cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A237651 G.f. satisfies: A(x) = (1+x+x^2) * A(x^2)^2.

Original entry on oeis.org

1, 1, 3, 2, 9, 7, 17, 10, 41, 31, 75, 44, 150, 106, 238, 132, 445, 313, 711, 398, 1251, 853, 1859, 1006, 3135, 2129, 4677, 2548, 7590, 5042, 10734, 5692, 16865, 11173, 23979, 12806, 36911, 24105, 50551, 26446, 75985, 49539, 104683, 55144, 155140, 99996, 207188, 107192, 300766, 193574, 403994
Offset: 0

Views

Author

Paul D. Hanna, May 04 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 2*x^3 + 9*x^4 + 7*x^5 + 17*x^6 + 10*x^7 +...
where:
A(x) = (1+x+x^2) * (1+x^2+x^4)^2 * (1+x^4+x^8)^4 * (1+x^8+x^16)^8 * (1+x^16+x^32)^16 *...* (1 + x^(2^n) + x^(2*2^n))^(2^n) *...

Crossrefs

Cf. A002487, A195586, A237650.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,#binary(n),A=(1+x+x^2)*subst(A^2,x,x^2) +x*O(x^n));polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x);A=prod(k=0,#binary(n),(1+x^(2^k)+x^(2*2^k)+x*O(x^n))^(2^k));polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

Formula

The odd-indexed bisection equals the self-convolution of this sequence.
The self-convolution cube yields A237650, the odd-indexed bisection of A195586.
G.f. A(x) satisfies:
(1) A(x) = Product_{n>=0} ( 1 + x^(2^n) + x^(2*2^n) )^(2^n).
(2) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2).

A237646 G.f.: exp( Sum_{n>=1} A163659(n^3)*x^n/n ), where x*exp(Sum_{n>=1} A163659(n)*x^n/n) = S(x) is the g.f. of Stern's diatomic series (A002487).

Original entry on oeis.org

1, 1, 8, 7, 63, 56, 329, 273, 1736, 1463, 7511, 6048, 32585, 26537, 124440, 97903, 475287, 377384, 1658881, 1281497, 5783960, 4502463, 18825023, 14322560, 61171649, 46849089, 188181672, 141332583, 577889023, 436556440, 1696298665, 1259742225, 4970284200, 3710541975, 14019036535, 10308494560
Offset: 0

Views

Author

Paul D. Hanna, May 03 2014

Keywords

Comments

Compare to the g.f. of A195586.

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 +...
where
log(A(x)) = x + 15*x^2/2 - 2*x^3/3 + 127*x^4/4 + x^5/5 - 30*x^6/6 + x^7/7 + 1023*x^8/8 +...+ A237649(n)*x^n/n +...
Bisections: let A(x) = B(x^2) + x*C(x^2), then:
B(x) = 1 + 8*x + 63*x^2 + 329*x^3 + 1736*x^4 + 7511*x^5 + 32585*x^6 +...
C(x) = 1 + 7*x + 56*x^2 + 273*x^3 + 1463*x^4 + 6048*x^5 + 26537*x^6 + 97903*x^7 + 377384*x^8 + 1281497*x^9 + 4502463*x^10 +...+ A237647(n)*x^n +...
Note that C(x)^(1/7) = (1+x+x^2) * C(x^2)^(4/7) is an integer series:
C(x)^(1/7) = 1 + x + 5*x^2 + 4*x^3 + 30*x^4 + 26*x^5 + 106*x^6 + 80*x^7 + 459*x^8 + 379*x^9 + 1451*x^10 + 1072*x^11 + 5210*x^12 +...+ A237648(n)*x^n +...
Also, C(x) / (1+x+x^2)^3 = A(x)^4:
A(x)^4 = 1 + 4*x + 38*x^2 + 128*x^3 + 817*x^4 + 2536*x^5 + 12890*x^6 +...
Further, C(x)*C(x^2)^3 = A(x)^7:
A(x)^7 = 1 + 7*x + 77*x^2 + 420*x^3 + 2954*x^4 + 13986*x^5 + 78414*x^6 +...
The g.f. may be expressed by the product:
A(x) = (1+x+x^2) * (1+x^2+x^4)^7 * (1+x^4+x^8)^28 * (1+x^8+x^16)^112 * (1+x^16+x^32)^448 *...* (1 + x^(2*2^n) + x^(4*2^n))^(7*4^n) *...

Crossrefs

Cf. A237647, A237648, A237649, A195586.

Programs

  • PARI
    {A163659(n)=if(n<1, 0, if(n%3, 1, -2)*sigma(2^valuation(n, 2)))}
{a(n)=polcoeff(exp(sum(k=1, n, A163659(k^3)*x^k/k)+x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} A237649(n)*x^n/n ), where A237649(n) = A163659(n^3).
G.f. A(x) satisfies:
(1) A(x) = (1+x+x^2) * (1+x^2+x^4)^3 * A(x^2)^4.
(2) A(x) = (1+x+x^2) * Product_{n>=0} ( 1 + x^(2*2^n) + x^(4*2^n) )^(7*4^n).
(3) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2).
Bisections: let A(x) = B(x^2) + x*C(x^2), then
(4) B(x) = (1+x) * C(x).
(5) C(x) = (1+x+x^2)^7 * C(x^2)^4.
(6) A(x) = (1+x+x^2) * C(x^2).
(7) A(x)^7 = C(x) * C(x^2)^3.
(8) A(x)^4 = C(x) / (1+x+x^2)^3.
(9) A(x)^3 = ( C(x)/A(x) - C(x^2)^4/A(x^2)^4 ) / (6*x + 14*x^3 + 6*x^5).

