A132463 -id:A132463 - cp's OEIS frontend

A032766 Numbers that are congruent to 0 or 1 (mod 3).

0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 102, 103

Offset: 0

Patrick De Geest, May 15 1998

Omitting the initial 0, a(n) is the number of 1's in the n-th row of the triangle in A118111. - Hans Havermann, May 26 2002
Binomial transform is A053220. - Michael Somos, Jul 10 2003
Smallest number of different people in a set of n-1 photographs that satisfies the following conditions: In each photograph there are 3 women, the woman in the middle is the mother of the person on her left and is a sister of the person on her right and the women in the middle of the photographs are all different. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
Partial sums of A000034. - Richard Choulet, Jan 28 2010
Starting with 1 = row sums of triangle A171370. - Gary W. Adamson, Feb 15 2010
a(n) is the set of values for m in which 6k + m can be a perfect square (quadratic residues of 6 including trivial case of 0). - Gary Detlefs, Mar 19 2010
For n >= 2, a(n) is the smallest number with n as an anti-divisor. - Franklin T. Adams-Watters, Oct 28 2011
Sequence is also the maximum number of floors with 3 elevators and n stops in a "Convenient Building". See A196592 and Erich Friedman link below. - Robert Price, May 30 2013
a(n) is also the total number of coins left after packing 4-curves patterns (4c2) into a fountain of coins base n. The total number of 4c2 is A002620 and voids left is A000982. See illustration in links. - Kival Ngaokrajang, Oct 26 2013
Number of partitions of 6n into two even parts. - Wesley Ivan Hurt, Nov 15 2014
Number of partitions of 3n into exactly 2 parts. - Colin Barker, Mar 23 2015
Nonnegative m such that floor(2*m/3) = 2*floor(m/3). - Bruno Berselli, Dec 09 2015
For n >= 3, also the independence number of the n-web graph. - Eric W. Weisstein, Dec 31 2015
Equivalently, nonnegative numbers m for which m*(m+2)/3 and m*(m+5)/6 are integers. - Bruno Berselli, Jul 18 2016
Also the clique covering number of the n-Andrásfai graph for n > 0. - Eric W. Weisstein, Mar 26 2018
Maximum sum of degeneracies over all decompositions of the complete graph of order n+1 into three factors. The extremal decompositions are characterized in the Bickle link below. - Allan Bickle, Dec 21 2021
Also the Hadwiger number of the n-cocktail party graph. - Eric W. Weisstein, Apr 30 2022
The number of integer rectangles with a side of length n+1 and the property: the bisectors of the angles form a square within its limits. - Alexander M. Domashenko, Oct 17 2024
The maximum possible number of 5-cycles in an outerplanar graph on n+4 vertices. - Stephen Bartell, Jul 10 2025

Indranil Ghosh, Table of n, a(n) for n = 0..10000
Allan Bickle, Nordhaus-Gaddum Theorems for k-Decompositions, Congr. Num. 211 (2012) 171-183.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Erich Friedman, Problem of the month November 2009
Z. Füredi, A. Kostochka, M. Stiebitz, R. Skrekovski, and D. West, Nordhaus-Gaddum-type theorems for decompositions into many parts, J. Graph Theory 50 (2005), 273-292.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 282. [Book's website]
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest, Problem #20. [Broken link; Cached copy]
Matroids Matheplanet, Number of d-generator groups of order 2^(d+1) and exponent-p class 2 (in German).
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche, Vol. 76, No. 1 (2021), pp. 277-310, see p. 302.
Kival Ngaokrajang, Illustration of initial terms (U).
Eric Weisstein's World of Mathematics, Andrásfai Graph
Eric Weisstein's World of Mathematics, Clique Covering Number
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Hadwiger Number
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Web Graph
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A006578 (partial sums), A000034 (first differences), A016789 (complement).
Essentially the same: A049624.
Column 1 (the second leftmost) of triangular table A026374.
Column 1 (the leftmost) of square array A191450.
Row 1 of A254051.
Row sums of A171370.
Cf. A066272 for anti-divisors.
Cf. A253888 and A254049 (permutations of this sequence without the initial zero).
Cf. A254103 and A254104 (pair of permutations based on this sequence and its complement).
Cf. A000035, A000217, A000982, A001651, A002620, A002717, A003945.
Cf. A004396, A004526, A007310, A007494, A008619, A030308, A035360.
Cf. A047270, A049615, A052928, A053220, A070893, A084056, A092942.
Cf. A118111, A132463, A133083, A196592, A249745.

