A002175 Excess of number of divisors of 12n+1 of form 4k+1 over those of form 4k+3. (Formerly M0416 N0159)

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

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number of ways to write n as an ordered sum of 2 generalized pentagonal numbers. - Ilya Gutkovskiy, Aug 14 2017

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

John Cerkan, Table of n, a(n) for n = 0..10000

J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (phi(-x^3) / chi(-x))^2 in powers of x where phi(), chi() are Ramanujan theta functions.

Expansion of q^(-1/12) * (eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)))^2 in powers of q. - Michael Somos, Sep 19 2005

Euler transform of period 6 sequence [ 2, 0, -2, 0, 2, -2, ...]. - Michael Somos, Sep 19 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258279. - Michael Somos, May 25 2015

From Michael Somos, Jun 02 2012: (Start)

a(n) = A008441(3*n) = A121363(3*n) = A122865(4*n) = A122856(8*n).

a(n) = A116604(6*n) = A125079(6*n) = A129447(6*n) = A138741(6*n).

a(n) = A(12*n+1) where A = A002654, A008442, A035154, A035181, A035184,

A112301, A113406, A113652, A121450, A122864, A125061, A129448, A132004,

A134013, A134015, A138746, A138950, A138952, A163746. (End)

Contribution from Michael Somos, May 25 2015: (Start)

a(n) = A258277(4*n) = A258278(8*n) = A258291(3*n).

a(n) = - A258210(12*n + 1) = A258228(12*n + 1) = A258256(12*n + 1).

2*a(n) = A258279(12*n + 1) = - A258292(12*n + 1). (End)

G.f.: (Sum_{k=-inf..inf} x^(k*(3*k-1)/2))^2. - Ilya Gutkovskiy, Aug 14 2017

Example

G.f. = 1 + 2*x + 3*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ...
G.f. = q + 2*q^13 + 3*q^25 + 2*q^37 + q^49 + 2*q^61 + 2*q^73 + 4*q^85 + 2*q^97 + ...

Mathematica

ed[n_]:=Module[{divs=Divisors[12n+1]}, Count[divs, _?(Mod[#, 4] == 1&)]- Count[divs, _?(Mod[#, 4]==3&)]]; Array[ed, 110, 0] (* Harvey P. Dale, Jul 01 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = 12 n + 1}, Sum[ KroneckerSymbol[ 4, d], {d, Divisors[m]}]]]; (* Michael Somos, Apr 23 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3]^2 / (QPochhammer[ x] QPochhammer[ x^6]))^2, {x, 0, n}]; (* Michael Somos, Apr 23 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] / QPochhammer[ x, x^2])^2, {x, 0, n}]; (* Michael Somos, May 25 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, n = 12*n + 1; sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Sep 19 2005 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)))^2, n))}; /* Michael Somos, Jun 02 2012 */

Crossrefs

Cf. A002654, A008441, A008442, A035154, A035181, A035184, A112301, A113406.

Cf. A113652, A116604, A121363, A121450, A122856, A122864, A122865, A125061.

Cf. A125079, A129447, A129448, A132004, A134013, A134015, A138741, A138746.

Cf. A138950, A138952, A163746, A258210, A258228, A258256, A258279, A258292.

Sequence in context: A265847 A260342 A281939 * A170823 A068073 A324504

Adjacent sequences: A002172 A002173 A002174 * A002176 A002177 A002178

Keyword

nonn

Author

N. J. A. Sloane

Status

approved