A096981 -id:A096981 - cp's OEIS frontend

A047273 Numbers that are congruent to {0, 1, 3, 5} mod 6.

0, 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 60, 61, 63, 65, 66, 67, 69, 71, 72, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 96, 97, 99, 101, 102, 103

Offset: 1

N. J. A. Sloane

Complement of A047235. - Reinhard Zumkeller, Oct 01 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A004524, A007310, A047228, A047235, A047241, A047261, A093719, A096981.
First differences of A281026.
See A301729 for an essentially identical sequence.

a047273 n = a047273_list !! (n-1)
a047273_list = 0 : 1 : 3 : 5 : map (+ 6) a047273_list
-- Reinhard Zumkeller, Feb 19 2013

[(6*n-6+(-1)^(n div 2)+(-1)^(-n div 2))/4: n in [1..100]]; // Wesley Ivan Hurt, May 20 2016

seq(2*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/4, n = 0..69); # Gary Detlefs, Mar 19 2010

LinearRecurrence[{2,-2,2,-1},{0,1,3,5},80] (* Harvey P. Dale, Jan 04 2015 *)

PARI
```
a(n)=n+(n+1)\4+(n+2)\4
```

[(lucas_number1(n+2, 0, 1)+3*n)/2 for n in range(0, 70)] # Zerinvary Lajos, Mar 09 2009

G.f.: x*(1+x+x^2)/((1-x)^2*(1+x^2)) = x*(1-x^2)*(1-x^3)/((1-x)^3*(1-x^4)).
a(n) = n + A004524(n+1) = -a(-n) for all n in Z.
Starting (1, 3, 5, ...) = partial sums of (1, 2, 2, 1, 1, 2, 2, 1, 1, ...). - Gary W. Adamson, Jun 19 2008
A093719(a(n)) = 1. - Reinhard Zumkeller, Oct 01 2008
a(n) = 2*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/4, with offset 0..a(0)=0. - Gary Detlefs, Mar 19 2010
a(n) = (3*n-3+cos(Pi*n/2))/2. - R. J. Mathar, Oct 08 2010
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>4.
a(n) = (6*n-6+(-1)^(n/2)+(-1)^(-n/2))/4. (End)
Euler transform of length 4 sequence [3, -1, -1, 1]. - Michael Somos, Jun 24 2017
Sum_{n>=2} (-1)^n/a(n) = log(2)/3 + log(3)/2. - Amiram Eldar, Dec 16 2021
E.g.f.: (2 + 3*exp(x)*(x - 1) + cos(x))/2. - Stefano Spezia, Jul 26 2024

A098884 Number of partitions of n into distinct parts in which each part is congruent to 1 or 5 mod 6.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 2, 1, 0, 1, 2, 3, 3, 2, 1, 1, 3, 5, 5, 3, 1, 2, 5, 7, 7, 5, 3, 3, 7, 11, 11, 7, 4, 6, 11, 15, 15, 11, 7, 8, 15, 22, 22, 15, 10, 13, 22, 30, 30, 23, 16, 18, 30, 42, 42, 31, 22, 27, 43, 56, 56, 44, 33, 37, 57, 77, 77, 59, 45, 53, 79, 101, 101, 82, 64, 71

Offset: 0

Noureddine Chair, Oct 14 2004

nonn

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

Convolution of A281244 and A280456. - Vaclav Kotesovec, Jan 18 2017

			E.g. a(25)=5 because 25=19+5+1=17+7+1=13+7+5=13+11+1.
G.f. = 1 + x + x^5 + x^6 + x^7 + x^8 + x^11 + 2*x^12 + 2*x^13 + x^14 + x^16 + ...
G.f. = q + q^13 + q^61 + q^73 + q^85 + q^97 + q^133 + 2*q^145 + 2*q^157 + q^169 + ...

Reinhard Zumkeller and Vaclav Kotesovec, Table of n, a(n) for n = 0..20000 (terms 0..300 from Reinhard Zumkeller)
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A109389, A227398.

