A080995 -id:A080995 - cp's OEIS frontend

A098151 Number of partitions of 2*n with no part divisible by 3 and all odd parts occurring with even multiplicities.

1, 2, 4, 6, 10, 16, 24, 36, 52, 74, 104, 144, 198, 268, 360, 480, 634, 832, 1084, 1404, 1808, 2316, 2952, 3744, 4728, 5946, 7448, 9294, 11556, 14320, 17688, 21780, 26740, 32736, 39968, 48672, 59122, 71644, 86616, 104484, 125768, 151072, 181104, 216684

Offset: 0

Noureddine Chair, Aug 29 2004

There are no partitions of 2n+1 in which all odd parts occur with even multiplicity. - Michael Somos, Apr 15 2012
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is also the number of Schur overpartitions of n, i.e., the number of overpartitions of n where adjacent parts differ by at least 3 if the smaller is overlined or divisible by 3 and adjacent parts differ by at least 6 if the smaller is overlined and divisible by 3. - Jeremy Lovejoy, Mar 23 2015
Let A(q) denote the g.f. of this sequence. Let m be a nonzero integer. The simple continued fraction expansions of the real numbers A(1/(2*m)) and A(1/(2*m+1)) may be predictable. For a given positive integer n, the sequence of the n-th partial denominators of the continued fractions are conjecturally polynomial or quasi-polynomial in m for m sufficiently large. An example is given below. Cf. A080054. - Peter Bala, Jun 09 2025

			a(4)=10 because 8 = 4+4 = 4+2+2=2+2+2+2 = 2+2+2+1+1 = 2+2+1+1+1+1 = 4+2+1+1 = 4+1+1+1+1 = 2+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1.
G.f. = 1 + 2*q + 4*q^2 + 6*q^3 + 10*q^4 + 16*q^5 + 24*q^6 + 36*q^7 + 52*q^8 + ...
From _Peter Bala_, Jun 09 2025: (Start)
G.f.: A(q) = f(q, q^2) / f(-q, -q^2).
Simple continued fraction expansions of A(1/(2*m)):
m =  2  [1;  1   9  1    5    8    45   4  1  2  1  1  1  3  3   2  2 ...]
m =  3  [1;  2  13  1   14   12   133   8  1  1  7  2  1  2  2   1  1 ...]
m =  4  [1;  3  17  1   27   16   297  12  2  2  1  1  1  2  2   2  2 ...]
m =  5  [1;  4  21  1   44   20   561  16  2  1  7  3  3  2  2  25  8 ...]
m =  6  [1;  5  25  1   65   24   949  20  3  2  1  1  1  3  4   2  1 ...]
m =  7  [1;  6  29  1   90   28  1485  24  3  1  7  4  5  2  1   1  6 ...]
m =  8  [1;  7  33  1  119   32  2193  28  4  2  1  1  1  4  6   2  1 ...]
m =  9  [1;  8  37  1  152   36  3097  32  4  1  7  5  7  2  1   1  3 ...]
m = 10  [1;  9  41  1  189   40  4221  36  5  2  1  1  1  5  8   2  1 ...]
...
The sequence of the 4th partial denominators [5, 14, 27, 44, ...] appears to be given by the polynomial (2*m + 1)*(m - 1) for m >= 2.
The sequence of the 6th partial denominators [45, 133, 297, 561, ...] appears to be given by the polynomial (2*m + 1)*(2*m^2 + 1) for m >= 2. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 14-16.
Byungchan Kim and Eunmi Kim, Partitions weighted by the number of two types of parts, Bull. Korean Math. Soc. (2024) Vol. 61, No. 6, 1677-1684. See p. 1679.
Jeremy Lovejoy, A theorem on seven-colored overpartitions and its applications, Int. J. Number Theory. 1 (2005) 215-224
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S24.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A015128, A001318, A080054, A080995, A010815, A103260, A215594.

series(product((1+x^k+x^(2*k))/(1-x^k+x^(2*k)),k=1..150),x=0,100);
# alternative program using expansion of f(q, q^2) / f(-q, -q^2):
with(gfun): series( add(x^(n*(3*n-1)/2),n = -8..8)/add((-1)^n*x^(n*(3*n-1)/2), n = -8..8), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^3]^2 / (QPochhammer[ q]^2 QPochhammer[ q^6]), {q, 0, n}] (* Michael Somos, Oct 23 2013 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(3*k-1)) * (1+x^(3*k-2)) / ((1-x^(3*k-1)) * (1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))} /* Michael Somos, Dec 04 2004 */