A195587 a(n) = A163659(n^2), where A163659 is the logarithmic derivative of Stern's diatomic series (A002487).

Original entry on oeis.org

1, 7, -2, 31, 1, -14, 1, 127, -2, 7, 1, -62, 1, 7, -2, 511, 1, -14, 1, 31, -2, 7, 1, -254, 1, 7, -2, 31, 1, -14, 1, 2047, -2, 7, 1, -62, 1, 7, -2, 127, 1, -14, 1, 31, -2, 7, 1, -1022, 1, 7, -2, 31, 1, -14, 1, 127, -2, 7, 1, -62, 1, 7, -2, 8191, 1, -14, 1, 31, -2, 7, 1, -254, 1, 7, -2, 31, 1, -14, 1, 511, -2, 7, 1, -62, 1, 7, -2, 127, 1, -14, 1, 31, -2, 7, 1, -4094
Offset: 1

Views

Author

Paul D. Hanna, Sep 20 2011

Keywords

Comments

Multiplicative because A163659 is. - Andrew Howroyd, Jul 26 2018

Examples

			L.g.f.: L(x) = x + 7*x^2/2 - 2*x^3/3 + 31*x^4/4 + x^5/5 - 14*x^6/6 + x^7/7 + 127*x^8/8 +...
where
exp(L(x)) = 1 + x + 4*x^2 + 3*x^3 + 15*x^4 + 12*x^5 + 37*x^6 + 25*x^7 +...

Links

Crossrefs

Cf. A195586, A163659, A237649.

Programs

  • Mathematica
    a[n_] := Times @@ (Function[{p, e}, Which[p == 2, 2^(e+1) - 1, p == 3, -2, True, 1]] @@@ FactorInteger[n^2]);
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019 *)
  • PARI
    {A163659(n)=if(n<1,0,if(n%3,1,-2)*sigma(2^valuation(n,2)))}
{a(n)=A163659(n^2)}
for(n=1, 64, print1(a(n), ", "))
  • PARI
    {a(n)=local(X=x+x*O(x^n), A); A=log(1+X+X^2) + sum(k=0, #binary(n), 3*2^k*log(1 + X^(2*2^k) + X^(4*2^k))); n*polcoeff(A, n)}
for(n=1, 64, print1(a(n), ", "))

Formula

L.g.f.: log(1+x+x^2) + Sum_{n>=0} 3*2^n * log(1 + x^(2*2^n) + x^(4*2^n)) = Sum_{n>=1} a(n)*x^n/n. - Paul D. Hanna, May 04 2014
G.f.: x*(1+2*x)/(1+x+x^2) + Sum_{n>=0} 6*4^n * x^(2*2^n) * (1 + 2*x^(2*2^n)) / (1 + x^(2*2^n) + x^(4*2^n)). - Paul D. Hanna, May 04 2014
Dirichlet g.f.: zeta(s) * (1 - 3^(1-s)) * (2^s + 2) / (2^s - 4). - Amiram Eldar, Oct 24 2023

A237650 G.f. satisfies: A(x) = (1+x+x^2)^3 * A(x^2)^2.

Original entry on oeis.org

1, 3, 12, 25, 75, 144, 357, 615, 1380, 2285, 4767, 7488, 14817, 22707, 43068, 63769, 116667, 169584, 301589, 427815, 741396, 1037149, 1761087, 2418432, 4025153, 5465955, 8956716, 11986009, 19330347, 25633296, 40835973, 53508711, 84129156, 109392269, 170278047, 219206976
Offset: 0

Views

Author

Paul D. Hanna, May 04 2014

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 25*x^3 + 75*x^4 + 144*x^5 + 357*x^6 +...
where:
A(x) = (1+x+x^2)^3 * (1+x^2+x^4)^6 * (1+x^4+x^8)^12 * (1+x^8+x^16)^24 * (1+x^16+x^32)^48 *...* (1 + x^(2^n) + x^(2*2^n))^(3*2^n) *...

Crossrefs

Cf. A195586, A237651, A237647.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,#binary(n),A=(1+x+x^2)^3*subst(A^2,x,x^2) +x*O(x^n));polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x);A=prod(k=0,#binary(n),(1+x^(2^k)+x^(2*2^k)+x*O(x^n))^(3*2^k));polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

Formula

The odd-indexed bisection of A195586.
The 3rd self-convolution of A237651.
G.f. A(x) satisfies:
(1) A(x) = Product_{n>=0} ( 1 + x^(2^n) + x^(2*2^n) )^(3*2^n).
(2) A(x) / A(-x) = (1+x+x^2)^3 / (1-x+x^2)^3.
Showing 1-4 of 4 results.

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