a032766 n = div n 2 + n  -- Reinhard Zumkeller, Dec 13 2014
(MIT/GNU Scheme) (define (A032766 n) (+ n (floor->exact (/ n 2)))) ;; Antti Karttunen, Jan 24 2015

&cat[ [n, n+1]: n in [0..100 by 3] ]; // Vincenzo Librandi, Nov 16 2014

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..69); # Zerinvary Lajos, Mar 16 2008
seq(floor(n/2)+n, n=0..69); # Gary Detlefs, Mar 19 2010
select(n->member(n mod 3,{0,1}), [$0..103]); # Peter Luschny, Apr 06 2014

a[n_] := a[n] = 2a[n - 1] - 2a[n - 3] + a[n - 4]; a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 4; Array[a, 60, 0] (* Robert G. Wilson v, Mar 28 2011 *)
Select[Range[0, 200], MemberQ[{0, 1}, Mod[#, 3]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
Flatten[{#,#+1}&/@(3Range[0,40])] (* or *) LinearRecurrence[{1,1,-1}, {0,1,3}, 100] (* or *) With[{nn=110}, Complement[Range[0,nn], Range[2,nn,3]]] (* Harvey P. Dale, Mar 10 2013 *)
CoefficientList[Series[x (1 + 2 x) / ((1 - x) (1 - x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 16 2014 *)
Floor[3 Range[0, 69]/2] (* L. Edson Jeffery, Jan 14 2017 *)
Drop[Range[0,110],{3,-1,3}] (* Harvey P. Dale, Sep 02 2023 *)

PARI
```
{a(n) = n + n\2}
```

concat(0, Vec(x*(1+2*x)/((1-x)*(1-x^2)) + O(x^100))) \\ Altug Alkan, Dec 09 2015

[int(3*n//2) for n in range(101)] # G. C. Greubel, Jun 23 2024

G.f.: x*(1+2*x)/((1-x)*(1-x^2)).
a(-n) = -A007494(n).
a(n) = A049615(n, 2), for n > 2.
From Paul Barry, Sep 04 2003: (Start)
a(n) = (6n - 1 + (-1)^n)/4.
a(n) = floor((3n + 2)/2) - 1 = A001651(n) - 1.
a(n) = sqrt(2) * sqrt( (6n-1) (-1)^n + 18n^2 - 6n + 1 )/4.
a(n) = Sum_{k=0..n} 3/2 - 2*0^k + (-1)^k/2. (End)
a(n) = 3*floor(n/2) + (n mod 2) = A007494(n) - A000035(n). - Reinhard Zumkeller, Apr 04 2005
a(n) = 2 * A004526(n) + A004526(n+1). - Philippe Deléham, Aug 07 2006
a(n) = 1 + ceiling(3*(n-1)/2). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
Row sums of triangle A133083. - Gary W. Adamson, Sep 08 2007
a(n) = (cos(Pi*n) - 1)/4 + 3*n/2. - Bart Snapp (snapp(AT)coastal.edu), Sep 18 2008
A004396(a(n)) = n. - Reinhard Zumkeller, Oct 30 2009
a(n) = floor(n/2) + n. - Gary Detlefs, Mar 19 2010
a(n) = 3n - a(n-1) - 2, for n>0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
a(n) = n + (n-1) - (n-2) + (n-3) - ... 1 = A052928(n) + A008619(n-1). - Jaroslav Krizek, Mar 22 2011
a(n) = a(n-1) + a(n-2) - a(n-3). - Robert G. Wilson v, Mar 28 2011
a(n) = Sum_{k>=0} A030308(n,k) * A003945(k). - Philippe Deléham, Oct 17 2011
a(n) = 2n - ceiling(n/2). - Wesley Ivan Hurt, Oct 25 2013
a(n) = A000217(n) - 2 * A002620(n-1). - Kival Ngaokrajang, Oct 26 2013
a(n) = Sum_{i=1..n} gcd(i, 2). - Wesley Ivan Hurt, Jan 23 2014
a(n) = 2n + floor((-n - (n mod 2))/2). - Wesley Ivan Hurt, Mar 31 2014
A092942(a(n)) = n for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n) = floor(3*n/2). - L. Edson Jeffery, Jan 18 2015
a(n) = A254049(A249745(n)) = (1+A007310(n)) / 2 for n >= 1. - Antti Karttunen, Jan 24 2015
E.g.f.: (3*x*exp(x) - sinh(x))/2. - Ilya Gutkovskiy, Jul 18 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) + log(3)/2. - Amiram Eldar, Dec 04 2021

