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Search: id:A092565
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| A092565 |
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Triangle of coefficients T(n,k) (n>=0, 0<=k<=2n), read by rows, where the n-th row polynomial equals the numerator of the n-th convergent of the continued fraction [1+x+x^2;1+x+x^2,1+x+x^2,...] for n>0, with the zeroth row defined as T(0,0)=1. |
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+0
2
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| 1, 1, 1, 1, 2, 2, 3, 2, 1, 3, 5, 8, 7, 6, 3, 1, 5, 10, 19, 22, 22, 16, 10, 4, 1, 8, 20, 42, 58, 69, 63, 49, 30, 15, 5, 1, 13, 38, 89, 142, 191, 206, 191, 146, 95, 50, 21, 6, 1, 21, 71, 182, 327, 491, 602, 637, 573, 447, 296, 167, 77, 28, 7, 1, 34, 130, 363, 722, 1191, 1626
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Diagonal forms A092566, row sums form A006190. Column T(n,0) forms Fibonacci numbers A000045, T(n,1) forms A001629.
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FORMULA
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n-th row polynomial R(n) = sum_{k=0..n} A037027(n, k)*x^k*(1+x)^k; R(n+1)/R(n) = [1+x+x^2;1+x+x^2, ...(n+1)times..., 1+x+x^2] for n>=0; R(0)=1.
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EXAMPLE
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Ratio of row polynomials R(3)/R(2) = (3+5*x+8*x^2+7*x^3+6*x^4+3*x^5+x^6)/(2+2*x+3*x^2+2*x^3+x^4) = [1+x+x^2;1+x+x^2,1+x+x^2].
Rows begin:
{1},
{1,1,1},
{2,2,3,2,1},
{3,5,8,7,6,3,1},
{5,10,19,22,22,16,10,4,1},
{8,20,42,58,69,63,49,30,15,5,1},
{13,38,89,142,191,206,191,146,95,50,21,6,1},
{21,71,182,327,491,602,637,573,447,296,167,77,28,7,1},
{34,130,363,722,1191,1626,1921,1958,1752,1366,931,546,273,112,36,8,1},...
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PROGRAM
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(PARI) T(n, k)=if(2*n<k|k<0, 0, polcoeff(contfracpnqn(vector(n, i, 1+x+x^2))[1, 1], k, x))
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CROSSREFS
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Cf. A092566, A037027.
Sequence in context: A096826 A116199 A162915 this_sequence A021452 A093420 A104897
Adjacent sequences: A092562 A092563 A092564 this_sequence A092566 A092567 A092568
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Feb 28 2004
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