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 A330889 a(n) is the total number of parts in all partitions of n into consecutive parts that differ by 3. 12
 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 1, 6, 5, 3, 4, 3, 5, 6, 1, 3, 8, 3, 1, 6, 5, 8, 4, 3, 5, 6, 6, 3, 8, 3, 1, 11, 5, 3, 4, 3, 10, 12, 1, 3, 8, 8, 1, 12, 5, 3, 9, 3, 5, 12, 1, 8, 8, 3, 1, 12, 17, 3, 4, 3, 5, 17, 1, 10, 8, 3, 6, 12, 5, 3, 11, 8, 5, 12, 1, 3, 13 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS The one-part partition n = n is included in the count. For the relation to pentagonal numbers see also A330888. LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 FORMULA Conjecture: G.f.: Sum_{n>=1} n*x^(n*(3*n-1)/2)/(1-x^n). Proof from Matthew C. Russell, Nov 21 2020: The summation variable n runs over the number of parts in the partition. For fixed n, the smallest such partition is: 1 + 4 + 7 + ... + (3n-2). The above sum is equal to n * (3*n-1) / 2. That's where the x^(n*(3*n-1)/2) factor comes from. Then we want to (add 1 to every part), (add 2 to every part), etc. to get 2 + 5 + 8 + ..., 3 + 6 + 9 + ..., which corresponds to adding n, 2*n, 3*n, etc. to the base partition. So we divide by (1 - x^n). Multiply by n (to count the total number of parts) and we are done. QED EXAMPLE For n = 21 there are three partitions of 21 into consecutive parts that differ by 3, including 21 as a partition. They are , [12, 9] and [10, 7, 4]. The number of parts of these partitions are 1, 2 and 3 respectively. The total number of parts is 1 + 2 + 3 = 6, so a(27) = 6. MAPLE A330889 := proc(n)     local a;     a := 0 ;     for k from 1 do         if n>= A000325(k) then             a := a+A330888(n, k);         else             return a;         end if;     end do: end proc: # R. J. Mathar, Oct 02 2020 MATHEMATICA nmax = 100; CoefficientList[Sum[n x^(n(3n-1)/2-1)/(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*(3*k-1)/2)/(1-x^k))) \\ Seiichi Manyama, Dec 04 2020 CROSSREFS Row sums of A330888. Column k=3 of A334466. Sequences of the same family whose consecutive parts differs by k are: A000203 (k=0), A204217 (k=1), A066839 (k=2), this sequence (k=3), A334464 (k=4), A334732 (k=5), A334949 (k=6). Cf. A338730. Sequence in context: A338730 A104740 A111736 * A035652 A205526 A233269 Adjacent sequences:  A330886 A330887 A330888 * A330890 A330891 A330892 KEYWORD nonn,easy AUTHOR Omar E. Pol, Apr 30 2020 EXTENSIONS More terms from R. J. Mathar, Oct 02 2020 STATUS approved

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Last modified August 3 08:21 EDT 2021. Contains 346435 sequences. (Running on oeis4.)