Better description from N. J. A. Sloane, Aug 01 1998

A003105 Schur's 1926 partition theorem: number of partitions of n into parts 6n+1 or 6n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23, 26, 30, 34, 38, 42, 47, 53, 60, 67, 74, 82, 91, 102, 114, 126, 139, 153, 169, 187, 207, 228, 250, 274, 301, 331, 364, 399, 436, 476, 520, 569, 622, 679, 739, 804, 875, 953, 1038, 1128, 1224, 1327

Offset: 0

N. J. A. Sloane, Herman P. Robinson

There are many (at least 8) equivalent definitions of this sequence (besides the comments below, see also Schur, Alladi, Andrews). - N. J. A. Sloane, Jun 17 2011
Coefficients of replicable function number 72e. - N. J. A. Sloane, Jun 10 2015
Also number of partitions of n into odd parts in which no part appears more than twice, cf. A070048 and A096938. - Vladeta Jovovic, Jan 18 2005
Also number of partitions of n into distinct parts congruent to 1 or 2 modulo 3. (Follows from second g.f.) - N. Sato, Jul 20 2005
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution of A262928 and A261612. - Vaclav Kotesovec, Jan 13 2017
Convolution of A109702 and A109701. - Vaclav Kotesovec, Jan 21 2017

			G.f: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...
T72e = 1/q + q^11 + q^23 + q^35 + q^47 + 2*q^59 + 2*q^71 + 3*q^83 + ...
The logarithm of the g.f. begins:
log(A(x)) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + x^6/6 + 8*x^7/7 + x^8/8 + x^9/9 + 6*x^10/10 + 12*x^11/11 + x^12/12 + ... + A186099(n)*x^n/n + ... . - _Paul D. Hanna_, Feb 17 2013

