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A103415
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Triangle read by rows: T(n,k)=P(n+1) - sum(i=1,k,DT(n,k)), where P denotes the Pell numbers and DT the triangle version of A008288.
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0
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1, 2, 1, 5, 4, 1, 12, 11, 6, 1, 29, 28, 21, 8, 1, 70, 69, 60, 35, 10, 1, 169, 168, 157, 116, 53, 12, 1, 408, 407, 394, 333, 204, 75, 14, 1, 985, 984, 969, 884, 653, 332, 101, 16, 1, 2378, 2377, 2360, 2247, 1870, 1189, 508, 131, 18, 1, 5741, 5740, 5721, 5576, 5001, 3712
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OFFSET
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0,2
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COMMENTS
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Triangle is generated from the product A*B of the infinite lower triangular matrices A =
1
1 1
1 3 1
1 5 5 1
... and B =
1
1 1
1 1 1
1 1 1 1
...
Absolute values of coefficients of characteristic polynomials of n-th matrix are the (n+1)th row of A007318 (Pascal's triangle). As they are:
x - 1
x^2 - 2*x + 1
x^3 - 3*x^2 + 3*x - 1
x^4 - 4*x^3 + 6*x^2 - 4*x + 1
x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1
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LINKS
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Table of n, a(n) for n=0..60.
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FORMULA
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a(n)=2a(n-1)+a(n-2)+2).
Third diagonal = 2*n^2 + 4*n + 5 (see also A093328(n+3)).
Fourth diagonal = 4/3*(n^3+5*n+3) for n>0.
Determinant(A*B) = 1 for all n.
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EXAMPLE
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Triangle begins:
1
2 1
5 4 1
12 11 6 1
29 28 21 8 1
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PROG
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(PARI) T(k, r) = if(r>k, 0, if(k==1, 1, if(k==2, 1, if(r==1||r==k, 1, T(k-1, r-1)+T(k-1, r)+T(k-2, r-1))))) ST(n, k) = sum(i=1, k, T(n, i)) P(n) = if(n==1, 1, if(n==2, 2, 2*P(n-1)+P(n-2))) BT(n, k) = P(n)-ST(n, k) for(i=1, 10, for(j=1, i, print1(BT(i, j-1), ", ")); print()) BM(n) = M=matrix(n, n); for(i=1, n, for(j=1, n, M[i, j]=T(i, j))); M IM(n) = M=matrix(n, n); for(i=1, n, for(j=1, n, if(j>i, M[i, j]=0, M[i, j]=1))); M BM(10)*IM(10)
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CROSSREFS
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First column = A000129(n+1) (Pell numbers).
Second column = A005409 (Number of polynomials of height n:
Row sums give A026937.
Cf. A008288, A000129, A005409, A093328, A007318, A103416.
Sequence in context: A299444 A110552 A129161 * A054456 A096164 A201166
Adjacent sequences: A103412 A103413 A103414 * A103416 A103417 A103418
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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.net) and Gary W. Adamson, Feb 04 2005
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STATUS
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approved
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