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 1-2 of 2 results.

A163659 L.g.f.: Sum_{n>=1} a(n)*x^n/n = log(S(x)/x) where S(x) is the g.f. of Stern's diatomic series (A002487).

Original entry on oeis.org

1, 3, -2, 7, 1, -6, 1, 15, -2, 3, 1, -14, 1, 3, -2, 31, 1, -6, 1, 7, -2, 3, 1, -30, 1, 3, -2, 7, 1, -6, 1, 63, -2, 3, 1, -14, 1, 3, -2, 15, 1, -6, 1, 7, -2, 3, 1, -62, 1, 3, -2, 7, 1, -6, 1, 15, -2, 3, 1, -14, 1, 3, -2, 127, 1, -6, 1, 7, -2, 3, 1, -30, 1, 3, -2, 7, 1, -6, 1, 31, -2, 3, 1, -14, 1
Offset: 1

Views

Author

Paul D. Hanna, Aug 02 2009

Keywords

Examples

			L.g.f.: log(S(x)/x) = x + 3*x^2/2 - 2*x^3/3 + 7*x^4/4 + x^5/5 - 6*x^6/6 +...
where S(x) is the g.f. of Stern's diatomic series (A002487):
S(x) = x + x^2 + 2*x^3 + x^4 + 3*x^5 + 2*x^6 + 3*x^7 + x^8 + 4*x^9 +...

Links

Crossrefs

Cf. A163658, A195587, A002487.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 2, 1, # == 2, 2^(#2 + 1) - 1, # == 3, -2, True, 1] & @@@ FactorInteger[n])]; (* Michael Somos, Jun 27 2017 *)
  • PARI
    A002487(n)=local(c=1, b=0); while(n>0, if(bitand(n, 1), b+=c, c+=b); n>>=1); b
{a(n)=n*polcoeff(log(sum(k=0,n,A002487(k+1)*x^k)+x*O(x^n)),n)}
for(n=1, 64, print1(a(n), ", "))
  • PARI
    {a(n) = if( n<1, 0, if( n%3, 1, -2) * sigma(2 ^ valuation(n, 2)))}; /* Michael Somos, Feb 14 2011 */
  • PARI
    a(n)=local(X=x+x*O(x^n), A); A=log(1+X+X^2) + sum(k=0, #binary(n), log(1 + X^(2*2^k) + X^(4*2^k))); n*polcoeff(A, n)
for(n=1, 64, print1(a(n), ", ")) \\ Paul D. Hanna, May 04 2014

Formula

a(2^n) = 2^(n+1) - 1 for n>=0.
a(n) is multiplicative with a(2^e) = 2^(e+1) - 1, a(3^e) = -2 if e>0, a(p^e) = 1 if p>3. - Michael Somos, Feb 14 2011
L.g.f.: Sum_{n>=0} log(1 + x^(2^n) + x^(2*2^n)) = Sum_{n>=1} a(n)*x^n/n. - Paul D. Hanna, May 04 2014
G.f.: Sum_{n>=0} 2^n * x^(2^n) * (1 + 2*x^(2^n)) / (1 + x^(2^n) + x^(2*2^n)). - Paul D. Hanna, May 04 2014
Dirichlet g.f.: zeta(s) * (1 - 3^(1-s)) / (1 - 2^(1-s)). - Amiram Eldar, Oct 24 2023

A195586 G.f.: exp( Sum_{n>=1} A163659(n^2)*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, 4, 3, 15, 12, 37, 25, 100, 75, 219, 144, 501, 357, 972, 615, 1995, 1380, 3665, 2285, 7052, 4767, 12255, 7488, 22305, 14817, 37524, 22707, 65775, 43068, 106837, 63769, 180436, 116667, 286251, 169584, 471173, 301589, 729404, 427815, 1169211, 741396, 1778545, 1037149
Offset: 0

Views

Author

Paul D. Hanna, Sep 20 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 3*x^3 + 15*x^4 + 12*x^5 + 37*x^6 + 25*x^7 +...
where
log(A(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 +...+ A195587(n)*x^n/n +...
Let C(x) be the odd bisection of g.f. A(x):
C(x) = 1 + 3*x + 12*x^2 + 25*x^3 + 75*x^4 + 144*x^5 + 357*x^6 + 615*x^7 + 1380*x^8 + 2285*x^9 + 4767*x^10 + 7488*x^11 + 14817*x^12 +...+ A237650(n)*x^n +...
then C(x) equals the cube of an integer series:
C(x)^(1/3) = 1 + x + 3*x^2 + 2*x^3 + 9*x^4 + 7*x^5 + 17*x^6 + 10*x^7 + 41*x^8 + 31*x^9 + 75*x^10 + 44*x^11 + 150*x^12 +...+ A237651(n)*x^n +...
which equals A(x)/C(x^2)^(1/3).
The g.f. may be expressed by the product:
A(x) = (1+x+x^2) * (1+x^2+x^4)^3 * (1+x^4+x^8)^6 * (1+x^8+x^16)^12 * (1+x^16+x^32)^24 *...* (1 + x^(2*2^n) + x^(4*2^n))^(3*2^n) *...

Links

Crossrefs

Cf. A195587, A163658, A163659, A237650, A002487; variant: A237646.

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^2)*x^k/k)+x*O(x^n)), n)}
for(n=0,50,print1(a(n),", "))
  • PARI
    /* G.f.: A(x) = (1+x+x^2) * (1+x^2+x^4) * A(x^2)^2: */
{a(n)=local(A=1+x); for(i=1, #binary(n), A=(1+x+x^2)*(1+x^2+x^4)*subst(A^2, x, x^2) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
  • PARI
    /* G.f.: (1+x+x^2) * Product_{n>=0} (1 + x^(2*2^n) + x^(4*2^n))^(3*2^n): */
{a(n)=local(A=1+x); A=(1+x+x^2)*prod(k=0, #binary(n), (1+x^(2*2^k)+x^(4*2^k)+x*O(x^n))^(3*2^k)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} A195587(n)*x^n/n ), where A195587(n) = A163659(n^2).
G.f. A(x) satisfies:
(1) A(x) = (1+x+x^2) * (1+x^2+x^4) * A(x^2)^2.
(2) A(x) = (1+x+x^2) * Product_{n>=0} ( 1 + x^(2*2^n) + x^(4*2^n) )^(3*2^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)^3 * C(x^2)^2.
(6) A(x) = (1+x+x^2) * C(x^2).
(7) A(x)^3 = C(x) * C(x^2).
(8) A(x)^2 = C(x) / (1+x+x^2).
(9) A(x) = ( C(x)/A(x) - C(x^2)^2/A(x^2)^2 ) / (2*x).

Extensions

Entry and formulas revised by Paul D. Hanna, May 04 2014
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.