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 A238863 Number of partitions of n where the difference between consecutive parts is at most 3. 10
 1, 1, 2, 3, 5, 7, 10, 13, 18, 24, 32, 41, 54, 68, 87, 111, 139, 174, 218, 269, 333, 410, 501, 611, 745, 902, 1090, 1315, 1578, 1891, 2263, 2695, 3205, 3805, 4503, 5322, 6277, 7384, 8673, 10172, 11904, 13908, 16227, 18894, 21971, 25516, 29578, 34245, 39597, 45717, 52720, 60721, 69842, 80243, 92091, 105559, 120865, 138248 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most three times (by taking conjugates). The g.f. for "max difference d" is 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^((d+1)*i))/(1-q^i) ) ), see cross references. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(4*i))/(1-q^i) ) ). a(n) = Sum_{k=0..3} A238353(n,k). - Alois P. Heinz, Mar 09 2014 MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,       add(b(n-i*j, i-1), j=0..min(3, n/i))))     end: g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,       add(b(n-i*j, i-1), j=1..n/i)))     end: a:= n-> add(g(n, k), k=0..n): seq(a(n), n=0..60);  # Alois P. Heinz, Mar 09 2014 MATHEMATICA b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1], {j, 0, Min[3, n/i]}]]]; g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1], {j, 1, n/i}]]]; a[n_] := Sum[g[n, k], {k, 0, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *) PROG (PARI) N=66;  q = 'q + O('q^N); Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1, k-1, (1-q^(4*i))/(1-q^i) ) ) ) CROSSREFS Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1), A034296 (d=1), A224956 (d=2), this sequence, A238864 (d=4), A238865 (d=5), A238866 (d=6), A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity). Sequence in context: A332283 A088318 A038083 * A060688 A005691 A035954 Adjacent sequences:  A238860 A238861 A238862 * A238864 A238865 A238866 KEYWORD nonn AUTHOR Joerg Arndt, Mar 08 2014 STATUS approved

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Last modified July 28 15:08 EDT 2021. Contains 346335 sequences. (Running on oeis4.)