The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A236970 The number of partitions of n into at least 3 parts from which we can form every partition of n into 3 parts by summing elements 3
 0, 0, 1, 2, 2, 3, 5, 6, 7, 13, 16, 19, 29, 38, 49, 72, 84, 108, 155, 195, 234, 331, 410, 501, 672, 824, 1006, 1341, 1621, 1981, 2583, 3111, 3740, 4846, 5819, 6957, 8787, 10582, 12606, 15840, 18762, 22386, 27851, 32934, 38824, 47961, 56633, 66577, 81168, 95612 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS The corresponding partitions with 2 in the definition instead of 3 are the complete partitions, which are counted by A126796. The qualifier 'into at least 3 parts' is only relevant for n = 1 or 2. It is included because otherwise the condition would be vacuously true for all partitions of 1 and 2. It seems neater to consider that there are no partitions of 1 and 2 of this form. LINKS Jack W Grahl, Haskell code for generating this sequence EXAMPLE The valid partitions of 5 are (2,1,1,1) and (1,1,1,1,1). Given any partition of 5 into 3 parts, it contains one part of at least 2. Therefore we can make any partition of 5 into 3 parts by joining (2,1,1,1) into three sums. (3, 1, 1) is not a valid partition, since (2,2,1) is a partition of 5 into 3 parts which cannot be made by summing elements from (3,1,1). Therefore a(5) = 2. MATHEMATICA ric[p_, {x_, y_}] := If[x==0, If[y > Total[p], False, y==0 || AnyTrue[ Reverse@ Union[p], y>=# && ric[ DeleteCases[p, #, 1, 1], {0, y-#}] &]], If[x >= Total[p], False, AnyTrue[ Reverse@ Union@ p, x>=# && ric[ DeleteCases[p, #, 1, 1], {x-#, y}] &]]]; chk[p_] := AllTrue[ Rest /@ IntegerPartitions[Plus @@ p, {3}], ric[p, #] &]; a[n_] := Length@ Select[ IntegerPartitions[n, {3, Infinity}], chk]; Array[a, 24] (* Giovanni Resta, Jul 18 2018 *) CROSSREFS Cf. A000041. A126796 is the case for 2 instead of 3, A236971 and A236972 are the cases for 4 and 5. Sequence in context: A308272 A035541 A187502 * A060966 A135279 A035631 Adjacent sequences: A236967 A236968 A236969 * A236971 A236972 A236973 KEYWORD nonn AUTHOR Jack W Grahl, Feb 02 2014 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 2 07:49 EST 2022. Contains 358493 sequences. (Running on oeis4.)