Revision History for A122432
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Showing entries 1-10
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#25 by Charles R Greathouse IV at Thu Sep 08 08:45:28 EDT 2022
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(MAGMAMagma) /* As triangle */ [[(-1)^(n-k)*Binomial(n-k+2, 2): k in [1..n]]: n in [1..10]]; // G. C. Greubel, Oct 29 2017
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Discussion
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| Thu Sep 08
| 08:45
| OEIS Server: https://oeis.org/edit/global/2944
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#24 by Michael De Vlieger at Tue Aug 09 14:17:42 EDT 2022
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#23 by Michel Marcus at Tue Aug 09 12:14:43 EDT 2022
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#22 by Michel Marcus at Tue Aug 09 12:14:41 EDT 2022
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From Wolfdieter Lang, Apr 05 2020:(: (Start)
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approved
editing
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#21 by Peter Luschny at Thu Apr 30 14:43:51 EDT 2020
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#20 by Peter Luschny at Thu Apr 30 11:30:57 EDT 2020
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#19 by Wesley Ivan Hurt at Mon Apr 20 00:23:35 EDT 2020
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#18 by Wesley Ivan Hurt at Mon Apr 20 00:22:36 EDT 2020
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having the k xX k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A127893. - Peter Bala, Jul 22 2014
The unsigned triangle, i.e., Tup(n, k) := (-1)^(n-k)*T(n-1,k-1) = binomial(n-k+2, 2) with n >= 1, k = 1..n, gives the number of triangles of length k (in some units), for k = 1..n, in the matchstick arrangement (or tower of cards, with n cards as basis) with an enclosing triangle of length n, but only triangles with orientation (up) like the enclosing triangle are counted. The total number of matchsticks (cards) is 3*A000217(n). (See the comment by Andrew Howroyd in A085691. ). Recurrence: Tup(n., , k) = 0 for n < k, Tup(1, 1) = 1, and Tup(n. , k) = Tup(n-1, k) + n - k + 1, for n >= 2, k = 1..n. Row sums give A000292(n). (End)
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| FORMULA
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Number triangle T(n, k)=[) = [k<=n]*(-1)^(n-k)*binomial(n-k+2, 2).
Recurrence: T(n, k) = - T(n-1, k) + (-1)^(n-k)*(n-k+1), for n >= 0, and k = = 0..n. - Wolfdieter Lang, Apr 06 2020
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proposed
editing
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#17 by Wolfdieter Lang at Fri Apr 17 08:27:09 EDT 2020
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#16 by Wolfdieter Lang at Mon Apr 06 13:04:57 EDT 2020
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The unsigned triangle, i.e., Tup(n, k) := (-1)^(n-k)*T(n-1,k) := -1) = binomial(n-k+2, 2) with n >= 1, k = 1..n, gives the number of triangles of length k (in some units), for k = 1..n, in the matchstick arrangement (or tower of cards, with n cards as basis) with an enclosing triangle of length n, but only triangles with orientation (up) like the enclosing triangle are counted. The total number of matchsticks (cards) is 3*A000217(n). (See the comment by Andrew Howroyd in A085691. Recurrence: Tup(n., k) = 0 for n < k, Tup(1, 1) = 1, and Tup(n. k) = Tup(n-1, k) + n - k + 1, for n >= 2, k = 1..n. Row sums give A000292(n). (End)
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Discussion
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| Mon Apr 13
| 19:01
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