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Search: id:A101447
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| A101447 |
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Triangle read by rows: T(n,k) = (2*k+1)*(n+1-k), 0<=k<n. |
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1
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| 1, 2, 3, 3, 6, 5, 4, 9, 10, 7, 5, 12, 15, 14, 9, 6, 15, 20, 21, 18, 11, 7, 18, 25, 28, 27, 22, 13, 8, 21, 30, 35, 36, 33, 26, 15, 9, 24, 35, 42, 45, 44, 39, 30, 17, 10, 27, 40, 49, 54, 55, 52, 45, 34, 19, 11, 30, 45, 56, 63, 66, 65, 60, 51, 38, 21, 12, 33, 50, 63, 72, 77, 78, 75, 68
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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The triangle is generated from the product of matrix A and matrix B, I.e. A * B where A = the infinite lower triangular matrix:
1 0 0 0 0 ...
1 1 0 0 0 ...
1 1 1 0 0 ...
1 1 1 1 0 ...
1 1 1 1 1 ...
... and B = the infinite lower triangular matrix:
1 0 0 0 0...
1 3 0 0 0...
1 3 5 0 0...
1 3 5 7 0...
1 3 5 7 9...
...
Row sums give the square pyramidal numbers A000330.
T(n+0,0)=1*n=A000027(n+1); T(n+1,1)=3*n=A008585(n); T(n+2,2)=5*n=A008587(n); T(n+3,3)=7*n=A008589(n); etc. So T(n,0)*T(n,1)=3*n*(n+1)=A028896(n) (6 times triangular numbers.) T(n,1)*T(n,2)/10=3*n*(n+1)/2=A045943(n) for n>0 T(n,2)*T(n,3)/10=7/2*n*(n+1)=A024966(n) for n>1 (7 times triangular numbers.) etc.
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MATHEMATICA
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t[n_, k_] := If[n < k, 0, (2*k + 1)*(n - k + 1)]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (from Robert G. Wilson v Jan 20 2005)
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PROGRAM
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(PARI) T(n, k)=if(n<k, 0, (2*k+1)*(n-k+1)) for(i=0, 15, for(j=0, i, print1(T(i, j), ", ")); print())
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CROSSREFS
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Cf. A094728 (triangle generated by B*A), A000330.
Sequence in context: A045892 A115973 A057047 this_sequence A119322 A014498 A023821
Adjacent sequences: A101444 A101445 A101446 this_sequence A101448 A101449 A101450
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KEYWORD
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nonn,tabl
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AUTHOR
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Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 19 2005
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