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A138137
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First differences of A006128.
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79
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1, 2, 3, 6, 8, 15, 19, 32, 42, 64, 83, 124, 157, 224, 288, 395, 502, 679, 854, 1132, 1422, 1847, 2307, 2968, 3677, 4671, 5772, 7251, 8908, 11110, 13572, 16792, 20439, 25096, 30414, 37138, 44798, 54389, 65386, 78959, 94558, 113687, 135646, 162375, 193133
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OFFSET
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1,2
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COMMENTS
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Number of parts in the last section of the set of partitions of n (see A135010, A138121).
Sum of largest parts in all partitions in the head of the last section of the set of partitions of n. - Omar E. Pol, Nov 07 2011
a(n) is also the total number of parts in the n-th section of the set of partitions of any positive integer >= n.
a(n) is also the total number of divisors of all terms in the n-th row of triangle A336811. These divisors are also all parts in the last section of the set of partitions of n. (End)
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LINKS
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FORMULA
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a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(6*n/Pi^2)) / (8*sqrt(3)*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 21 2016
G.f.: Sum_{i>=1} i*x^i * Product_{j=2..i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 04 2017
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EXAMPLE
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Illustration of initial terms (n = 1..6) as sums of the first columns from the last sections of the first six natural numbers (or from the first six sections of 6):
. 6
. 3+3
. 4+2
. 2+2+2
. 5 1
. 3+2 1
. 4 1 1
. 2+2 1 1
. 3 1 1 1
. 2 1 1 1 1
. 1 1 1 1 1 1
. --- ----- ------- --------- ----------- --------------
. 1, 2, 3, 6, 8, 15,
...
Also, we can see that the sequence gives the number of parts in each section. For the number of odd/even parts (and more) see A207031, A207032 and also A206563. (End)
The geometric model looks like this:
. _ _ _ _ _ _
. |_ _ _ _ _ _|
. |_ _ _|_ _ _|
. |_ _ _ _|_ _|
. _ _ _ _ _ |_ _|_ _|_ _|
. |_ _ _ _ _| |_|
. _ _ _ _ |_ _ _|_ _| |_|
. |_ _ _ _| |_| |_|
. _ _ _ |_ _|_ _| |_| |_|
. _ _ |_ _ _| |_| |_| |_|
. _ |_ _| |_| |_| |_| |_|
. |_| |_| |_| |_| |_| |_|
.
. 1 2 3 6 8 15
.
(End)
On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the divisors of these terms are [1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]. There are 15 divisors so a(6) = 15. - Omar E. Pol, Jul 27 2021
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MAPLE
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b:= proc(n, i) option remember; local f, g;
if n=0 then [1, 0]
elif i<1 then [0, 0]
elif i>n then b(n, i-1)
else f:= b(n, i-1); g:= b(n-i, i);
[f[1]+g[1], f[2]+g[2] +g[1]]
fi
end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
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MATHEMATICA
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b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]]+g[[1]], f[[2]]+g[[2]]+g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
Table[PartitionsP[n - 1] + Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 45}] (* Robert Price, May 01 2020 *)
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CROSSREFS
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Cf. A000005, A000041, A002865, A006128, A135010, A138121, A182703, A336811, A336812, A338156, A340035, A341062.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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