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 A334464 a(n) is the total number of parts in all partitions of n into consecutive parts that differ by 4. 7
 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 1, 6, 1, 3, 4, 7, 1, 6, 1, 7, 4, 3, 1, 10, 1, 3, 4, 7, 1, 6, 1, 7, 9, 3, 1, 10, 1, 8, 4, 7, 1, 6, 6, 7, 4, 3, 1, 15, 1, 3, 4, 7, 6, 12, 1, 7, 4, 8, 1, 16, 1, 3, 9, 7, 1, 12, 1, 12, 4, 3, 1, 16, 6, 3, 4, 7, 1, 17, 8, 7, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS The one-part partition n = n is included in the count. For the relation to hexagonal numbers see also A334462. LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 FORMULA G.f.: Sum_{n>=1} n*x^(n*(2*n-1))/(1-x^n). (For proof, see A330889. - N. J. A. Sloane, Nov 21 2020) EXAMPLE For n = 28 there are three partitions of 28 into consecutive parts that differ by 4, including 28 as a valid partition. They are [28], [16, 12] and [13, 9, 5, 1]. The number of parts of these partitions are 1, 2, 4 respectively. The total number of parts is 1 + 2 + 4 = 7, so a(28) = 7. MATHEMATICA nmax = 100; CoefficientList[Sum[n x^(n(2n-1)-1)/(1-x^n), {n, 1, nmax}]+O[x]^nmax, x] (* Jean-François Alcover, Nov 30 2020 *) PROG (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, k*x^(k*(2*k-1))/(1-x^k))) \\ Seiichi Manyama, Dec 04 2020 CROSSREFS Row sums of A334462. Column k=4 of A334466. Cf. A000384. Sequences of the same family whose consecutive parts differs by k are: A000203 (k=0), A204217 (k=1), A066839 (k=2), A330889 (k=3), this sequence (k=4), A334732 (k=5), A334949 (k=6). Cf. A334461. Sequence in context: A307193 A111742 A178220 * A316780 A338730 A104740 Adjacent sequences: A334461 A334462 A334463 * A334465 A334466 A334467 KEYWORD nonn AUTHOR Omar E. Pol, May 05 2020 STATUS approved

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Last modified August 4 01:12 EDT 2024. Contains 374905 sequences. (Running on oeis4.)