K. Alladi, Refinements of Rogers-Ramanujan type identities. In Special Functions, q-Series and Related Topics (Toronto, ON, 1995), 1-35, Fields Inst. Commun., 14, Amer. Math. Soc., Providence, RI, 1997.
G. E. Andrews, Schur's theorem, partitions with odd parts and the Al-Salam-Carlitz polynomials. In q-Series From a Contemporary Perspective (South Hadley, MA, 1998), 45-56, Contemp. Math., 254, Amer. Math. Soc., Providence, RI, 2000.
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
I. Schur, Zur Additiven Zahlentheorie, Ges. Abh., Vol. 2, Springer, pp. 43-50.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 201 terms from R. Zumkeller)
K. Alladi and B. Gordon, Generalizations of Schur's partition theorem, Manuscr. Math. 79 (1993) 113-126.
K. Alladi and B. Gordon, Schur's partition theorem, companions, refinements and generalizations, Trans. Amer. Math. Soc. 347 (1995) 1591-1608.
G. E. Andrews, K. Alladi, B. Gordon, Generalizations and refinements of a partition theorem of Göllnitz, J. Reine Angew. Math. 460 (1995) 165-188.
D. M. Bressoud, A combinatorial proof of Schur's 1926 partition theorem, Proc. Amer, Math. Soc. 79 (1980) 338-340.
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. Vol. 225 (1967), 154-190.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12.
Padmavathamma, R. Raghavendra and B. M. Chandrashekara, A new bijective proof of a partition theorem of K. Alladi, Discrete Math., 237 (2004), 125-128.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Schur's Partition Theorem
James J. Y. Zhao, A Bijective Proof of the Alladi-Andrews-Gordon Partition Theorem, Elect. J. Combin, Volume 22, Issue 1 (2015) Paper #P1.68.
Index entries for McKay-Thompson series for Monster simple group

Cf. A000041, A001651, A003114, A000726, A109389, A109697, A132462, A132463, A285219, A304047.

Cf. A186099 (log).

a003105 n = p 1 n where
   p k m | m == 0 = 1 | m < k = 0 | otherwise = q k (m-k) + p (k+2) m
   q k m | m == 0 = 1 | m < k = 0 | otherwise = p (k+2) (m-k) + p (k+2) m
-- Reinhard Zumkeller, Nov 12 2011

with(combinat);
A:=proc(n) local i, j, t3, t2, t1;
    t2:=0;
    t1:=firstpart(n);
    for j from 1 to numbpart(n)+2 do
        t3:=1;
        for i from 1 to nops(t1) do
            if (t1[i] mod 6) <> 1 and (t1[i] mod 6) <> 5 then t3:=0; fi;
        od;
        if t3=1 then t2:=t2+1; fi;
        if nops(t1) = 1 then RETURN(t2); fi;
        t1:=nextpart(t1);
    od;
end;
# brute-force Maple program from N. J. A. Sloane, Jun 17 2011

max = 63; f[x_] := 1/Product[1 - x^k + x^(2k), {k, 0, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 01 2011, after Vladeta Jovovic *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] / QPochhammer[ -x^3, x^3], {x, 0, n}]; (* Michael Somos, Jul 05 2014 *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 3] != 0, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 13 2017 *)
nmax = 63; kmax = nmax/6;
s = Flatten[{Range[0, kmax]*6 + 1}~Join~{Range[kmax]*6 - 1}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Jan 09 2005 */

{S(n,x)=sumdiv(n,d,d*(1-x^d)^(n/d))}
{a(n)=polcoeff(exp(sum(k=1,n,S(k,x)*x^k/k)+x*O(x^n)),n)}
for(n=0,60,print1(a(n),", "))
/* Paul D. Hanna, Feb 17 2013 */

