A008653 -id:A008653 - cp's OEIS frontend

A205967 a(n) = Fibonacci(n)*A008653(n) for n>=1, with a(0)=1, where A008653 is the theta series of direct sum of 2 copies of hexagonal lattice.

1, 12, 36, 24, 252, 360, 288, 1248, 3780, 408, 11880, 12816, 12096, 39144, 108576, 43920, 367164, 344952, 93024, 1003440, 3409560, 1050816, 7651152, 8253216, 8346240, 27909300, 61182072, 2357016, 213568992, 185122440, 179720640, 516967296, 1646801604, 507539232

Offset: 0

Paul D. Hanna, Feb 04 2012

nonn

Compare g.f. to the Lambert series of A008653: 1 + 12*Sum_{n>=1} Chi(n,3)*n*x^n/(1-x^n).

Here Chi(n,3) = principal Dirichlet character of n modulo 3.

			G.f.: A(x) = 1 + 12*x + 36*x^2 + 24*x^3 + 252*x^4 + 360*x^5 + 288*x^6 +...
where A(x) = 1 + 1*12*x + 1*36*x^2 + 2*12*x^3 + 3*84*x^4 + 5*72*x^5 + 8*36*x^6 +...+ Fibonacci(n)*A008653(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 12*( 1*1*x/(1-x-x^2) + 1*2*x^2/(1-3*x^2+x^4) + 3*4*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 13*7*x^7/(1-29*x^7-x^14) + 21*8*x^8/(1-47*x^8-x^16)  +...).
The values of the Dirichlet character Chi(n,3) repeat [1,1,0, ...].

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

Cf. A008653, A205966, A205968, A203847, A000204 (Lucas).

Cf. A209447 (Pell variant).

terms = 34; s = 1 + 12*Sum[Fibonacci[n]*KroneckerSymbol[n, 3]^2*n*(x^n/(1 - LucasL[n]*x^n + (-1)^n*x^(2*n))), {n, 1, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 05 2017 *)
b[n_] := If[n < 1, Boole[n == 0], 12 Sum[If[Mod[d, 3] > 0, d, 0], {d, Divisors@n}]]; Table[If[n == 0, 1, b[n]*Fibonacci[n]], {n, 0, 50}] (* G. C. Greubel, Jul 17 2018 *)

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 12*sum(m=1,n,fibonacci(m)*kronecker(m,3)^2*m*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,50,print1(a(n),", "))

G.f.: 1 + 12*Sum_{n>=1} Fibonacci(n)*Chi(n,3)*n*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

A209447 a(n) = Pell(n)*A008653(n) for n>=1, with a(0)=1, where A008653 is the theta series of direct sum of 2 copies of hexagonal lattice.

Original entry on oeis.org

1, 12, 72, 60, 1008, 2088, 2520, 16224, 73440, 11820, 513648, 826704, 1164240, 5621448, 23265216, 14041800, 175149504, 245524824, 98791560, 1590026160, 8061191712, 3706940640, 40272058656, 64816900128, 97801149600, 487966581012, 1596075244848, 91744440540

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to the Lambert series of A008653: 1 + 12*Sum_{n>=1} Chi(n,3)*n*x^n/(1-x^n).

Here Chi(n,3) = principal Dirichlet character of n modulo 3.

			G.f.: A(x) = 1 + 12*x + 72*x^2 + 60*x^3 + 1008*x^4 + 2088*x^5 + 2520*x^6 +...
where A(x) = 1 + 1*12*x + 2*36*x^2 + 5*12*x^3 + 12*84*x^4 + 29*72*x^5 + 70*36*x^6 +...+ Pell(n)*A008653(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 12*( 1*1*x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 12*4*x^4/(1-34*x^4+x^8) + 29*5*x^5/(1-82*x^5-x^10) + 169*7*x^7/(1-478*x^7-x^14) + 408*8*x^8/(1-1154*x^8-x^16)  +...).
The values of the Dirichlet character Chi(n,3) repeat [1,1,0, ...].

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

Cf. A008653, A205967, A209446, A209448, A204270, A000129 (Pell), A002203.

A008653[n_]:= If[n < 1, Boole[n == 0], 12*Sum[If[Mod[d, 3] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A008653[n], {n, 1, 1000}]] (* G. C. Greubel, Jan 02 2017 *)

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1 + 12*sum(m=1,n,Pell(m)*kronecker(m,3)^2*m*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,50,print1(a(n),", "))

G.f.: 1 + 12*Sum_{n>=1} Pell(n)*Chi(n,3)*n*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A008655 Theta series of direct sum of 4 copies of hexagonal lattice.