Expansion of phi(-q^3) / phi(-q) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Apr 15 2012
Expansion of f(q, q^2) / f(-q, -q^2) in powers of q where f(,) is the Ramanujan two-variable theta function. - Michael Somos, Apr 15 2012
Expansion of eta(q^2) * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q.
G.f. = (Sum_{n = -oo..oo} (-1)^n*q^(3*n^2)) / (Sum_{n = -oo..oo} (-1)^n*q^(n^2)). - N. J. A. Sloane, Aug 09 2016
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u^2) * (u^2 + v^4) - 4 * u^2*v^4. - Michael Somos, Apr 15 2012
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v + 3 * u*v^2 - 3 * u^2*v^3. - Michael Somos, Dec 04 2004
Euler transform of period 6 sequence [2, 1, 0, 1, 2, 0, ...]. - Vladeta Jovovic, Sep 24 2004
Taylor series of product_{k=1..inf}(1+x^k+x^(2*k))/(1-x^k+x^(2*k))= product_{k=1..inf}(1+x^k)(1-x^(3k))/((1-x^k)(1+x^(3k)))=Theta_4(0, x^3)/theta_4(0, x)
a(n) ~ Pi * BesselI(1, Pi*sqrt(2*n/3)) / (3*sqrt(2*n)) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/4) * 3^(3/4) * n^(3/4)) * (1 - 3*sqrt(3)/(8*Pi*sqrt(2*n)) - 45/(256*Pi^2*n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 09 2017
Convolution of A000726 and A003105. - R. J. Mathar, Nov 17 2017
From Peter Bala, Jun 09 2025: (Start)
G.f.: A(q) = Sum_{n = -oo..oo} q^(n*(3*n+1)/2) / Sum_{n = -oo..oo} (-1)^n * q^(n*(3*n+1)/2).
Recurrences:
a(n) - a(n-1) - a(n-2) + a(n-5) + a(n-7) - a(n-12) - a(n-15) + + - - ... = f(n), where [0, 1, 2, 5, 7, 12, 15, ...] is the sequence of generalized pentagonal numbers A001318, a(n) is set equal to 0 for negative n and f(n) = 1 if n is a generalized pentagonal number, otherwise f(n) = 0 (see A080995). Compare with the recurrence for the partition function p(n) = A000041(n).
a(n) - 2*a(n-1) + 2*a(n-4) - 2*a(n-9) + 2*a(n-16) - 2*a(n-25) + - ... = g(n), where g(n) = 2*(-1)^k if n is of the form 3*(k^2), otherwise g(n) = 0. (End)

A214263 Expansion of f(x^1, x^7) in powers of x where f() is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

Michael Somos and Omar E. Pol, Jul 09 2012

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

Characteristic function of A074377: a(n) = 1 if and only if n is in A074377.

			G.f. = 1 + x + x^7 + x^10 + x^22 + x^27 + x^45 + x^52 + x^76 + x^85 + x^115 + ...
G.f. = q^9 + q^25 + q^121 + q^169 + q^361 + q^441 + q^729 + q^841 + q^1225 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for characteristic functions.

Cf. A047621, A074377, A115359.
A000122, A080995, A010054, A133100, A089801 have g.f. of f(x,x^k) for k=1..5.
Cf. A000122, A000700, A121373.

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; Table[SeriesCoefficient[f[q, q^7], {q, 0, n}], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)

PARI
```
{a(n) = issquare(16*n + 9)};
```

Euler transform of period 16 sequence [ 1, -1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, -1, 1, -1, ...].
G.f.: f(x, x^7) = sum_{k in Z} x^(4*k^2 - 3*k).
a(n) = A010054(2*n + 1) = A115359(2*n).
Sum_{k=1..n} a(k) ~ sqrt(n). - Amiram Eldar, Jan 13 2024

A133100 Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Sep 11 2007

nonn

			G.f. = 1 + x + x^4 + x^7 + x^13 + x^18 + x^27 + x^34 + x^46 + x^55 + x^70 + ...
G.f. = q^9 + q^49 + q^169 + q^289 + q^529 + q^729 + q^1089 + q^1369 + q^1849 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for characteristic functions.