G.f.: 1/Product_{k>=0} (1-x^(6*k+1))*(1-x^(6*k+5)) = Product_{k>=0} (1+x^(3*k+1))*(1+x^(3*k+2)) = 1/Product_{k>=0} (1-x^k+x^(2*k)). - Vladeta Jovovic, Jun 08 2003
Expansion of chi(-x^3) / chi(-x) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Mar 04 2012
Expansion of f(x, x^2) / f(-x^3) = f(-x^6) / f(-x, -x^5) in powers of x where f() is Ramanujan theta function. - Michael Somos, Jul 05 2014
Expansion of q^(1/12) * eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Jan 09 2005
Euler transform of period 6 sequence [1, 0, 0, 0, 1, 0, ...]. - Michael Somos, Jan 09 2005
Given g.f. A(x), then B(q) = (A(q^12) / q)^4 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v^4 + (1 - u^3) * v^3 + 6*u^2*v^2 + (u^4 - u)*v + u^3. - Michael Somos, Jan 09 2005
The logarithmic derivative equals A186099. - Paul D. Hanna, Feb 17 2013
G.f.: exp( Sum_{n>=1} A186099(n) * x^n/n ) where A186099(n) = sum of divisors of n congruent to 1 or 5 mod 6. - Paul D. Hanna, Feb 17 2013
G.f.: exp( Sum_{n>=1} S(n,x) * x^n/n ) where S(n,x) = Sum_{d|n} d*(1-x^d)^(n/d). - Paul D. Hanna, Feb 17 2013
a(n) ~ Pi*sqrt(2) / sqrt(3*(12*n-1)) * BesselI(1, Pi*sqrt(12*n-1) / (3*sqrt(6))) ~ exp(Pi*sqrt(2*n)/3) / (2^(5/4) * sqrt(3) * n^(3/4)) * (1 - (9/(8*Pi) + Pi/36)/sqrt(2*n) + (5 - 135/(4*Pi^2) + Pi^2/81)/(64*n)). - Vaclav Kotesovec, Aug 23 2015, extended Jan 09 2017
a(n) = (1/n)*Sum_{k=1..n} A186099(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017

More terms from Vladeta Jovovic, Jun 08 2003

A132462 Number of partitions of n into distinct parts congruent to 0 or 2 modulo 3.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 1, 3, 2, 2, 5, 2, 4, 7, 4, 7, 10, 6, 11, 14, 9, 17, 19, 14, 25, 26, 21, 36, 35, 31, 50, 47, 45, 69, 63, 64, 93, 84, 89, 125, 111, 124, 165, 147, 169, 216, 194, 227, 281, 254, 303, 363, 332, 400, 466, 432, 523, 595, 559, 680, 756, 721, 876, 956, 926, 1121

Offset: 0

Reinhard Zumkeller, Aug 22 2007

nonn

			a(8)=3 because we have 8, 6+2 and 5+3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..200

Cf. A007494, A035361, A003105, A132463.

g:=product((1+x^(3*k))*(1+x^(3*k-1)),k=1..30): gser:=series(g,x=0,100): seq(coeff(gser,x,n),n=0..70); # Emeric Deutsch, Aug 30 2007

nmax = 40; CoefficientList[Series[Product[((1+x^(3*k))*(1+x^(3*k-1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2015 *)

G.f.: Product_{k>=1} (1+x^(3*k))*(1+x^(3*k-1)). - Emeric Deutsch, Aug 30 2007

a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(23/12) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Aug 24 2015

a(0)=1 prepended by Vaclav Kotesovec, Aug 24 2015

A035360 Number of partitions of n into parts 3k or 3k+1.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 7, 8, 11, 15, 17, 23, 30, 35, 44, 57, 66, 82, 103, 121, 146, 181, 211, 253, 308, 360, 425, 513, 596, 700, 834, 969, 1127, 1333, 1541, 1786, 2093, 2415, 2781, 3241, 3723, 4273, 4946, 5669, 6476, 7461, 8519, 9705, 11123, 12669, 14379, 16418

Offset: 0

Olivier Gérard

nonn

Euler transform of period 3 sequence [ 1, 0, 1, ...]. - Kevin T. Acres, Apr 28 2018

			1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 8*x^8 + 11*x^9 + ...

Robert Price, Table of n, a(n) for n = 0..1000

Cf. A032766, A132463, A000726, A035361.

nmax = 50; CoefficientList[Series[Product[1/((1 - x^(3*k))*(1 - x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 23 2015 *)
nmax = 52; kmax = nmax/3;
s = Flatten[{Range[0, kmax]*3}~Join~{Range[0, kmax]*3 + 1}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

a(n) ~ Gamma(1/3) * exp(2*Pi*sqrt(n)/3) / (4 * sqrt(3) * Pi^(2/3) * n^(11/12)). - Vaclav Kotesovec, Aug 23 2015

A374064 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-1)).

