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.

A161805 A trisection of A161804: a(n) = A161804(3n) for n>=0.

Original entry on oeis.org

1, 12, 66, 255, 903, 2970, 8571, 23001, 59763, 145164, 333693, 751059, 1623009, 3363576, 6872307, 13677228, 26351985, 50309910, 94392525, 172538934, 313558506, 563064207, 988996095, 1730456433, 3001805067, 5106353439
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2009

Keywords

Comments

G.f. of A161804 is exp( Sum_{n>=1} A002129(n) * 3*A038500(n) * q^n/n ),
where A002129 forms the l.g.f. of log[ Sum_{n>=0} x^(n(n+1)/2) ], and
A038500(n) is the highest power of 3 dividing n.

Examples

			G.f.: T_0(q) = 1 + 12*q + 66*q^2 + 255*q^3 + 903*q^4 + 2970*q^5 +...

Crossrefs

Cf. A161804, other trisections: A161806 (T_1), A161807 (T_2).

Programs

  • PARI
    {a(n)=local(L=sum(m=1, 3*n,3*3^valuation(m,3)*sumdiv(m, d, -(-1)^d*d)*x^m/m)+x*O(x^(3*n))); polcoeff(exp(L), 3*n)}

A161806 A trisection of A161804: a(n) = A161804(3n+1) for n>=0.

Original entry on oeis.org

3, 30, 141, 513, 1815, 5727, 15882, 42417, 108165, 255831, 585258, 1302966, 2762349, 5705829, 11577633, 22708053, 43675938, 83011398, 153929484, 281210994, 509494515, 905832642, 1591395774, 2778237765, 4776943011
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2009

Keywords

Comments

G.f. of A161804 is exp( Sum_{n>=1} A002129(n) * 3*A038500(n) * q^n/n ),
where A002129 forms the l.g.f. of log[ Sum_{n>=0} x^(n(n+1)/2) ], and
A038500(n) is the highest power of 3 dividing n.

Examples

			G.f.: T_1(q) = 3 + 30*q + 141*q^2 + 513*q^3 + 1815*q^4 + 5727*q^5 +...
Terms are divisible by 3:
A/3=[1,10,47,171,605,1909,5294,14139,36055,85277,195086,434322,...].

Crossrefs

Cf. A161804, other trisections: A161805 (T_0), A161807 (T_2).

Programs

  • PARI
    {a(n)=local(L=sum(m=1, 3*n+1,3*3^valuation(m,3)*sumdiv(m, d, -(-1)^d*d)*x^m/m)+x*O(x^(3*n+1))); polcoeff(exp(L), 3*n+1)}

A161807 A trisection of A161804: a(n) = A161804(3n+2) for n>=0.

Original entry on oeis.org

3, 27, 111, 378, 1356, 4131, 10881, 29106, 73500, 167643, 382053, 849339, 1754061, 3605094, 7330311, 14094945, 26980563, 51481332, 93965784, 170910270, 311155296, 545970024, 955201653, 1676274750, 2849709768, 4831999623
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2009

Keywords

Comments

G.f. of A161804 is exp( Sum_{n>=1} A002129(n) * 3*A038500(n) * q^n/n ),
where A002129 forms the l.g.f. of log[ Sum_{n>=0} x^(n(n+1)/2) ], and
A038500(n) is the highest power of 3 dividing n.

Examples

			G.f.: T_2(q) = 3 + 27*q + 111*q^2 + 378*q^3 + 1356*q^4 + 4131*q^5 +...
Terms are divisible by 3:
A/3=[1,9,37,126,452,1377,3627,9702,24500,55881,127351,283113,...].

Crossrefs

Cf. A161804, other trisections: A161805 (T_0), A161806 (T_1).

Programs

  • PARI
    {a(n)=local(L=sum(m=1, 3*n+2,3*3^valuation(m,3)*sumdiv(m, d, -(-1)^d*d)*x^m/m)+x*O(x^(3*n+2))); polcoeff(exp(L), 3*n+2)}

A161808 G.f.: A(q) = exp( Sum_{n>=1} A162552(n) * 3*A038500(n) * q^n/n ).

Original entry on oeis.org

1, 3, 3, 3, 9, 12, 12, 27, 36, 57, 141, 165, 135, 321, 450, 399, 780, 1068, 1308, 2913, 3537, 2736, 5940, 8430, 7173, 13251, 18267, 17661, 35007, 45051, 31866, 58506, 85890, 65694, 102000, 145293, 101547, 140574, 203781, 114765, 93051, 161754
Offset: 0

Views

Author

Paul D. Hanna, Jul 21 2009

Keywords

Comments

A162552 forms the l.g.f. of log[ Sum_{n>=0} x^(n^2) ], and
A038500(n) is the highest power of 3 dividing n.
The first negative term is a(43) = -162729.

Examples

			G.f.: A(q) = 1 + 3*q + 3*q^2 + 3*q^3 + 9*q^4 + 12*q^5 + 12*q^6 +...
log(A(q)) = 3*q - 3*q^2/2 + 9*q^3/3 + 9*q^4/4 - 12*q^5/5 + 45*q^6/6 - 18*q^7/7 +...
Compare to: q - q^2/2 + q^3/3 + 3*q^4/4 - 4*q^5/5 + 5*q^6/6 - 6*q^7/7 +...
which equals log( Sum_{n>=0} q^(n^2) ) as described by A162552.

Links

Crossrefs

Cf. A161804 (variant).

Programs

  • PARI
    {a(n)=local(Q=sum(m=0,n,x^(m^2))+x*O(x^n),A); A=exp(sum(k=1,n,polcoeff(log(Q),k)*3*3^valuation(k,3)*x^k)+x*O(x^n));polcoeff(A,n)}
Showing 1-4 of 4 results.

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