Cf. A007310, A056970, A096981, A097451.

a098884 = p a007310_list where
   p _  0     = 1
   p (k:ks) m = if k > m then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Feb 19 2013

series(product((1+x^(6*k-1))*(1+x^(6*k-5)),k=1..100),x=0,100);

a[ n_] := SeriesCoefficient[ Product[ 1 - (-x)^k + x^(2 k), {k, n}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^k + x^(2 k), {k, 1, n, 2}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}] / Product[ 1 + x^k, {k, 3, n, 6}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 6}] Product[ 1 + x^k, {k, 5, n, 6}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^6] QPochhammer[ -x^5, x^6], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ -x^3, x^6], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)

{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^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2), n))}; /* Michael Somos, Jun 26 2005 */

{a(n) = my(A, m); if( n<0, 0, A = x * O(x^n); m = sqrtint(3*n + 1); polcoeff( sum(k= -((m-1)\3), (m+1)\3, x^(k * (3*k - 2)), A) / eta(x^6 + A), n))}; /* Michael Somos, Sep 20 2013 */

Expansion of chi(x) / chi(x^3) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 20 2013
Expansion of f(x^1, x^5) / f(-x^6) in powers of x where f(,) is a Ramanujan theta function. - Michael Somos, Sep 20 2013
Expansion of G(x^6) * H(-x) + x * G(-x) * H(x^6) where G() (A003114), H() (A003106) are Rogers-Ramanujan functions.
Expansion of q^(-1/12) * eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)^2) in powers of q.
Euler transform of period 12 sequence [ 1, -1, 0, 0, 1, 0, 1, 0, 0, -1, 1, 0, ...]. - Michael Somos, Jun 26 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227398. - Michael Somos, Sep 20 2013
G.f.: Product_{k>0} (1 - (-x)^k + x^(2*k)).
G.f.: 1 / Product_{k>0} (1 - x^(2*k - 1) + x^(4*k - 2)).
G.f.: 1 / Product_{k>0} ((1 + x^(6*k - 3)) / (1 + x^(2*k - 1))).
G.f.: Product_{k>0} ((1 + x^(6*k - 1)) * (1 + x^(6*k - 5))).
G.f.: 1 / Product_{k>0} (1  + (-x)^(3*k - 1)) * (1 + (-x)^(3*k - 2)).
G.f.: (Sum_{k in Z} x^(k * (3*k - 2))) / (Sum_{k in Z} (-1)^k * x^(3*k * (3*k-1))).
A109389(n) = (-1)^n * a(n). Convolution inverse of A227398.
a(n) ~ exp(sqrt(n)*Pi/3)/ (2*sqrt(6)*n^(3/4)) * (1 + (Pi/72 - 9/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 30 2015, extended Jan 18 2017

Typo in Maple program fixed by Vaclav Kotesovec, Nov 15 2016

A097451 Number of partitions of n into parts congruent to {2, 3, 4} mod 6.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 5, 4, 7, 6, 11, 9, 15, 14, 22, 20, 31, 29, 43, 41, 58, 57, 80, 78, 106, 107, 142, 143, 188, 191, 247, 253, 321, 332, 418, 432, 537, 561, 690, 721, 880, 924, 1118, 1178, 1412, 1493, 1781, 1884, 2231, 2370, 2789, 2965, 3472, 3698, 4309, 4596

Offset: 0

Vladeta Jovovic, Aug 23 2004

Number of partitions of n in which no part is 1, no part appears more than twice and no two parts differ by 1. Example: a(6)=3 because we have [6],[4,2] and [3,3]. - Emeric Deutsch, Feb 16 2006

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

			a(8)=5 because we have [8],[44],[422],[332] and [2222].
G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 4*x^9 + ...
G.f. = q^7 + q^55 + q^79 + 2*q^103 + q^127 + 3*q^151 + 2*q^175 + 5*q^199 + ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, Exercise 7.9.

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A070047.

Cf. A047228, A056970, A096981, A098884.

a097451 n = p a047228_list n where
   p _  0         = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 16 2012

g:=1/product((1-x^(2+6*j))*(1-x^(3+6*j))*(1-x^(4+6*j)),j=0..15): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=0..67); # Emeric Deutsch, Feb 16 2006

a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - Boole[ OddQ[ Quotient[ k + 1, 3]]] x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Sep 24 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^3] QPochhammer[ x^6] / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Sep 24 2013 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - ( (k+1)\3 % 2) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Sep 24 2013 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 24 2013 */

Euler transform of period 6 sequence [ 0, 1, 1, 1, 0, 0, ...].
G.f.: 1/Product_{j>=0} ((1-x^(2+6j))(1-x^(3+6j))(1-x^(4+6j))). - Emeric Deutsch, Feb 16 2006
Expansion of psi(x^3) / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions. - Michael Somos, Sep 24 2013
Expansion of q^(-7/24) * eta(q^6)^2 / (eta(q^2) * eta(q^3)) in powers of q. - Michael Somos, Sep 24 2013
a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(3/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
Expansion of f(-x, -x^5) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 06 2015

