*x^k), x*O(x^n)), n))}

{a(n)=if(n<1, 0, polcoeff( sum(k=1,n, x^k/(1-(-x^2)^k), x*O(x^n)), n))}

Moebius transform is period 8 sequence [1, 0, -1, 0, 1, 2, -1, 0, ...].
G.f.: Sum_{k>0} x^k/(1-(-x^2)^k) = Sum_{k>0} x^k/(1+x^(2k))+2x^(6k)/(1-x^(8k)) = Sum_{k>0} -(-1)^k x^(2k-1)/(1+(-1)^k*x^(2k-1)).
a(4n+3) = 0.
a(n) = A001826(n) + (-1)^n * A001842(n). - David Spies, Sep 26 2012

A199015 G.f.: 1/(1-x) * Product_{n>=1} (1 - x^(2n))^2/(1 - x^(2n-1))^2.

Original entry on oeis.org

1, 3, 4, 6, 8, 8, 11, 13, 13, 15, 17, 19, 20, 22, 22, 24, 28, 28, 30, 30, 31, 35, 37, 37, 39, 41, 41, 43, 45, 47, 48, 52, 52, 52, 54, 54, 58, 60, 62, 64, 64, 64, 67, 69, 69, 71, 75, 75, 77, 79, 79, 83, 83, 83, 83, 87, 90, 92, 94, 94, 96, 98, 98, 98, 100, 102, 106, 108, 108, 110, 112, 112, 115, 117, 117, 117, 121, 121, 123

Offset: 0

Paul D. Hanna, Nov 02 2011

Equals the partial sums of A008441, where A008441(n) is the number of ways of writing n as the sum of 2 triangular numbers.

			G.f.: A(x) = 1 + 3*x + 4*x^2 + 6*x^3 + 8*x^4 + 8*x^5 + 11*x^6 + 13*x^7 + ...
where the g.f. equals the product:
A(x) = 1/(1-x) * (1-x^2)^2/(1-x)^2 * (1-x^4)^2/(1-x^3)^2 * (1-x^6)^2/(1-x^5)^2 * ...
Illustrate the limit a(n)/n = Pi/2:
a(10)/10 = 1.7, a(10^2)/10^2 = 1.58, a(10^3)/10^3 = 1.574, a(10^4)/10^4 = 1.5704, a(10^5)/10^5 = 1.57086, a(10^6)/10^6 = 1.570784, a(10^7)/10^7 = 1.5707972, ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A008441, A008443.

CoefficientList[Series[EllipticTheta[2, 0, Sqrt[x]]^2/(4*x^(1/4)*(1 - x)), {x, 0, 50}], x] (* G. C. Greubel, Aug 13 2018 *)
a = Compile[{{n, Integer}}, Block[{c = 0}, Do[ c ++, {i, 0, (1 + Sqrt[1 + 8 n])/4}, {j, 0, (1 + Sqrt[1 + 8(n - i (i + 1))])/2}]; c]]; Array[a, 79, 0] (* _Robert G. Wilson v, Aug 28 2025 *)

{a(n)=sum(k=0,n,sumdiv(4*k+1, d, (-1)^(d\2)))}

{a(n)=polcoeff(1/(1-x)*prod(m=1,n\2+1,(1-x^(2*m))/(1-x^(2*m-1)+x*O(x^n)))^2,n)}

Lim_{n->infinity} a(n)/n = Pi/2.
a(n) = Sum_{k=0..n} Sum_{d|4*k+1} (-1)^floor(d/2). - Michael Somos [see A008441]
G.f.: 1/(1-x) * Sum_{n>=0} x^n/(1 - x^(4*n + 1)). - Michael Somos [see A008441]
G.f.: theta_2(sqrt(x))^2/(4*x^(1/4)*(1 - x)), where theta_2() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

A224928 Numbers of pairs {x, y} such that x <= y and triangular(x) + triangular(y) = 2^n.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 1, 0, 3, 0, 2, 0, 4, 0, 1, 0, 8, 0, 2, 0, 4, 0, 4, 0, 8, 0, 2, 0, 24, 0, 2, 0, 8, 0, 8, 0, 8, 0, 2, 0, 32, 0, 4, 0, 16, 0, 4, 0, 32, 0, 4, 0, 32, 0, 4, 0, 4, 0, 8, 0, 16, 0, 2, 0, 32, 0, 6, 0, 48, 0, 16, 0, 16, 0, 8, 0, 384, 0, 4, 0, 16, 0, 16, 0, 16, 0, 8, 0, 768, 0, 2, 0, 8, 0, 4, 0, 32, 0, 32, 0, 256

Offset: 0

Alex Ratushnyak, May 08 2013

nonn

Conjectures:
1. a(n) = 0 for odd n > 1.
2. a(n) is even for even n > 14.

			2^1 = 1 + 1, the only representation of 2 as a sum of two triangular numbers, so a(1)=1.
2^4 = 16 = 1+15 = 6+10, two representations, so a(4) = 2.
2^8 = 256 = 3+253 = 66+190 = 120+136, so a(8) = 3.
2^12 = 4096 = 1+4095 = 91+4005 = 1540+2556 = 2016+2080, so a(12) = 4.

Antti Karttunen, Table of n, a(n) for n = 0..329

Cf. A000217, A052343, A006307, A225437.

#include 
#include 
typedef unsigned long long U64;
U64 isTriangular(U64 a) {      // ! Must be a <= (1<<63)
    U64 s = sqrt(a*2);
    if (a>=(1ULL<<63)) {
        if (a==(1ULL<<63)) return 0;
        printf("Error: a = %llu\n", a), exit(1);
    }
    return (s*(s+1)/2 == a);
}
int main() {
  U64 c, n, x, tx;
  for (n = 1; n; n+=n) {
    for (c = x = tx = 0; tx*2 <= n; ++x, tx+=x)
      if (isTriangular(n - tx))
        ++c;//, printf("(%llu+%llu) ", tx, n-tx);
    printf("%llu, ", c);
  }
  return 0;
}

A008441(n) = if(!n,n,sumdiv(4*n + 1, d, (d%4==1) - (d%4==3)));
A052343(n) = if(!n,1,my(u=A008441(n)); ((u\2)+(u%2)));
A224928(n) = A052343(2^n); \\ Antti Karttunen, May 24 2021

a(n) = A052343(2^n).

More terms from Antti Karttunen, May 24 2021

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