Original entry on oeis.org

1, 24, 216, 888, 1752, 3024, 7992, 8256, 14040, 24216, 27216, 31968, 64824, 52752, 74304, 111888, 112344, 117936, 217944, 164640, 220752, 305472, 287712, 292032, 519480, 378024, 474768, 654072

Offset: 0

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.

Convolution of A008654 and A004016. Convolution square of A008653. - R. J. Mathar, Feb 22 2021

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

A008655 := proc(n)
        add( A004016(i)*x^i,i=0..n) ;
        coeftayl(%^4,x=0,n) ;
end proc: # R. J. Mathar, Feb 22 2021

terms = 28; s = (EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(81*EllipticTheta[3, 0, q^3]^4) + O[q]^(2 terms); CoefficientList[s, q^2][[1 ;; terms]] (* Jean-François Alcover, Jul 07 2017, from LatticeData(A2) *)

Expansion of (theta_3(z)*theta_3(3z) + theta_2(z)*theta_2(3z))^4.

A186100 Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, -12, -12, -12, -12, -72, -12, -96, -12, -12, -72, -144, -12, -168, -96, -72, -12, -216, -12, -240, -72, -96, -144, -288, -12, -372, -168, -12, -96, -360, -72, -384, -12, -144, -216, -576, -12, -456, -240, -168, -72, -504, -96, -528, -144, -72, -288

Offset: 0

Michael Somos, Feb 12 2011

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 - 12*q - 12*q^2 - 12*q^3 - 12*q^4 - 72*q^5 - 12*q^6 - 96*q^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008653, A098098, A111932, A121443, A131943, A131946, A186099.

a[ n_] := If[ n < 1, Boole[n == 0], -12 DivisorSum[ n, # Boole[ 1 == GCD[#, 6]] &]]; (* Michael Somos, Jul 07 2015 *)
a[ n_] := SeriesCoefficient[(EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^3])^2 - 1/2 (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)])^2, {x, 0, n}]; (* Michael Somos, Jul 07 2015 *)

{a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * (1 == gcd( d, 6) ) ) )};

{a(n) = if( n<1, n==0, -12 * direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n])};

Expansion of b(x) * b(x^2) - c(x) * c(x^2) in powers of x where b(), c() are cubic AGM functions.
Expansion of (phi(-x) * phi(-x^3))^2 - 8 * x * (psi(x) * psi(x^3))^2 in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of (P(q) - 2*P(q^2) - 3*P(q^3) + 6*P(q^6)) / 2 in powers of q where P() is a Ramanujan Eisenstein series. - Michael Somos, Jul 07 2015
a(n) = -12 * A186099(n) if n>0. a(2*n) = a(n). a(2*n + 1) = - A008653(2*n + 1). a(n) = 2 * A008653(n) - A008653(2*n) = A131946(n) - 8 * A111932(n) = A131943(n) - 9 * A121443(n).
a(3*n) = a(n). a(6*n + 5) = -72 * A098098(n).- Michael Somos, Jul 07 2015

A227131 Sum of divisors of n that are not divisible by 25. a(0) = 1.

Original entry on oeis.org

1, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 6, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 18, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 195, 74, 114, 24, 140

Offset: 0

Michael Somos, Jul 02 2013

			G.f. = 1 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 15*q^8 + 13*q^9 + ...
75 has six divisors: 1, 3, 5, 15, 25, 75, but both 25 and 75 are divisible by 25, thus not counted, and we have a(75) = 1+3+5+15 = 24. - _Antti Karttunen_, Nov 23 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to sums of divisors.

Cf. A000118, A004011, A008653, A028594, A028887, A096726, A107505.

A := Basis( ModularForms( Gamma0(25), 2), 66); A[1] + A[2] + 3*A[3] + 4*A[4] + 7*A[5]; /* Michael Somos, Jun 12 2014 */

a[ n_] := If[ n < 1, Boole[ n == 0], Sum[ If[ Mod[ d, 25] > 0, d, 0], {d, Divisors @ n}]];

{a(n) = if( n<1, n==0, sumdiv( n, d, if( d%25, d)))};

{a(n) = if( n<1, n==0, 1 * (sigma(n) - if( n%25==0, 25 * sigma( n / 25))))};

A = ModularForms( Gamma0(25), 2, prec=66) . basis(); A[0] + A[1] + 3*A[2] + 4*A[3] + 7*A[4];

a(n) is multiplicative with a(0) = 1, a(5^e) = 6 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (25 t)) = 25 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: 1 + Sum_{k>0} k * x^k / (1 - x^k) - Sum_{k>0} 25 * k * x^(25*k) / (1 - x^(25*k)).
Sum_{k=1..n} a(k) ~ (2*Pi^2/25) * n^2. - Amiram Eldar, Oct 04 2022

