A028594 -id:A028594 - cp's OEIS frontend

A205976 a(n) = Fibonacci(n)A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in (theta_3(x)theta_3(7x)+theta_2(x)theta_2(7*x))^2.

1, 4, 12, 32, 84, 120, 384, 52, 1260, 1768, 3960, 4272, 16128, 13048, 4524, 58560, 122388, 114984, 403104, 334480, 1136520, 175136, 2550384, 2751072, 11128320, 9303100, 20394024, 31426880, 8898708, 61707480, 239627520, 172322432, 548933868, 676718976, 1231823592

Offset: 0

Paul D. Hanna, Feb 04 2012

nonn

Compare g.f. to the Lambert series of A028594:
1 + 4*Sum_{n>=1} Chi(n,7)*n*x^n/(1-x^n).
Here Chi(n,7) = principal Dirichlet character of n modulo 7.

			G.f.: A(x) = 1 + 4*x + 12*x^2 + 32*x^3 + 84*x^4 + 120*x^5 + 384*x^6 + 52*x^7 +...
where A(x) = 1 + 1*4*x + 1*12*x^2 + 2*16*x^3 + 3*28*x^4 + 5*24*x^5 + 8*48*x^6 + 13*4*x^7 + 21*60*x^8 + 34*52*x^9 +...+ Fibonacci(n)*A028594(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 4*( 1*1*x/(1-x-x^2) + 1*2*x^2/(1-3*x^2+x^4) + 2*3*x^3/(1-4*x^3-x^6) + 3*4*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 8*6*x^6/(1-18*x^6+x^12) + 0*13*7*x^7/(1+29*x^7-x^14) +...).
The values of the Dirichlet character Chi(n,7) repeat [1,1,1,1,1,1,0, ...].

Cf. A028594, A205975, A203847, A000204 (Lucas).

Cf. A209456 (Pell variant).

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 4*sum(m=1,n,fibonacci(m)*kronecker(m,7)^2*m*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,60,print1(a(n),", "))

G.f.: 1 + 4*Sum_{n>=1} Fibonacci(n)*Chi(n,7)*n*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

A209456 a(n) = Pell(n)A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in (theta_3(x)theta_3(7x)+theta_2(x)theta_2(7*x))^2.

Original entry on oeis.org

1, 4, 24, 80, 336, 696, 3360, 676, 24480, 51220, 171216, 275568, 1552320, 1873816, 969384, 18722400, 58383168, 81841608, 428096760, 530008720, 2687063904, 617823440, 13424019552, 21605633376, 130401532800, 162655527004, 532025081616, 1223259207200

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to the Lambert series of A028594:
1 + 4*Sum_{n>=1} Chi(n,7)*n*x^n/(1-x^n).
Here Chi(n,7) = principal Dirichlet character of n modulo 7.

			G.f.: A(x) = 1 + 4*x + 24*x^2 + 80*x^3 + 336*x^4 + 696*x^5 + 3360*x^6 +...
where A(x) = 1 + 1*4*x + 2*12*x^2 + 5*16*x^3 + 12*28*x^4 + 29*24*x^5 + 70*48*x^6 + 169*4*x^7 + 408*60*x^8 + 985*52*x^9 +...+ Pell(n)*A028594(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 4*( 1*1*x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 5*3*x^3/(1-14*x^3-x^6) + 12*4*x^4/(1-34*x^4+x^8) + 29*5*x^5/(1-82*x^5-x^10) + 70*6*x^6/(1-198*x^6+x^12) + 0*169*7*x^7/(1+478*x^7-x^14) +...).
The values of the Dirichlet character Chi(n,7) repeat [1,1,1,1,1,1,0, ...].

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

Cf. A028594, A205976, A209455, A204270, A000129 (Pell), A002203.

A028594[n_]:= If[n < 1, Boole[n == 0], 4*Sum[If[Mod[d, 7] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A028594[n], {n,1,50}]] (* G. C. Greubel, Jan 03 2018 *)

{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 + 4*sum(m=1,n,Pell(m)*kronecker(m,7)^2*m*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,35,print1(a(n),", "))

G.f.: 1 + 4*Sum_{n>=1} Pell(n)*Chi(n,7)*n*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A113957 Sum of the divisors of n which are not divisible by 7.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 1, 15, 13, 18, 12, 28, 14, 3, 24, 31, 18, 39, 20, 42, 4, 36, 24, 60, 31, 42, 40, 7, 30, 72, 32, 63, 48, 54, 6, 91, 38, 60, 56, 90, 42, 12, 44, 84, 78, 72, 48, 124, 1, 93, 72, 98, 54, 120, 72, 15, 80, 90, 60, 168, 62, 96, 13, 127, 84, 144, 68, 126, 96, 18, 72

Offset: 1

Michael Somos, Nov 10 2005

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 467, Entry 5(i).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A028594(n)=4*a(n) if n>0.