Cf. f(x, x^k) for k=2..11: A080995, A010054, A133100, A089801, A274179, A214263, A281814, A205988, A281815, A186742.

Cf. A085787, A113429.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^5] QPochhammer[ -x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)
a[ n_] := SquaresR[ 1, 40 n + 9] / 2; (* Michael Somos, Jan 30 2017 *)
a[ n_] := If[n < 0, 0, Boole @ IntegerQ @ Sqrt @ (40 n + 9)]; (* Michael Somos, Jan 30 2017 *)

{a(n) = if( n<0, 0, polcoeff( prod( k=1,n, 1 + x^k*[-1, 1, 0, 0, 1][k%5 + 1], 1 + x * O(x^n)), n))};

PARI
```
{a(n) = issquare( 40*n + 9)};
```

f(x,x^m) = 1 + Sum_{k=1..oo} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - N. J. A. Sloane, Jan 30 2017
The characteristic function of A085787 generalized heptagonal numbers.
Euler transform of period 10 sequence [1, -1, 0, 1, -1, 1, 0, -1, 1, -1, ...].
G.f.: Prod_{k>0} (1 - x^(5*k)) * (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)) = Sum_{k in Z} x^((5*k^2 + 3*k) / 2).
a(n) = |A113429(n)|. a(3*n + 2) = 0.
Sum_{k=1..n} a(k) ~ 2 * sqrt(2/5) * sqrt(n). - Amiram Eldar, Jan 13 2024

A143062 Expansion of false theta series variation of Euler's pentagonal number series in powers of x.

Original entry on oeis.org

1, -1, 1, 0, 0, -1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jul 21 2008

a(n) = sum over all partitions of n into distinct parts of number of partitions with even largest part minus number with odd largest part.

In the Berndt reference replace {a -> 1, q -> x} in equation (3.1) to get g.f. Replace {a -> x, q -> x} to get f(x). G.f. is 1 - f(x) * x / (1 + x).

			a(5) = -1 +1 -1 = -1 since 5 = 4 + 1 = 3 + 2. a(7) = -1 +1 -1 +1 +1 = 1 since 7 = 6 + 1 = 5 + 2 = 4 + 3 = 4 + 2 + 1.
G.f. = 1 - x + x^2 - x^5 + x^7 - x^12 + x^15 - x^22 + x^26 - x^35 + x^40 + ...
G.f. = q - q^25 + q^49 - q^121 + q^169 - q^289 + q^361 - q^529 + q^625 - q^841 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See Section 9.4, pp. 232-236.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, see p. 41, 10th equation numerator.

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.
I. Pak, On Fine's partition theorems, Dyson, Andrews and missed opportunities, Math. Intelligencer, 25 (No. 1, 2003), 10-16.

Cf. A010815, A026837, A026838, A080995, A121373, A199918, A203568, A268539.

a[ n_] := If[ SquaresR[ 1, 24 n + 1] == 2, (-1)^Quotient[ Sqrt[24 n + 1], 3], 0];
a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, (-1)^Quotient[ m, 3], 0]]; (* Michael Somos, Nov 18 2015 *)

{a(n) = if( issquare( 24*n + 1, &n), (-1)^(n \ 3) )};