More terms from Antti Karttunen, Nov 23 2017

A359101 a(n) = phi(5 * n)/4.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 6, 4, 6, 5, 10, 4, 12, 6, 10, 8, 16, 6, 18, 10, 12, 10, 22, 8, 25, 12, 18, 12, 28, 10, 30, 16, 20, 16, 30, 12, 36, 18, 24, 20, 40, 12, 42, 20, 30, 22, 46, 16, 42, 25, 32, 24, 52, 18, 50, 24, 36, 28, 58, 20, 60, 30, 36, 32, 60, 20, 66, 32, 44, 30, 70, 24, 72, 36, 50, 36, 60, 24, 78

Offset: 1

Seiichi Manyama, Dec 16 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Moebius Transform.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A195459, A359102.

Cf. A008653, A359100.

Array[EulerPhi[5 #]/4 &, 79] (* Michael De Vlieger, Dec 16 2022 *)

PARI
```
a(n) = eulerphi(5*n)/4;
```

my(N=80, x='x+O('x^N)); Vec(-sum(k=1, N, moebius(5*k)*x^k/(1-x^k)^2))

G.f.: -Sum_{k>=1} mu(5 * k) * x^k / (1 - x^k)^2, where mu() is the Moebius function (A008683).
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(5^e) = 5^e, and a(p^e) = (p-1)*p^(e-1) if p != 5.
Dirichlet g.f.: zeta(s-1)/(zeta(s)*(1-1/5^s)).
Sum_{k=1..n} a(k) ~ (25/(8*Pi^2)) * n^2. (End)

A359102 a(n) = phi(7 * n)/6.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 4, 6, 4, 10, 4, 12, 7, 8, 8, 16, 6, 18, 8, 14, 10, 22, 8, 20, 12, 18, 14, 28, 8, 30, 16, 20, 16, 28, 12, 36, 18, 24, 16, 40, 14, 42, 20, 24, 22, 46, 16, 49, 20, 32, 24, 52, 18, 40, 28, 36, 28, 58, 16, 60, 30, 42, 32, 48, 20, 66, 32, 44, 28, 70, 24, 72, 36, 40, 36, 70, 24, 78

Offset: 1

Seiichi Manyama, Dec 16 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Moebius Transform.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A195459, A359101.

Cf. A008653, A359099.

Array[EulerPhi[7 #]/6 &, 79] (* Michael De Vlieger, Dec 16 2022 *)

PARI
```
a(n) = eulerphi(7*n)/6;
```

my(N=80, x='x+O('x^N)); Vec(-sum(k=1, N, moebius(7*k)*x^k/(1-x^k)^2))

G.f.: -Sum_{k>=1} mu(7 * k) * x^k / (1 - x^k)^2, where mu() is the Moebius function (A008683).
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(7^e) = 7^e, and a(p^e) = (p-1)*p^(e-1) if p != 7.
Dirichlet g.f.: zeta(s-1)/(zeta(s)*(1-1/7^s)).
Sum_{k=1..n} a(k) ~ (49/(16*Pi^2)) * n^2. (End)

A242874 Expansion of b(q)^2 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -6, 9, 12, -42, 18, 36, -48, 45, 12, -108, 36, 84, -84, 72, 72, -186, 54, 36, -120, 126, 96, -216, 72, 180, -186, 126, 12, -336, 90, 216, -192, 189, 144, -324, 144, 84, -228, 180, 168, -540, 126, 288, -264, 252, 72, -432, 144, 372, -342, 279, 216, -588

Offset: 0

Michael Somos, May 26 2014

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 6*q + 9*q^2 + 12*q^3 - 42*q^4 + 18*q^5 + 36*q^6 - 48*q^7 + 45*q^8 + ...

O. Kolberg, The coefficients of j(tau) modulo powers of 3, Acta Univ. Bergen., Series Math., Arbok for Universitetet I Bergen, Mat.-Naturv. Serie, 1962 No. 16, pp. 1-7. See t, page 1.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)

Cf. A005928, A008653, A033686, A144614.

A := Basis( ModularForms( Gamma0(9), 2), 53); A[1] - 6*A[2] + 9*A[3]; /* Michael Somos, Sep 27 2016 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 / QPochhammer[ q^3])^2, {q, 0, n}];

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

A = ModularForms( Gamma0(9), 2, prec=53) . basis(); A[0] - 6*A[1] + 9*A[2];

Expansion of (eta(q)^3 / eta(q^3))^2 in powers of q.
Euler transform of period 3 sequence [-6, -6, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 243 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033686.
G.f.: Product_{k>0} ( (1 - x^k)^3 / (1 - x^(3*k)) )^2.
a(3*n) = A008653(n). a(3*n + 1) = -6 * A144614(n). a(3*n + 2) = 9 * A033686(n).
Convolution square of A005928.

