A102422 -id:A102422 - cp's OEIS frontend

A102420 Number of partitions of n into exactly k = 5 parts with each part p <= 5.

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Thomas Wieder, Jan 09 2005

There are only 26 nonzero terms.

a(n) is also the number of partitions of n-1 into exactly 4 parts with each part p in the range 1 <= p <= 6; i.e. the number of ways of arriving at a total of n-1 with 4 6-sided dice. - Toby Gottfried, Feb 19 2009

			a(8) = 3 because we can write 8=1+1+1+2+3 or 1+1+1+1+4 or 1+1+2+2+2.

Cf. A000041, A102422, A036606.

Table[Count[IntegerPartitions[n,{5}],?(Max[#]<6&)],{n,0,110}] (* _Harvey P. Dale, Nov 29 2012 *)

G.f.:z^5+z^6+2*z^7+3*z^8+5*z^9+6*z^10+8*z^11+9*z^12+11*z^13 +11*z^14 +12*z^15+ 11*z^16+11*z^17+9*z^18+8*z^19+6*z^20+5*z^21+3*z^22 +2*z^23 +z^24 +z^25.

A107110 Square array by antidiagonals where T(n,k) is the number of partitions of k into no more than n parts each no more than n. Visible version of A063746.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 3, 2, 1, 1, 0, 0, 0, 3, 3, 2, 1, 1, 0, 0, 0, 3, 5, 3, 2, 1, 1, 0, 0, 0, 3, 5, 5, 3, 2, 1, 1, 0, 0, 0, 2, 7, 7, 5, 3, 2, 1, 1, 0, 0, 0, 1, 7, 9, 7, 5, 3, 2, 1, 1, 0, 0, 0, 1, 8, 11, 11, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 7, 14, 13, 11, 7, 5, 3, 2

Offset: 0

Henry Bottomley, May 12 2005

			Rows start 1,0,0,0,...; 1,1,0,0,0,...; 1,1,2,1,1,0,0,0,...; 1,1,2,3,3,3,3,2,1,1,0,0,0,...; 1,1,2,3,5,5,7,7,8,7,7,5,5,3,2,1,1,0,0,0,...; etc.
T(4,6)=7 since 6 can be written seven ways with no more than 4 parts each no more than 4: 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, or 2+2+1+1.

Henry Bottomley, Partition and composition calculator.

Cf. A063746. Fifth row is A102422.

See A063746 for formulas. T(n, k)=A000041(k) if n>=k. T(n, k)=T(n, n^2-k). T(n, [n^2/2])=A029895(n); T(2n, 2n^2)=A063074(n). Row sums are A000984.

A127500 On the triangular peg solitaire board of side n, the shortest solution to any problem beginning with one peg missing and ending with one peg.

Original entry on oeis.org

5, 9, 9, 12, 13, 16, 18

Offset: 4

George Bell (gibell(AT)comcast.net), Mar 31 2007

Shortest means the minimum number of moves, where a move is one or more jumps by the same peg. The reference calculates a(n) up to n=10 and gives the bounds 19<=a(11)<=28, 21<=a(12)<=29, as well as an upper bound for n a multiple of 12. A trivial upper bound is a(n)<=T(n)-2, where T(n) is the n-th triangular number.

			a(4)=5, the 10-hole triangular board can be solved in 5 moves (and always 8 jumps).

Martin Gardner, Penny Puzzles, in Mathematical Carnival, p. 12-26, Alfred A. Knopf, Inc., 1975

George I. Bell, Triangular Peg Solitaire.
George I. Bell, Solving Triangular Peg Solitaire [arXiv:math/0703865v4]
G. I. Bell, Solving Triangular Peg Solitaire, JIS 11 (2008) 08.4.8

Cf. A000217, A102422.

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A102420 Number of partitions of n into exactly k = 5 parts with each part p <= 5.

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A107110 Square array by antidiagonals where T(n,k) is the number of partitions of k into no more than n parts each no more than n. Visible version of A063746.

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A127500 On the triangular peg solitaire board of side n, the shortest solution to any problem beginning with one peg missing and ending with one peg.

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