A002479 -id:A002479 - cp's OEIS frontend

A129134 Expansion of (1 - phi(-q) * phi(-q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

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

Offset: 1

Michael Somos, Mar 30 2007

sign

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

For n nonzero, a(n) is nonzero if and only if n is in A002479.

			G.f. = q + q^2 - 2*q^3 - q^4 + 2*q^6 - q^8 + 3*q^9 - 2*q^11 - 2*q^12 - q^16 + ...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002325, A002479, A082564, A258747, A258764.

a[ n_] := If[ n < 1, 0, (-1)^Quotient[ n - 1, 2] DivisorSum[n, KroneckerSymbol[-2, #] &]]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2]) / 2 , {q, 0, n}]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4]) / 2 , {q, 0, n}]; (* Michael Somos, Jun 09 2015 *)

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

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

Expansion of (1 - eta(q)^2 * eta(q^2) / eta(q^4)) / 2 in powers of q.
G.f.: (1 - Product_{k>0} (1 - x^k)^2 / (1 + x^(2*k)) )/2.
a(n) = A002325(n) * (-1)^floor((n-1)/2). A082564(n) = -2 * a(n) unless n=0.
a(3*n + 1) = A258747(n). a(3*n + 2) = A258764(n). - Michael Somos, Jun 09 2015

A139093 Expansion of phi(q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, -2, -4, 2, 0, -4, 0, 2, 6, 0, -4, 4, 0, 0, 0, 2, 4, -6, -4, 0, 0, -4, 0, 4, 2, 0, -8, 0, 0, 0, 0, 2, 8, -4, 0, 6, 0, -4, 0, 0, 4, 0, -4, 4, 0, 0, 0, 4, 2, -2, -8, 0, 0, -8, 0, 0, 8, 0, -4, 0, 0, 0, 0, 2, 0, -8, -4, 4, 0, 0, 0, 6, 4, 0, -4, 4, 0, 0, 0, 0, 10, -4, -4, 0, 0, -4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, -2, -12, 2, 0, -8, 0

Offset: 0

Michael Somos, Apr 08 2008

sign

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

a(n) is nonzero if and only if n is in A002479.

			G.f. = 1 + 2*q - 2*q^2 - 4*q^3 + 2*q^4 - 4*q^6 + 2*q^8 + 6*q^9 - 4*q^11 + ...

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. A002479, A033715, A033761, A082564, A112603, A125095, A133692.

A := Basis( ModularForms( Gamma1(32), 1), 105); A[1] + 2*A[2] - 2*A[3] - 4*A[4] + 2*A[5] - 4*A[7] + 2*A[9] + 6*A[10] - 4*A[12] + 4*A[13] + 4*A[16]; /* Michael Somos, Aug 29 2014 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^7 / (QPochhammer[ q]^2 QPochhammer[ q^4]^3), {q, 0, n}]; (* Michael Somos, Feb 18 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 (-1)^Quotient[n, 2] Sum[ JacobiSymbol[ -2, d], {d, Divisors @ n}]]; (* Michael Somos, Feb 18 2015 *)

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

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

Expansion of eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3) in powers of q.
Euler transform of period 4 sequence [ 2, -5, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112603.
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k-1))^2 / (1 + x^(2*k)).
a(8*n + 5) = a(8*n + 7) = 0.
a(n) = (-1)^n * A082564(n). a(2*n) = A133692(n). a(2*n + 1) = 2 * A125095(n). a(4*n) = a(8*n) = A033715(n). a(8*n + 1) = 2 * A112603(n). a(8*n + 3) = -4 * A033761(n).

A308547 Number of ways to write n as a^2 + 2b^2 + 2^c3^d, where a,b,c,d are nonnegative integers.

Original entry on oeis.org

1, 2, 3, 4, 4, 4, 4, 4, 5, 7, 5, 8, 6, 5, 6, 4, 6, 8, 7, 10, 8, 6, 4, 8, 8, 8, 10, 10, 6, 9, 6, 4, 10, 9, 11, 14, 8, 8, 9, 10, 8, 11, 8, 9, 13, 6, 5, 8, 9, 10, 11, 13, 7, 14, 8, 10, 13, 9, 11, 16, 7, 7, 13, 4, 12, 12, 10, 12, 10, 13, 5, 14, 13, 9, 17, 12, 7, 12, 6, 10

Offset: 1

Zhi-Wei Sun, Jun 06 2019

nonn

As 3*(a^2 + 2*b^2 + 2^c*3^d) = (a+2*b)^2 + 2*(a-b)^2 + 2^c*3^(d+1), we have a(3*n) > 0 if a(n) > 0.