A136747 Expansion of a(q)^2 * (b(q) * c(q) / 3)^3 in powers of q where a(), b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, 6, -27, -92, 390, -162, -64, -1320, 729, 2340, -948, 2484, -5098, -384, -10530, 3856, 28386, 4374, -8620, -35880, 1728, -5688, -15288, 35640, 73975, -30588, -19683, 5888, 36510, -63180, -276808, 192096, 25596, 170316, -24960, -67068, 268526, -51720, 137646

Offset: 1

Michael Somos, Jan 20 2008

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = q + 6*q^2 - 27*q^3 - 92*q^4 + 390*q^5 - 162*q^6 - 64*q^7 - 1320*q^8 + ...

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.] See page 30, line -3.
LMFDB, Newform orbit 3.8.a.a.

Cf. A007332, A008653.

Basis( CuspForms( Gamma0(3), 8), 40)[1]; /* Michael Somos, Oct 12 2015 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^3])^6 ((QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / QPochhammer[ q^3])^2, {q, 0, n}]; (* Michael Somos, May 28 2013 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^6 * sum(k=1, n, 12 * (sigma(3*k) - 3 * sigma(k)) * x^k, 1 + A), n))};

CuspForms( Gamma0(3), 8, prec=40).0; # Michael Somos, May 28 2013

Expansion of (eta(q) * eta(q^3))^6 * ((eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3))^2 in powers of q.
a(n) is multiplicative with a(3^e) = (-27)^e, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) unless p = 3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^4*w + 512*u^3*w^2 + 131072*u^2*w^3 + 16777216*u*w^4 - 24*u^3*v*w - 9216*u^2*v*w^2 - 1572864*u*v*w^3 + 288*u^2*v^2*w + 73728*u*v^2*w^2 - u^2*v^3 - 1984*w*v^3*u - 65536*w^2*v^3 + 12*v^4*u + 3072*w*v^4 - 36*v^5.
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 81 (t/i)^8 f(t) where q = exp(2 Pi i t).
G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(3*k)))^6 * (Sum_{j,k in Z} x^(j*j + j*k + k*k))^2.
Convolution of A007332 and A008653.

A266288 Expansion of a(q)^2 * (c(q)/3)^3 in powers of q where a(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, 15, 81, 241, 624, 1215, 2402, 3855, 6561, 9360, 14640, 19521, 28562, 36030, 50544, 61681, 83520, 98415, 130322, 150384, 194562, 219600, 279840, 312255, 390001, 428430, 531441, 578882, 707280, 758160, 923522, 986895, 1185840, 1252800, 1498848, 1581201

Offset: 1

Michael Somos, Dec 26 2015

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Convolution of A008653 and A106402.

			G.f. = x + 15*x^2 + 81*x^3 + 241*x^4 + 624*x^5 + 1215*x^6 + 2402*x^7 + ...

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

Cf. A004016, A005882, A005928, A008653, A106402, A344778.

A := Basis( ModularForms( Gamma1(3), 5),37);  A[2];

a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (With[{s = {1, -1, 0}[[Mod[#, 3, 1]]]}, ((#^4)^(#2 + 1) - s^(#2 + 1)) / (#^4 - s)] & @@@ FactorInteger[n])];

{a(n) = my(A, U1, u3, U9); if( n<1, 0, n--; A = x * O(x^n); U1 = eta(x + A)^3; u3 = eta(x^3 + A); U9 = eta(x^9 + A)^3; polcoeff( U1 * u3^7 * (1 + 9*x*U9/U1)^2, n))};

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

a(n) is multiplicative with a(p^e) = ((p^4)^(e+1) - s^(e+1)) / (p^4 - s) where s = 0 if p = 3, s = 1 if p == 1 (mod 3), s = -1 if p == 2 (mod 3).

Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 4*Pi^5/(729*sqrt(3)) = 0.9694405... (A344778). - Amiram Eldar, Nov 09 2023

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A205967 a(n) = Fibonacci(n)*A008653(n) for n>=1, with a(0)=1, where A008653 is the theta series of direct sum of 2 copies of hexagonal lattice.

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A209447 a(n) = Pell(n)*A008653(n) for n>=1, with a(0)=1, where A008653 is the theta series of direct sum of 2 copies of hexagonal lattice.

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A008655 Theta series of direct sum of 4 copies of hexagonal lattice.

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Maple

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A186100 Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.

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A227131 Sum of divisors of n that are not divisible by 25. a(0) = 1.

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Magma

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PARI

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A359101 a(n) = phi(5 * n)/4.

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A359102 a(n) = phi(7 * n)/6.

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