a(n) = b(24*n + 1) where b() is multiplicative with b(p^(2*e)) = (-1)^e if p = 5 (mod 6), b(p^(2*e)) = +1 if p = 1 (mod 6) and b(p^(2*e-1)) = b(2^e) = b(3^e) = 0 if e>0.
G.f.: Sum_{k>=0} x^((3*k^2 + k) / 2) * (1 - x^(2*k + 1)) = 1 - Sum_{k>0} x^((3*k^2 - k) / 2) * (1 - x^k).
G.f.: 1 - x / (1 + x) + x^3 / ((1 + x) * (1 + x^2)) - x^6 / ((1 + x) * (1 + x^2) * (1 + x^3)) + ...
G.f.: 1 - x / (1 + x^2) + x^2 / ((1 + x^2) * (1 + x^4)) - x^3 / ((1 + x^2 ) * (1 + x^4) * (1 + x^6)) + ...
|a(n)| = |A010815(n)| = |A080995(n)| = |A199918(n)| = |A121373(n)|.
From Joerg Arndt, Jun 24 2013: (Start)
a(n) = A026838(n) - A026837(n) (Fine's theorem), see the Pak reference.
a(n)=1 if n = k(3k+1)/2, a(n)=-1 if n = k(3k-1)/2, a(n)=0 otherwise.
G.f.: Sum_{n >= 0} (-q)^n * (Product_{k = 1..n-1} 1 + q^k). (End)
a(n) = - A203568(n) unless n=0. a(0) = 1. - Michael Somos, Jul 12 2015
From Peter Bala, Feb 04 2021: (Start)
A conjectural g.f: 1 + Sum_{n >= 0} (-1)^n*x^(2*n-1)/Product_{k = 1..n} 1 + x^(2*k-1).
G.f.: 1 - Sum_{n >= 1} x^(n*(2*n-1))/Product_{k = 1..2*n} 1 + x^k [added Dec 19 2024: see Berndt et al., Entry 9.44]. (End)
Conjectural g.f.: (1/(1 + x)) * (2 - Sum_{n >= 0} (-1)^n * x^(3*n)/Product_{k = 1..n} 1 + x^(2*k)). - Peter Bala, Jan 19 2025

A193832 Irregular triangle read by rows in which row n lists 2n-1 copies of 2n-1 and n copies of 2n, for n >= 1.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14

Offset: 1

Omar E. Pol, Aug 22 2011

Sequence of successive positive integers k in which if k is odd then k appears k times, otherwise if k is even then k appears k/2 times.
Note that an arrangement of the blocks of this sequence shows the growth of the generalized pentagonal numbers A001318 (see example).
The sums of each block give the positive integers of A129194: 1, 2, 9, 8, 25, 18, 49,...
Partial sums of A080995. - Paolo P. Lava, Aug 23 2011.
Concatenations of rows of triangles A001650 and A111650; also, seen as a flat list, the row lengths of triangle A260672 and the first differences of its row sums (cf. A260706). - Reinhard Zumkeller, Nov 17 2015
Also a(n) = number of squares in the arithmetic progression {24k + 1: 0 <= k <= n-1} [Granville]. - N. J. A. Sloane, Dec 13 2017

			a) If written as a triangle the initial rows are
  1, 2,
  3, 3, 3, 4, 4,
  5, 5, 5, 5, 5, 6, 6, 6,
  7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8,
  9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10,
  ...
Row sums give A126587.
b) An application using the blocks of this sequence: the illustration of the growth of an arrangement which represents the generalized pentagonal numbers A001318. For example; the first 9 positive initial terms: 1, 2, 5, 7, 12, 15, 22, 26, 35.
.
.         9
.       8 9
.     8 7 9
.   8 6 7 9
. 8 6 5 7 9
. 6 4 5 7 9
. 4 3 5 7 9
. 2 3 5 7 9
. 1 3 5 7 9
...

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Andrew Granville, Squares in arithmetic progressions and infinitely many primes, arXiv:1708.06951 [math.NT], 2017.
Andrew Granville, Squares in arithmetic progressions and infinitely many primes, The American Mathematical Monthly, 124.10 (2017): 951-954. See p. 952.

Cf. A001318, A080995, A126587, A129194, A221671, A221672.

Cf. A001650, A111650, A260672, A260706.

a193832 n k = a193832_tabf !! (n-1) !! (k-1)
a193832_row n = a193832_tabf !! (n-1)
a193832_tabf = zipWith (++) a001650_tabf a111650_tabl
a193832' n = a193832_list !! (n - 1)
a193832_list = concat a193832_tabf
-- Reinhard Zumkeller, Nov 15 2015

Array[Join @@ MapIndexed[ConstantArray[#, #/(1 + Boole[First@ #2 == 2])] &, {2 # - 1, 2 #}] &, 7] // Flatten (* or *)
Table[If[k <= 2 n - 1, 2 n - 1, 2 n], {n, 7}, {k, 3 n - 1}] // Flatten (* Michael De Vlieger, Dec 14 2017 *)

a(n) = sqrt(8n/3) plus or minus 1 [Granville] - N. J. A. Sloane, Dec 13 2017

If 8 <= n <= 52, then a(n-1) < a(n) if and only if n is in A221672. - Jonathan Sondow, Dec 14 2017

Edited by N. J. A. Sloane, Dec 13 2017

A089806 Expansion of Jacobi theta function (theta_3(q^(1/3))-theta_2(q^3))/2/q^(1/12).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Eric W. Weisstein, Nov 12 2003

			1 + q^2 + q^4 + q^10 + q^14 + q^24 + q^30 + q^44 + q^52 + ...