The first positive integer n with a(n) = 0 is 139571911. We also have a(142991573) = 0.

			a(1) = 1 with 1 = 0^2 + 2*0^2 + 2^0*3^0.
a(2) = 2 with 2 = 0^2 + 2*0^2 + 2^1*3^0 = 1^2 + 2*0^2 + 2^0*3^0.
a(1117) = 2 with 1117 = 10^2 + 2*12^2 + 2^0*3^6 = 19^2 + 2*18^2 + 2^2*3^3.
a(78373) = 1 with 78373 = 271^2 + 2*48^2 + 2^2*3^4.
a(448159) = 1 with 448159 = 610^2 + 2*195^2 + 2^0*3^2.
a(82816213) = 2 with 82816213 = 4353^2 + 2*5651^2 + 2^1*3^0 = 3681^2 + 2*5885^2 + 2^1*3^0.
a(90685253) = 2 with 90685253 = 7007^2 + 2*4560^2 + 2^2*3^0 = 607^2 + 2*6720^2 + 2^2*3^0.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000079, A000244, A000290, A002479, A302984, A303637, A303656, A308411.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-3^k*2^m-2x^2],r=r+1],{k,0,Log[3,n]},{m,0,Log[2,n/3^k]},{x,0,Sqrt[(n-3^k*2^m)/2]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A034027 Numbers of the form x^2+2y^2 with x >= y >= 0.

Original entry on oeis.org

0, 1, 3, 4, 6, 9, 11, 12, 16, 17, 18, 24, 25, 27, 33, 34, 36, 38, 43, 44, 48, 49, 51, 54, 57, 64, 66, 67, 68, 72, 75, 81, 82, 83, 86, 89, 96, 99, 100, 102, 108, 113, 114, 118, 121, 123, 129, 131, 132, 136, 139, 144, 146, 147, 150, 152, 153, 162, 169, 171

Offset: 1

N. J. A. Sloane

nonn

Cf. A002479.

Definition corrected by N. J. A. Sloane, Apr 30 2015 (thanks to Harvey P. Dale for pointing out that something was wrong).

A035172 a(n) = Sum_{d|n} Kronecker(-18, d).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

For n nonzero, a(n) is nonzero if and only if n is in A002479.

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 81, Eq. (32.51).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Alexander Berkovich and Hamza Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2006-2007.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002479, A093825, A122071 (odd bisection).

Cf. A000122, A000700, A010054, A121373.

a[n_] := DivisorSum[n, KroneckerSymbol[-18, #]&]; Array[a, 105] (* Jean-François Alcover, Nov 14 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -18, d)))}

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -18, p) * X))[n])}

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^24 + A) / eta(x + A) / eta(x^8 + A), n))}

Expansion of q * psi(-q^3) * psi(-q^6) * chi(-q^4) / chi(-q) in powers of q where psi(), chi() are Ramanujan theta functions.
From Michael Somos, Apr 25 2003: (Start)
G.f.: x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(24*k)) * (1 + x^k) / (1 + x^(4*k)).
Euler transform of period 24 sequence [ 1, 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -2, 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -2, ...]. (End)
Moebius transform is period 24 sequence [ 1, 0, 0, 0, -1, 0, -1 ,0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, ...]. - Michael Somos, Jan 28 2006
From Michael Somos, Aug 04 2006: (Start)
Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1,3 (mod 8), a(p^e) = (1 + (-1)^e)/2 if p == 5,7 (mod 8).
G.f.: Sum_{k>0} x^k * (1 - x^(4*k)) * (1 - x^(6* k)) / (1 + x^(12*k)). (End)
Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -18.
G.f.: 1 + Sum{n = -infinity...infinity} (q^n - q^(5n)) / (1 + q^(12n)) (see Berkovich/Yesilyurt). - Ralf Stephan, May 14 2007
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(2)) = 0.7404804... (A093825). - Amiram Eldar, Nov 16 2023

A302641 Number of nonnegative integers k such that n^2 - 32^k can be written as x^2 + 2y^2 with x and y integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 10 2018

nonn

The author's Square Conjecture in A301471 implies that a(n) > 0 for all n > 1.

We have a(2^n) = 1 for all n > 0. In fact, (2^n)^2 = (2^(n-1))^2 + 2*0^2 + 3*2^(2*n-2). If k > 2*n-2 then 3*2^k >= 6*2^(2*n-2) > (2^n)^2. If 0 <= k < 2*n-2, then 2*n-k is at least 3 and hence (2^n)^2 - 3*2^k = 2^k*(2^(2*n-k)-3) cannot be written as x^2 + 2*y^2 with x and y integers.

			a(2) = 1 with 2^2 = 1^2 + 2*0^2 + 3*2^0.
a(3) = 2 with 3^2 = 2^2 + 2*1^2 + 3*2^0 = 1^2 + 2*1^2 + 3*2^1.
a(2857932461) = 1 since 3 is the only nonnegative integer k such that 2857932461^2 - 3*2^k has the form x^2 + 2*y^2 with x and y integers.
a(4428524981) = 2 since 3 and 8 are the only nonnegative integers k such that 4428524981^2 - 3*2^k has the form x^2 + 2*y^2 with x and y integers.
a(4912451281) = 3 since 3, 6 and 7 are the only nonnegative integers k with 4428524981^2 - 3*2^k = x^2 + 2*y^2 for some integers x and y.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.
Zhi-Wei Sun, My square conjecture with prize, a message to Number Theory List, April 7, 2018.

