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A060964 Table by antidiagonals where T(n,k) = n*T(n,k-1) - T(n,k-2) with T(n,0) = 2 and T(n,1) = n. 2
2, 0, 2, -2, 1, 2, 0, -1, 2, 2, 2, -2, 2, 3, 2, 0, -1, 2, 7, 4, 2, -2, 1, 2, 18, 14, 5, 2, 0, 2, 2, 47, 52, 23, 6, 2, 2, 1, 2, 123, 194, 110, 34, 7, 2, 0, -1, 2, 322, 724, 527, 198, 47, 8, 2, -2, -2, 2, 843, 2702, 2525, 1154, 322, 62, 9, 2, 0, -1, 2, 2207, 10084, 12098, 6726, 2207, 488, 79, 10, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
FORMULA
For all m, T(n, k) = T(n, |m|)*T(n, |k - m|) - T(n, |k - 2m|).
T(n, 2k) = T(n, k)^2 - 2.
T(n, 2k + 1) = T(n, k)*T(n, k + 1) - n.
T(n, 3k) = T(n, k)^3 - 3*T(n, k).
T(n, 4k) = T(n, k)^4 - 4*T(n, k)^2 + 2.
T(n, 5k) = T(n, k)^5 - 5*T(n, k)^3 + 5*T(n, k) etc.
T(n, -k) = T(n, k).
T(-n, k) = T(-n, -k) = (-1)^k * T(n, k).
T(n, k) = ( n*( ((n + sqrt(n^2 -4))/2)^k - ((n - sqrt(n^2 -4))/2)^k ) - 2*( ((n + sqrt(n^2 -4))/2)^(k-1) - ((n - sqrt(n^2 -4))/2)^(k-1) ) )/sqrt(n^2 -4).
T(n, k) = n*ChebyshevU(k-1, n/2) - 2*ChebyshevU(k-2, n/2). - G. C. Greubel, Jan 15 2020
EXAMPLE
Square array begins as:
2, 0, -2, 0, 2, 0, -2, ...
2, 1, -1, -2, -1, 1, 2, ...
2, 2, 2, 2, 2, 2, 2, ...
2, 3, 7, 18, 47, 123, 322, ...
2, 4, 14, 52, 194, 724, 2702, ...
2, 5, 23, 110, 527, 2525, 12098, ...
MAPLE
seq(seq( simplify(k*ChebyshevU(n-k-1, k/2) -2*ChebyshevU(n-k-2, k/2)), k=0..n), n=0..12); # G. C. Greubel, Jan 15 2020
MATHEMATICA
Table[k*ChebyshevU[n-k-1, k/2] -2*ChebyshevU[n-k-2, k/2], {n, 0, 12}, {k, 0, n} ]//Flatten
PROG
(PARI) T(n, k) = n*polchebyshev(k-1, 2, n/2) -2*polchebyshev(k-2, 2, n/2);
for(n=0, 12, for(k=0, n, print1(T(k, n-k), ", "))) \\ G. C. Greubel, Jan 15 2020
(Magma)
function T(n, k)
if k eq 0 then return 2;
elif k eq 1 then return n;
else return n*T(n, k-1) - T(n, k-2);
end if; return T; end function;
[T(k, n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 15 2020
(Sage) [[k*chebyshev_U(n-k-1, k/2) -2*chebyshev_U(n-k-2, k/2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 15 2020
(GAP)
T:= function(n, k)
if k=0 then return 2;
elif k=1 then return n;
else return n*T(n, k-1) - T(n, k-2);
fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(k, n-k) ))); # G. C. Greubel, Jan 15 2020
CROSSREFS
Columns include A001477, A007395, A008865, A058794.
Sequence in context: A339697 A023604 A219660 * A360022 A118206 A029314
KEYWORD
sign,tabl
AUTHOR
Henry Bottomley, May 09 2001
STATUS
approved

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Last modified March 28 07:33 EDT 2024. Contains 371235 sequences. (Running on oeis4.)