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 1-6 of 6 results.

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

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).

A237649 a(n) = A163659(n^3), where A163659 is the logarithmic derivative of Stern's diatomic series (A002487).

Original entry on oeis.org

1, 15, -2, 127, 1, -30, 1, 1023, -2, 15, 1, -254, 1, 15, -2, 8191, 1, -30, 1, 127, -2, 15, 1, -2046, 1, 15, -2, 127, 1, -30, 1, 65535, -2, 15, 1, -254, 1, 15, -2, 1023, 1, -30, 1, 127, -2, 15, 1, -16382, 1, 15, -2, 127, 1, -30, 1, 1023, -2, 15, 1, -254, 1, 15, -2, 524287, 1, -30, 1, 127
Offset: 1

Views

Author

Paul D. Hanna, May 03 2014

Keywords

Comments

Multiplicative because A163659 is. - Andrew Howroyd, Jul 27 2018

Examples

			L.g.f.: L(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 +...+ A163659(n^3)*x^n/n +...
where
exp(L(x)) = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 + 1736*x^8 +...+ A237646(n)*x^n +...

Links

Crossrefs

Cf. A237646, A163659.

Programs

  • PARI
    {A163659(n)=if(n<1, 0, if(n%3, 1, -2)*sigma(2^valuation(n, 2)))}
{a(n)=A163659(n^3)}
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),7*4^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} 7*4^n * log(1 + x^(2*2^n) + x^(4*2^n)) = Sum_{n>=1} a(n)*x^n/n.
G.f.: x*(1+2*x)/(1+x+x^2) + Sum_{n>=0} 14*8^n * x^(2*2^n) * (1 + 2*x^(2*2^n)) / (1 + x^(2*2^n) + x^(4*2^n)).

A163658 G.f.: A(x) = 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, 5, 6, 26, 30, 95, 115, 347, 412, 1076, 1308, 3277, 3941, 9081, 11050, 24694, 29834, 63067, 76711, 158127, 191360, 379032, 460448, 893441, 1081113, 2035189, 2468182, 4565994, 5520070, 9970503, 12068315, 21475803, 25926236, 45246532
Offset: 0

Views

Author

Paul D. Hanna, Aug 02 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 6*x^3 + 26*x^4 + 30*x^5 + 95*x^6 +...
log(A(X)) = x + 3^2*x^2/2 + 2^2*x^3/3 + 7^2*x^4/4 + x^5/5 + 6^2*x^6/6 +...
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 +...

Crossrefs

Cf. A163659, A002487, A156302 (variant).

Programs

  • PARI
    {A002487(n)=local(c=1, b=0); while(n>0, if(bitand(n, 1), b+=c, c+=b); n>>=1); b}
{A163659(n)=n*polcoeff(log(sum(k=0,n,A002487(k+1)*x^k)+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A163659(k)^2*x^k/k)+x*O(x^n)), n)}

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

A185954 G.f.: A(x) = exp( Sum_{n>=1} A163659(2n)*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, 3, 8, 13, 23, 32, 49, 59, 80, 93, 127, 144, 185, 203, 256, 269, 319, 328, 401, 419, 504, 525, 639, 656, 761, 763, 904, 917, 1063, 1064, 1241, 1227, 1368, 1317, 1503, 1480, 1681, 1659, 1928, 1909, 2143, 2080, 2393, 2371, 2696, 2653, 3055, 2992, 3305, 3147
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2011

Keywords

Comments

Compare with g.f. of A171238: exp( Sum_{n>=1} A163659(3n)*x^n/n ).

Examples

			G.f.: A(x) = 1 + 3*x + 8*x^2 + 13*x^3 + 23*x^4 + 32*x^5 + 49*x^6 +...
log(A(x)) = 3*x + 7*x^2/2 - 6*x^3/3 + 15*x^4/4 + 3*x^5/5 - 14*x^6/6 + 3*x^7/7 + 31*x^8/8 - 6*x^9/9 +...+ A163659(2n)*x^n/n +...

Crossrefs

Cf. A163658, A163659, A171238, A002487.

Programs

  • PARI
    {A002487(n)=local(c=1, b=0); while(n>0, if(bitand(n, 1), b+=c, c+=b); n>>=1); b}
{A163659(n)=n*polcoeff(log(sum(k=0, n, A002487(k+1)*x^k)+x*O(x^n)), n)}
{a(n)=polcoeff(exp(sum(k=1, n, A163659(2*k)*x^k/k)+x*O(x^n)), n)}

Formula

G.f. satisfies: A(x) = A(x^2)*(1+x)*(1-x^3)^2/[(1-x)^2*(1+x^3)].
Showing 1-6 of 6 results.

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