Cf. A000079, A000290, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301472, A301479, A301579, A301640.

f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[(Mod[Part[Part[f[n],i],1],8]==5||Mod[Part[Part[f[n],i],1],8]==7)&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[n^2-3*2^k],r=r+1],{k,0,Log[2,n^2/3]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A034031 Numbers that are primitively but not imprimitively represented by x^2+2y^2.

Original entry on oeis.org

1, 2, 3, 6, 11, 17, 19, 22, 33, 34, 38, 41, 43, 51, 57, 59, 66, 67, 73, 82, 83, 86, 89, 97, 102, 107, 113, 114, 118, 123, 129, 131, 134, 137, 139, 146, 163, 166, 177, 178, 179, 187, 193, 194, 201, 209, 211, 214, 219, 226, 227

Offset: 1

N. J. A. Sloane

nonn

Cf. A002479, A097700, A057127, A034030-A034034.

Corrected by N. J. A. Sloane, Apr 30 2015

A034033 Both primitively and imprimitively represented by x^2+2y^2.

Original entry on oeis.org

9, 18, 27, 54, 81, 99, 121, 153, 162, 171, 198, 242, 243, 289, 297, 306, 342, 361, 363, 369, 387, 459, 486, 513, 531, 578, 594, 603, 657, 722, 726, 729, 738, 747, 774, 801, 867, 873, 891, 918, 963, 1017, 1026, 1062, 1083, 1089

Offset: 1

N. J. A. Sloane

nonn

Cf. A002479, A097700, A057127, A034030-A034034.

Corrected by N. J. A. Sloane, Apr 30 2015

A124452 Expansion of psi(-q) * psi(-q^2) * chi(q^3) * chi(q^6) in powers of q where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Nov 02 2006

sign

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

a(n) is nonzero if and only if n is in A002479.

			1 - q - q^2 + q^3 - q^4 + q^6 - q^8 + 3*q^9 - 2*q^11 + q^12 - q^16 - 2*q^17 + ...

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. A002479.

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]* eta[q^6]*eta[q^8]*eta[q^12]/(eta[q^3]*eta[q^24]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 08 2018 *)

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p==3, 1 - 2*e, if( p%8<4, e+1, !(e%2)))))))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^8 + A) * eta(x^12 + A) / (eta(x^3 + A) * eta(x^24 + A)), n))}

q='q+O('q^99); Vec(eta(q)*eta(q^6)*eta(q^8)*eta(q^12)/(eta(q^3)*eta(q^24))) \\ Altug Alkan, Mar 09 2018

Expansion of eta(q) * eta(q^6) * eta(q^8) * eta(q^12) / (eta(q^3) * eta(q^24)) in powers of q.
a(n) = -b(n) where b(n) is multiplicative with b(2^e) = 1, b(3^e) = 1 - 2*e, b(p^e) = 1+e if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8).
Euler transform of period 24 sequence [ -1, -1, 0, -1, -1, -1, -1, -2, 0, -1, -1, -2, -1, -1, 0, -2, -1, -1, -1, -1, 0, -1, -1, -2, ...].
Moebius transform is period 24 sequence [ -1, 0, 2, 0, 1, 0, 1, 0, 2, 0, -1, 0, 1, 0, -2, 0, -1, 0, -1, 0, -2, 0, 1, 0, ...].
a(8*n + 5) = a(8*n + 7) = 0. a(2*n) = a(n). a(3*n) >= 0.

A133693 Expansion of (1 - phi(-q) * phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 20 2007

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

For n nonzero, a(n) is nonzero if and only if n is in A002479.

			G.f. = q - q^2 + 2*q^3 - q^4 - 2*q^6 - q^8 + 3*q^9 + 2*q^11 - 2*q^12 - q^16 + ...

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. A002325, A002479, A113411, A133692.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Oct 30 2015 *)

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

Expansion of (1 - eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2)) / 2 in powers of q.
Moebius transform is period 16 sequence [ 1, -2, 1, 0, -1, -2, -1, 0, 1, 2, 1, 0, -1, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8), a(p^e) = e + 1 if p == 1, 3 (mod 8).
a(8*n + 5) = a(8*n + 7) = 0. A133692(n) = -2 * a(n) unless n=0. a(n) = -(-1)^n * A002325(n). a(2*n + 1) = A113411(n).

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A129134 Expansion of (1 - phi(-q) * phi(-q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

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A139093 Expansion of phi(q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function.

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A308547 Number of ways to write n as a^2 + 2*b^2 + 2^c*3^d, where a,b,c,d are nonnegative integers.

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A034027 Numbers of the form x^2+2y^2 with x >= y >= 0.

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A035172 a(n) = Sum_{d|n} Kronecker(-18, d).

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PARI

PARI

PARI

Formula

A302641 Number of nonnegative integers k such that n^2 - 3*2^k can be written as x^2 + 2*y^2 with x and y integers.

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A034031 Numbers that are primitively but not imprimitively represented by x^2+2y^2.

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A034033 Both primitively and imprimitively represented by x^2+2y^2.

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A308547 Number of ways to write n as a^2 + 2b^2 + 2^c3^d, where a,b,c,d are nonnegative integers.

A302641 Number of nonnegative integers k such that n^2 - 32^k can be written as x^2 + 2y^2 with x and y integers.