Cf. A244600.

f[p_, e_] := If[p == 7, 1, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

a(n)=if(n<1, 0, sigma(n/7^valuation(n,7)))

a(n) is multiplicative and a(p^e) = 1, if p=7, a(p^e) = (p^(e+1)-1)/(p-1) otherwise.
G.f.: ((theta_3(z)*theta_3(7z) + theta_2(z)*theta_2(7z))^2-1)/4.
L.g.f.: log(Product_{k>=1} (1 - x^(7*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018
Sum_{k=1..n} a(k) ~ (Pi^2/14) * n^2. - Amiram Eldar, Oct 04 2022
Dirichlet g.f. (1-7^(1-s))*zeta(s)*zeta(s-1). - R. J. Mathar, May 17 2023

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

A366091 a(n) is the number of ways to write n = i^2 + 2j^2 + 3k^2 with i,j,k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 0, 2, 4, 1, 2, 2, 2, 1, 3, 2, 2, 4, 2, 1, 2, 2, 0, 4, 3, 2, 5, 2, 1, 3, 2, 2, 7, 2, 2, 5, 0, 2, 0, 2, 4, 4, 3, 1, 4, 3, 3, 5, 3, 2, 7, 1, 2, 6, 0, 3, 6, 2, 2, 4, 2, 2, 6, 3, 2, 4, 3, 3, 3, 2, 0, 7, 5, 2, 6, 3, 2, 8, 2, 2, 11, 2, 5, 2, 2, 3, 0, 4, 3, 7, 3, 2, 2, 3, 3

Offset: 0

Robert Israel, Sep 28 2023

nonn

			a(9) = 3 because 9 = 3^2 + 2*0^2 + 3*0^2 = 1^2 + 2*2^2 + 3*0^2 = 2^2 + 2*1^2 + 3*1^2.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A028594 (allows any integer i,j,k), A055042 (a(n) = 0)

g:= (1+JacobiTheta3(0,z))*(1+JacobiTheta3(0,z^2))*(1+JacobiTheta3(0,z^3))/8:
S:= series(g,z,101):
seq(coeff(S,z,j),j=0..100);

from itertools import count
from sympy.ntheory.primetest import is_square
def A366091(n):
    c = 0
    for k in count(0):
        if (a:=3*k**2)>n:
            break
        for j in count(0):
            if (b:=a+(j**2<<1))>n:
                break
            if is_square(n-b):
                c += 1
    return c # Chai Wah Wu, Sep 29 2023

G.f. (1 + theta_3(0,z)) * (1 + theta_3(0,z^2)) * (1 + theta_3(0,z^3))/8 where theta_3 is a Jacobi theta function.

A028596 Expansion of (theta_3(z)theta_3(7z) + theta_2(z)theta_2(7z))^4.

Original entry on oeis.org

1, 8, 40, 128, 328, 656, 1216, 1864, 2856, 3560, 5392, 6368, 9856, 10640, 17000, 16832, 22600, 23760, 32776, 32576, 43792, 52864, 57568, 58560, 78528, 76024, 94864, 98432, 137864, 116720, 152512, 143040, 179240, 179072, 212112, 237328, 265768, 242352, 296704, 295232

Offset: 0

N. J. A. Sloane

nonn

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Cf. A002652, A002653, A028594.

a(n) = polcoeff((1 + 2*x*Ser(qfrep([2, 1; 1, 4], n, 1)))^4, n); \\ Jinyuan Wang, Feb 21 2020

More terms from Jinyuan Wang, Feb 21 2020

cp's OEIS Frontend

A205976 a(n) = Fibonacci(n)*A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in (theta_3(x)*theta_3(7*x)+theta_2(x)*theta_2(7*x))^2.

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A209456 a(n) = Pell(n)*A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in (theta_3(x)*theta_3(7*x)+theta_2(x)*theta_2(7*x))^2.

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Mathematica

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A113957 Sum of the divisors of n which are not divisible by 7.

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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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A366091 a(n) is the number of ways to write n = i^2 + 2*j^2 + 3*k^2 with i,j,k >= 0.

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Maple

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A028596 Expansion of (theta_3(z)*theta_3(7z) + theta_2(z)*theta_2(7z))^4.

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A205976 a(n) = Fibonacci(n)A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in (theta_3(x)theta_3(7x)+theta_2(x)theta_2(7*x))^2.

A209456 a(n) = Pell(n)A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in (theta_3(x)theta_3(7x)+theta_2(x)theta_2(7*x))^2.

A366091 a(n) is the number of ways to write n = i^2 + 2j^2 + 3k^2 with i,j,k >= 0.

A028596 Expansion of (theta_3(z)theta_3(7z) + theta_2(z)theta_2(7z))^4.