More terms from Emeric Deutsch, Feb 16 2006

A056970 Number of partitions of n into distinct parts congruent to 2, 4 or 5 mod 6.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 11, 13, 13, 15, 16, 17, 20, 21, 23, 25, 27, 30, 33, 36, 38, 42, 45, 49, 54, 57, 62, 67, 72, 79, 85, 92, 98, 106, 114, 123, 133, 141, 152, 163, 175, 189, 202, 216, 231, 248, 265, 284, 304, 323

Offset: 0

Eric W. Weisstein

Also number of partitions of n into parts equal to 2,5, or 11 mod 12 (Gollnitz's theorem). Example: a(18)=4 because we have [14,2,2], [11,5,2], [5,5,2,2,2,2] and [2,2,2,2,2,2,2,2,2]. - Emeric Deutsch, Apr 18 2006

			a(18)=4 because we have [16,2], [14,4], [11,5,2] and [10,8].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
K. Alladi, Going beyond the partition theorem of Goellnitz
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 101.
G. E. Andrews, K. Alladi, and B. Gordon, Generalizations and refinements of a partition theorem of Göllnitz, Journal für die reine und angewandte Mathematik (1995), Volume: 460, page 165-188.
H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. Vol. 225 (1967), 154-190.
Eric Weisstein's World of Mathematics, Göllnitz's Theorem.

Cf. A047261, A096981, A097451, A098884.

a056970 n = p a047261_list n where
   p _  0     = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Nov 16 2012

g:=product((1+x^(2+6*j))*(1+x^(4+6*j))*(1+x^(5+6*j)),j=0..30): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..67); # Emeric Deutsch, Apr 18 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(irem(d, 12) in [2, 5, 11], d, 0)
      , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Oct 27 2015

max = 70; g[x_] := Product[(1+x^(2+6j))(1+x^(4+6j))(1+x^(5+6j)), {j, 0, Floor[max/6]}]; CoefficientList[ Series[g[x], {x, 0, max}], x](* Jean-François Alcover, Nov 16 2011, after Emeric Deutsch *)
a[n_] := a[n] = If[n==0, 1, Sum[Sum[If[MatchQ[Mod[d, 12], 2|5|11], d, 0], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Dec 23 2015, after Alois P. Heinz *)

{a(n)= if(n<0, 0, polcoeff( 1/prod(k=1, n, 1-(k%3==2)*(k%12!=8)*x^k, 1+x*O(x^n)), n))} /* Michael Somos, Jul 24 2007 */

From Emeric Deutsch, Apr 18 2006: (Start)
G.f.: Product_{j >= 0} (1+x^(2+6j))(1+x^(4+6j))(1+x^(5+6j)).
G.f.: 1/Product_{j >= 0} (1-x^(2+12j))(1-x^(5+12j))(1-x^(11+12j)).
(End)
Euler transform of period 12 sequence [ 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, ...]. - Michael Somos, Jul 24 2007
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(25/12) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 30 2015

A279479 Expansion of f(-x, -x^5) / f(-x^24)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 2, -2, 0, 0, 0, -2, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, -2, 0, 0, 5, -5, 0, 0, 0, -5, 0, 0, 6, -2, 0, 0, 0, 0, 0, 0, 7, -1, 0, 0, 0, -5, 0, 0, 10, -10, 0, 0, 0, -10, 0, 0, 12

Offset: 0

Michael Somos, Dec 12 2016

sign

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

			G.f. = 1 - x - x^5 + x^8 + x^16 - x^21 + 2*x^24 - 2*x^25 - 2*x^29 + ...
G.f. = q^-5 - q^-2 - q^10 + q^19 + q^43 - q^58 + 2*q^67 - 2*q^70 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A096981, A279476.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6] / QPochhammer[ x^24]^2 , {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2 x^(3/2) QPochhammer[ x, x^2] QPochhammer[ -x^3, x^6] / EllipticTheta[ 2, 0, x^6] , {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^24 + A)^2), n))};

Expansion of chi(-x) * chi(x^3) / psi(x^12) in  powers of x where chi(), psi() are Ramanujan theta functions.
Euler transform of period 24 sequence [ -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 1, ...].
G.f. is a period 1 Fourier series that satisfies f(-1 / (864 t)) = 3^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A279476.
a(4*n) = a(4*n + 1) = a(8*n + 2) = 0. a(8*n) = A096981(n).

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A047273 Numbers that are congruent to {0, 1, 3, 5} mod 6.

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A098884 Number of partitions of n into distinct parts in which each part is congruent to 1 or 5 mod 6.

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A097451 Number of partitions of n into parts congruent to {2, 3, 4} mod 6.

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A056970 Number of partitions of n into distinct parts congruent to 2, 4 or 5 mod 6.

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A279479 Expansion of f(-x, -x^5) / f(-x^24)^2 in powers of x where f(, ) is Ramanujan's general theta function.

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