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 A182714 Number of 4's in the outer shell of the set of partitions of n. 18
 0, 0, 0, 1, 0, 1, 1, 3, 2, 5, 5, 10, 10, 17, 19, 31, 34, 51, 60, 86, 100, 139, 165, 223, 265, 349, 418, 543, 648, 827, 992, 1251, 1495, 1866, 2230, 2758, 3289, 4033, 4803, 5852, 6949, 8411, 9973, 12005, 14194, 17002, 20060, 23919, 28153, 33426, 39256, 46438 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Zero together with the first differences of A024788. Also number of 4's in all partitions of n that do not contain 1 as a part. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 FORMULA It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3) + a(n+4), n >= 0. - Omar E. Pol, Feb 04 2012 EXAMPLE a(8) = 3 counts the 4's in 8 = 4+4 = 4+2+2. The 4's in 8 = 4+3+1 = 4+2+1+1 = 4+1+1+1+1 do not count. Contribution from Omar E. Pol, _Oct 25 2012_ (Start): -------------------------------------- .                             Number Outer shell of 8              of 4's -------------------------------------- 8 .............................. 0 4 + 4 .......................... 2 5 + 3 .......................... 0 6 + 2 .......................... 0 3 + 3 + 2 ...................... 0 4 + 2 + 2 ...................... 1 2 + 2 + 2 + 2 .................. 0 .   1 .......................... 0 .       1 ...................... 0 .       1 ...................... 0 .           1 .................. 0 .       1 ...................... 0 .           1 .................. 0 .           1 .................. 0 .               1 .............. 0 .           1 .................. 0 .               1 .............. 0 .               1 .............. 0 .                   1 .......... 0 .                   1 .......... 0 .                       1 ...... 0 .                           1 .. 0 ------------------------------------ .           6 - 3 =              3 The difference between the sum of the fourth column and the sum of the fifth column of the outer shell of 8 is 6 - 3 = 3 and equals the number of 4's in the outer shell of the set of partitions of 8, so a(8) = 3 (see also A024788). (End) MAPLE b:= proc(n, i) option remember; local g, h;       if n=0 then [1, 0]     elif i<2 then [0, 0]     else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));          [g[1]+h[1], g[2]+h[2]+`if`(i=4, h[1], 0)]       fi     end: a:= n-> b(n, n)[2]: seq (a(n), n=1..70);  # Alois P. Heinz, Mar 19 2012 PROG (Sage) A182714 = lambda n: sum(list(p).count(4) for p in Partitions(n) if 1 not in p) CROSSREFS Column 4 of A194812. Cf. A015739, A024788, A135010, A138121, A182703, A182712, A182713. Sequence in context: A186545 A008623 A035546 * A198755 A134237 A099889 Adjacent sequences:  A182711 A182712 A182713 * A182715 A182716 A182717 KEYWORD nonn AUTHOR Omar E. Pol, Nov 13 2011 STATUS approved

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