Antti Karttunen, Table of n, a(n) for n = 0..10034
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.
Index entries for characteristic functions.

Cf. A080995(n) = a(2n).

Cf. A020832.

nmax = 100; CoefficientList[Series[Product[(1+x^(2*k)) * (1-x^(6*k)) / (1+x^(6*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 05 2016 *)
Table[If[IntegerQ[Sqrt[12*n + 1]], 1, 0], {n, 0, 100}] (* Vaclav Kotesovec, Dec 29 2023 *)

a(n)=issquare(12*n+1) /* Michael Somos, Apr 13 2005 */

lista(nn) = {q='q+O('q^nn); Vec(eta(q^4)*eta(q^6)^2/(eta(q^2)*eta(q^12)))} \\ Altug Alkan, Mar 22 2018

Euler transform of period 12 sequence [0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, ...]. - Michael Somos, Apr 13 2005
a(n) = b(12n+1) where b(n) is multiplicative and b(3^e)=0^e, b(p^e)=(1+(-1)^e)/2 if p<>3. - Michael Somos, Jun 06 2005
Expansion of q^(-1/12)(eta(q^4)eta(q^6)^2)/(eta(q^2)eta(q^12)) in powers of q.
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 2/sqrt(3) = 1.1547005... (10 * A020832). - Amiram Eldar, Dec 29 2023

A033683 a(n) = 1 if n is an odd square not divisible by 3, otherwise 0.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

			G.f. = x + x^25 + x^49 + x^121 + x^169 + x^289 + x^361 + x^529 + x^625 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 105, Eq. (41).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for characteristic functions.

Cf. A098108, A033684.

Cf. A010052, A010872, A104777.

a033683 n = fromEnum $ odd n && mod n 3 > 0 && a010052 n == 1
-- Reinhard Zumkeller, Nov 14 2015

Basis( ModularForms( Gamma0(144), 1/2), 106)[2]; /* Michael Somos, Dec 07 2019 */

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x^4] - EllipticTheta[ 2, 0, x^36])/2, {x, 0, n}] // PowerExpand; (* Michael Somos, Dec 07 2019 *)
Table[If[OddQ[n]&&IntegerQ[Sqrt[n]]&&Mod[n,3]!=0,1,0],{n,0,120}] (* Harvey P. Dale, Sep 06 2020 *)

{a(n) = if( n%24 == 1, issquare(n), 0)}; /* Michael Somos, Jan 26 2008 */

Essentially the series psi_6(z)=(1/2)(theta_2(z/9)-theta_2(z)).
a(A104777(n)) = 1.
A080995(n) = a(24n+1).
Multiplicative with a(p^e) = 1 if 2 divides e and p > 3, 0 otherwise. - Mitch Harris, Jun 09 2005
Euler transform of a period 144 sequence. - Michael Somos, Jan 26 2008
a(n) = A033684(n) * A000035(n).
Dirichlet g.f.: zeta(2*s) *(1-2^(-2s)) *(1-3^(-2s)). - R. J. Mathar, Mar 10 2011
G.f.: Sum_{k in Z} x^(6*k+1)^2. - Michael Somos, Dec 07 2019
Sum_{k=1..n} a(k) ~ sqrt(n)/3. - Amiram Eldar, Jan 14 2024

A089810 Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q.

Original entry on oeis.org

1, 1, 0, 0, -1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0

Offset: 0

Eric W. Weisstein, Nov 12 2003

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^5, b = -x. - Michael Somos, Jul 12 2012
Convolution square is A258279. - Michael Somos, May 25 2015
Number 8 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 + q - q^4 - 2*q^9 - q^16 + q^25 + 2*q^36 + q^49 - q^64 - 2*q^81 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Quintuple Product Identity.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.

Cf. A002448, A080995, A089801, A089802, A089807, A089812, A204843, A204853, A258279.