Original entry on oeis.org

1, 0, -1, 0, 1, -1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -2, 2, 1, -3, 1, 3, -3, 0, 3, -3, -1, 4, -3, -1, 5, -3, -3, 7, -3, -5, 7, -1, -7, 8, 0, -8, 8, 1, -11, 10, 3, -14, 9, 8, -17, 8, 10, -18, 6, 14, -22, 6, 19, -24, 1, 26, -26, -3, 30, -25, -9, 37, -27, -13, 42, -26, -23, 51, -25, -31, 56

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

sign

Cf. A035386, A081360, A081362, A109389, A132463, A261612, A262928, A284315, A374065.

nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

a(0) = 1; a(n) = -Sum_{k=1..n} A262928(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132463(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A261612(n-k).

A301504 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-3)).

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 1, 0, 1, 3, 2, 0, 2, 5, 4, 1, 2, 7, 7, 2, 3, 10, 11, 4, 4, 14, 17, 8, 6, 19, 25, 13, 8, 25, 36, 21, 12, 33, 50, 33, 18, 43, 69, 49, 26, 56, 93, 71, 38, 72, 124, 102, 55, 92, 163, 142, 79, 118, 212, 195, 112, 151, 273, 265, 157, 193, 350, 354, 217, 246, 444

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 0 or 1 mod 4.

			a(9) = 3 because we have [9], [8, 1] and [5, 4].

Index entries for sequences related to partitions

Cf. A035362, A035451, A035457, A042948, A108932, A132463, A261734, A301505, A301507, A301508.

nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k)) (1 + x^(4 k - 3)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[x^3 QPochhammer[-1, x^4] QPochhammer[-x^(-3), x^4]/(2 (1 + x) (1 - x + x^2)), {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 1}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A042948(k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

A374058 Expansion of Product_{k>=1} (1 - x^(3k-2)) (1 - x^(3*k)).

Original entry on oeis.org

1, -1, 0, -1, 0, 1, -1, 1, 0, 0, 1, 0, -1, 1, -1, 1, 0, -1, 0, 0, -1, 1, 0, -1, 1, -1, 0, 1, -1, 0, 1, -1, 1, 1, -1, 0, 0, -1, 2, 0, -1, 1, 0, -1, 2, -2, 0, 1, -1, 0, 1, -1, 0, 1, -2, 1, 1, -2, 1, 0, -2, 2, 0, -2, 2, -1, 0, 2, -1, -1, 1, -1, -1, 3, -2, 0, 2, -2, 1, 2, -3, 1, 1, -2, 2, 1

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

sign

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A010815, A035360, A035386, A081362, A082051, A132463, A137569, A274719, A284312, A374060.

nmax = 85; CoefficientList[Series[Product[(1 - x^(3 k - 2)) (1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Plus @@ Select[Divisors[k], Mod[#, 3] != 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A082051(k) * a(n-k).
a(0) = 1; a(n) = -Sum_{k=1..n} A035360(k) * a(n-k).
a(n) = Sum_{k=0..n} A010815(k) * A035386(n-k).

cp's OEIS Frontend

A032766 Numbers that are congruent to 0 or 1 (mod 3).

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A003105 Schur's 1926 partition theorem: number of partitions of n into parts 6n+1 or 6n-1.

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A132462 Number of partitions of n into distinct parts congruent to 0 or 2 modulo 3.

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A035360 Number of partitions of n into parts 3k or 3k+1.

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A374064 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-1)).

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A301504 Expansion of Product_{k>=1} (1 + x^(4*k))*(1 + x^(4*k-3)).

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A374058 Expansion of Product_{k>=1} (1 - x^(3*k-2)) * (1 - x^(3*k)).

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A301504 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-3)).

A374058 Expansion of Product_{k>=1} (1 - x^(3k-2)) (1 - x^(3*k)).