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A162514
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Triangle of coefficients of polynomials defined by the Binet form P(n,x) = U^n+L^n, where U=(x+d)/2, L=(x-d)/2, d=(4 + x^2)^(1/2).
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4
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2, 1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 4, 0, 2, 1, 0, 5, 0, 5, 0, 1, 0, 6, 0, 9, 0, 2, 1, 0, 7, 0, 14, 0, 7, 0, 1, 0, 8, 0, 20, 0, 16, 0, 2, 1, 0, 9, 0, 27, 0, 30, 0, 9, 0, 1, 0, 10, 0, 35, 0, 50, 0, 25, 0, 2, 1, 0, 11, 0, 44, 0, 77, 0, 55, 0, 11, 0, 1, 0, 12, 0, 54, 0, 112, 0, 105, 0, 36, 0, 2, 1, 0, 13, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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FORMULA
| P(n,x) = x*P(n-1,x) + P(n-2,x) for n>=2, P(0,x)=2, P(1,x)=x.
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EXAMPLE
| Triangle begins
2; == 2
1, 0; == x + 0
1, 0, 2; == x^2 + 2
1, 0, 3, 0; == x^3 + 3*x + 0
1, 0, 4, 0, 2;
1, 0, 5, 0, 5, 0;
1, 0, 6, 0, 9, 0, 2;
1, 0, 7, 0, 14, 0, 7, 0;
1, 0, 8, 0, 20, 0, 16, 0, 2;
1, 0, 9, 0, 27, 0, 30, 0, 9, 0;
1, 0, 10, 0, 35, 0, 50, 0, 25, 0, 2; ...
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PROG
| (PARI)
P(n)=
{
local(U, L, d, r, x);
if ( n<0, return(0) );
x = 'x+O('x^(n+1));
d=(4 + x^2)^(1/2);
U=(x+d)/2; L=(x-d)/2;
r = U^n+L^n;
r = truncate(r);
return( r );
}
for (n=0, 10, print(Vec(P(n))) ); /* show triangle */
/* Joerg Arndt, Jul 24 2011 */
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CROSSREFS
| A000032, A162515, A162516, A162517
Sequence in context: A115672 A079694 A068906 * A166347 A055300 A156256
Adjacent sequences: A162511 A162512 A162513 * A162515 A162516 A162517
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KEYWORD
| nonn,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Jul 05 2009
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