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, 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

Offset

0,1

Links

G. C. Greubel, Antidiagonals n = 0..100, flattened

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

Rows include A003499, A003500, A003501, A005248, A007395, A056854, A056918, A057079.

Columns include A001477, A007395, A008865, A058794.

Sequence in context: A339697 A023604 A219660 * A360022 A118206 A029314

Adjacent sequences: A060961 A060962 A060963 * A060965 A060966 A060967

Keyword

sign,tabl

Author

Henry Bottomley, May 09 2001

Status

approved