Cf. A000700, A000122, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/6, q], {q, 0, n}]; (* Michael Somos, Nov 14 2011 *)
a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 4, 0, q^9] - EllipticTheta[ 4, 0, q])  /2, {q, 0, n}]; (* Michael Somos, Nov 14 2011 *)
QP = QPochhammer; s = QP[q^2]^2*(QP[q^3] / (QP[q]*QP[q^6])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *)

{a(n) = my(x); if( n<1, n==0, issquare(n, &x) * (1 + (n%3==0)) * (-1)^((1 + x) \ 3))}; /* Michael Somos, Nov 05 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Jan 26 2008 */

Expansion of Jacobi theta function (3theta_4(q^9) - theta_4(q)) / 2 in powers of q.
a(n) is multiplicative with  a(0)=1, a(2^e) = -(1 + (-1)^e)/2, if e>0, a(3^e) = -2(1 + (-1)^e)/2 if e>0, a(p^e) = (1 + (-1)^e)/2 otherwise.
From Michael Somos, Nov 05 2005: (Start)
Euler transform of period 6 sequence [ 1, -1, 0, -1, 1, -1, ...].
G.f.: (Sum_{k in Z} 3 * (-x)^((3*k)^2) - (-x)^(k^2)) / 2 = Product_{k>0} (1 - x^(2*k)) / ((1 - x^(6*k - 1)) * (1 - x^(6*k-5))).
Expansion of eta(q^2)^2 * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. (End)
Expansion of psi(q) * chi(-q^3) in powers of q where psi(), chi() are Ramanujan theta functions. - Michael Somos, Sep 16 2007
Expansion of (3 * phi(-q^9) - phi(-q)) / 2 in powers of q where phi() is a Ramanujan theta function.
From Michael Somos, Sep 17 2007: (Start)
Expansion of Jacobi theta function theta_3(Pi/6, q) in powers of q.
Expansion of f(x*w, x/w) in powers of x where w is a primitive sixth root of unity and f() is Ramanujan's two-variable theta function. (End)
From Michael Somos, Jan 26 2008: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 72^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A080995.
G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^k + x^(2*k)). (End)
a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 3) = a(8*n + 5) = a(9*n + 3) = a(9*n + 6) = 0. a(3*n + 1) = A089802(n). a(4*n) = A089807(n). a(9*n) = A002448(n).
a(n) = (floor(sqrt(n))-floor(sqrt(n-1)))*(abs(2-4*sin((floor(sqrt(n))+1)*Pi/3)^2) - 4*sin((floor(sqrt(n))+2)*Pi/3)^2)*(-1)^floor(floor(sqrt(n)-1)/3). - Mikael Aaltonen, Jan 17 2015
From Michael Somos, May 25 2015: (Start)
a(n) = (-1)^n * A089807(n) = A204843(4*n) = A204853(4*n).
a(8*n + 1) = A089812(n). a(12*n + 4) = - A089801(n). (End)
Sum_{k=1..n} abs(a(k)) ~ (4/3)*sqrt(n). - Amiram Eldar, Jan 27 2024

A235963 n appears (n+1)/(1 + (n mod 2)) times.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13

Offset: 0

Mircea Merca, Jan 17 2014

n appears A001318(n+1) - A001318(n) = A026741(n+1) times.
Sum_{k=0...a(n)} (-1)^ceiling(k/2)*p(n-G(k)) = 0 for n>0, where p(n)=A000041(n) is the partition function, and G(k)=A001318(k) denotes the generalized pentagonal numbers.
Row lengths of A238442, n >= 1. - Omar E. Pol, Dec 22 2016

			As a triangle:
  0;
  1;
  2, 2, 2;
  3, 3;
  4, 4, 4, 4, 4;
  5, 5, 5;
  6, 6, 6, 6, 6, 6, 6;
  7, 7, 7, 7;
  8, 8, 8, 8, 8, 8, 8, 8, 8;
  ...

G. C. Greubel, Table of n, a(n) for n = 0..5000

First differences are A080995.

Cf. A000041, A001318, A026741, A085141, A180447, A238442.

