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A024790 Number of 6's in all partitions of n. 13
0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 16, 24, 33, 47, 63, 89, 117, 159, 209, 278, 360, 474, 607, 786, 1001, 1280, 1615, 2049, 2565, 3222, 4011, 4998, 6180, 7653, 9407, 11571, 14154, 17308, 21063, 25630, 31044, 37586, 45339, 54646, 65646, 78804, 94305, 112761, 134473 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

The sums of six successive terms give A000070. - Omar E. Pol, Jul 12 2012

a(n) is also the difference between the sum of 6th largest and the sum of 7th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A181187(n,6) - A181187(n,7). - Omar E. Pol, Oct 25 2012

From Peter Bala, Dec 26 2013: (Start)

a(n+6) - a(n) = A000041(n). a(n) + a(n+3) = A024787(n).

a(n) + a(n+2) + a(n+4) = A024786(n).

O.g.f.: x^6/(1 - x^6) * product {k >= 1} 1/(1 - x^k) = x^6 + x^7 + 2*x^8 + 3*x^9 + ....

Asymptotic result: log(a(n)) ~ 2*sqrt(Pi^2/6)*sqrt(n) as n -> inf. (End)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (12*Pi*sqrt(2*n)) * (1 - 73*Pi/(24*sqrt(6*n)) + (73/48 + 3601*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016

MAPLE

b:= proc(n, i) option remember; local g;

      if n=0 or i=1 then [1, 0]

    else g:= `if`(i>n, [0$2], b(n-i, i));

         b(n, i-1) +g +[0, `if`(i=6, g[1], 0)]

      fi

    end:

a:= n-> b(n, n)[2]:

seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

MATHEMATICA

Table[ Count[ Flatten[ IntegerPartitions[n]], 6], {n, 1, 52} ]

b[n_, i_] := b[n, i] = Module[{g}, If [n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 6, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-Fran├žois Alcover, Oct 09 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A000041, A066633, A024786, A024787, A024788, A024789, A024791, A024792, A024793, A024794.

Sequence in context: A105930 A122622 A266775 * A275592 A319635 A179822

Adjacent sequences:  A024787 A024788 A024789 * A024791 A024792 A024793

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified April 19 10:38 EDT 2019. Contains 322255 sequences. (Running on oeis4.)