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 A035382 Number of partitions of n into parts congruent to 1 mod 3. 23
 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 26, 29, 33, 38, 42, 48, 54, 61, 68, 77, 85, 96, 107, 119, 132, 148, 163, 181, 201, 223, 245, 272, 299, 330, 363, 400, 438, 483, 529, 580, 635, 697, 760, 832, 908, 992, 1081, 1180, 1283, 1399, 1521 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS a(n) = A116373(3*n). - Reinhard Zumkeller, Feb 15 2006 LINKS Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz) Joerg Arndt, Matters Computational (The Fxtbook), p. 350. FORMULA a(n) = 1/n*Sum_{k=1..n} A078181(k)*a(n-k), a(0) = 1. G.f.: 1/prod(j>=0, 1-x^(1+3*j) ). - Emeric Deutsch, Mar 30 2006 From Joerg Arndt, Oct 02 2012: (Start) G.f.: sum(n>=0, q^n/prod(k=1..n, 1-q^(3*k)) ); this is the special case of R=1, M=3 of the g.f. sum(n>=0, q^(R*n)/prod(k=1..n, 1-q^(M*k) ) ) for partitions into parts R mod M (where R!=0). G.f. sum(n>=0, q^(3*n^2-2*n) / prod(k=0..n-1, (1-q^(3*k+3))*(1-q^(3*k+1))) ); this is the special case of R=1, M=3 of the g.f. sum(n>=0, q^(M*n^2+(R-M)*n) / prod(k=0..n-1, (1-q^(M*k+M))*(1-q^(M*k+R))) ) for partitions into parts R mod M (where R!=0). (See Fxtbook link) (End) a(n) ~ Gamma(1/3) * exp(sqrt(2*n)*Pi/3) / (2*sqrt(3) * (2*Pi*n)^(2/3)) * (1 + (Pi/72 - 2/(3*Pi)) / sqrt(2*n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017 Euler transform of period 3 sequence [ 1, 0, 0, ...]. - Kevin T. Acres, Apr 28 2018 EXAMPLE a(3) = 1 because we have [1,1,1]; a(4) = 2 because we have [1,1,1,1] and [4]; a(9) = 4 because we have [7,1,1], [4,4,1], [4,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1]. 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ... MAPLE g:= 1/product(1-x^(1+3*j), j=0..50): gser:= series(g, x=0, 64): seq(coeff(gser, x, n), n=0..61); # Emeric Deutsch, Mar 30 2006 # second Maple program b:= proc(n, i) option remember; `if`(n=0, 1,       `if`(i<1, 0, b(n, i-3) +`if`(i>n, 0, b(n-i, i))))     end: a:= n-> b(n, 3*iquo(n, 3)+1): seq(a(n), n=0..100);  # Alois P. Heinz, Oct 03 2012 MATHEMATICA b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-3] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[n, 3*Quotient[n, 3]+1]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *) nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = -1; Do[If[Mod[k, 3] == 1, Do[poly[[j + 1]] -= poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly2 = Take[poly, {2, nmax + 1}]; poly3 = 1 + Sum[poly2[[n]]*x^n, {n, 1, Length[poly2]}]; CoefficientList[Series[1/poly3, {x, 0, Length[poly2]}], x] (* Vaclav Kotesovec, Jan 13 2017 *) nmax = 50; s = Range[0, nmax/3]*3 + 1; Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *) CROSSREFS Cf. A035386, A035451, A261612. Sequence in context: A029075 A029052 A131795 * A094988 A173911 A076269 Adjacent sequences:  A035379 A035380 A035381 * A035383 A035384 A035385 KEYWORD nonn AUTHOR STATUS approved

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Last modified July 23 14:33 EDT 2021. Contains 346259 sequences. (Running on oeis4.)