T:= n-> n$(n+1)/(n mod 2+1):
seq(T(n), n=0..13);  # Alois P. Heinz, Nov 23 2024

Table[Table[n, {(n + 1)/(1 + Mod[n, 2])}], {n, 0, 14}]//Flatten (* T. D. Noe, Jan 29 2014 *)

from math import isqrt
def A235963(n): return (m:=isqrt((n+1<<3)//3))-(n+1<=(m*(3*m+4)+1 if m&1 else m*(3*m+2))>>3) # Chai Wah Wu, Nov 23 2024

A235963=lambda n: ((s:=isqrt(24*n+1))+1)//6+(s-1)//6 # Natalia L. Skirrow, May 13 2025

Let t = (sqrt(n*8/3 + 1) - 1)/2 + 1/3 and let k = floor(t); then a(n) = 2k if t - k < 2/3, 2k+1 otherwise. - Jon E. Schoenfield, Jun 13 2017
a(n) = m if n+1>A001318(m) and a(n) = m-1 otherwise where m = floor(sqrt(8(n+1)/3)). - Chai Wah Wu, Nov 23 2024
From Natalia L. Skirrow, May 13 2025: (Start)
a(n) = A180447(n) + A085141(n).
a(n) = floor((s+1)/6) + floor((s-1)/6) where s=floor(sqrt(24*n+1)).
G.f.: (f(x,x^2)-1)/(1-x), where f is Ramanujan's bivariate theta function. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8 + (2-sqrt(2))*log(2)/8 + log(2+sqrt(2))/(2*sqrt(2)). - Amiram Eldar, May 28 2025

A089807 Expansion of Jacobi theta function (3theta_3(q^9)-theta_3(q))/2.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 0

Eric W. Weisstein, Nov 12 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^5, b = x. - Michael Somos, Jul 12 2012
Number 11 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 - q - q^4 + 2*q^9 - q^16 - q^25 + 2*q^36 - q^49 - q^64 + 2*q^81 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Quintuple Product Identity.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.

Related to the 14 primitive eta-products which are holomorphic modular forms of weight 1/2: A000122, A002448, A010054, A010815, A080995, A089801, A089802, this sequence, A089810, A089812, A106459, A121373, A133985, A133988. - Seiichi Manyama, May 15 2017

Cf. A000700, A000122, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/3, q], {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 3, 0, q^9] - EllipticTheta[ 3, 0, q])/2, {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3, q^6] EllipticTheta[ 2, 0, Sqrt[ -q]] / (2 (-q)^(1/8)), {q, 0, n}] (* Michael Somos, Jul 12 2012 *);

{a(n) = if( n<1, n==0, issquare(n) * (3*(n%3==0) - 1))}; /* Michael Somos, Nov 05 2005 */

a(n) = -b(n) where b() is multiplicative with b(3^e) = -2(1 + (-1)^e) / 2 if e>0, b(p^e) = (1 + (-1)^e) / 2 otherwise.
From Michael Somos, Nov 05 2005: (Start)
Expansion of eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, -1, ...].
G.f.: (Sum_{k in Z} 3 * x^((3*k)^2) - x^(k^2)) / 2 = Product_{k>0} (1 - x^k) / ((1 - x^(12*k - 2)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 10))). (End)
Expansion of Jacobi theta function theta_3(Pi/3, q) in powers of q. - Michael Somos, Jan 26 2006
Expansion of chi(q^3) * psi(-q) in powers of q where chi(), psi() are Ramanujan theta functions. - Michael Somos, May 19 2007
Expansion of f(x*w, x/w) in powers of x where w is a primitive cube root of unity and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 17 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089801.
a(n) = (-1)^n * A089810(n). - Michael Somos, Jan 20 2012
For n > 0, a(n) = (floor(sqrt(n))-floor(sqrt(n-1)))*(2-4*sin(floor(sqrt(n))*Pi/3)^2). - Mikael Aaltonen, Jan 17 2015
Sum_{k=1..n} abs(a(k)) ~ (4/3)*sqrt(n). - Amiram Eldar, Jan 27 2024

cp's OEIS Frontend

A098151 Number of partitions of 2*n with no part divisible by 3 and all odd parts occurring with even multiplicities.

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A214263 Expansion of f(x^1, x^7) in powers of x where f() is Ramanujan's general theta function.

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A133100 Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

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A143062 Expansion of false theta series variation of Euler's pentagonal number series in powers of x.

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A193832 Irregular triangle read by rows in which row n lists 2n-1 copies of 2n-1 and n copies of 2n, for n >= 1.

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A089806 Expansion of Jacobi theta function (theta_3(q^(1/3))-theta_2(q^3))/2/q^(1/12).

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A033683 a(n) = 1 if n is an odd square not divisible by 3, otherwise 0.

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