A367559 Square array T(n, k) = 2^k - n, read by ascending antidiagonals.

1, 0, 2, -1, 1, 4, -2, 0, 3, 8, -3, -1, 2, 7, 16, -4, -2, 1, 6, 15, 32, -5, -3, 0, 5, 14, 31, 64, -6, -4, -1, 4, 13, 30, 63, 128, -7, -5, -2, 3, 12, 29, 62, 127, 256, -8, -6, -3, 2, 11, 28, 61, 126, 255, 512, -9, -7, -4, 1, 10, 27, 60, 125, 254, 511, 1024

Offset

0,3

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (antidiagonals 0..150, flattened).

Formula

G.f. of row n: 1/(1-2*x) - n/(1-x).

E.g.f. of row n: exp(2*x) - n*exp(x).

T(0, k) = 2^k = A000079(k).

T(1, k) = 2^k - 1 = A000225(k).

T(2, k) = 2^k - 2 = A000918(k).

T(3, k) = 2^k - 3 = A036563(k).

T(5, k) = 2^k - 5 = A168616(k).

T(9, k) = 2^k - 9 = A185346(k).

T(10, k) = 2^k - 10 = A246168(k).

T(n, k) = 3*T(n, k-1) - 2*T(n, k-2) for k > 1.

T(n+1, k) = T(n, k) + 1.

T(n, n) = 2^n - n = A000325(n).

Sum_{k = 0..n} T(n - k, k) = A084634(n).

a(n) = 2^A002262(n) - A025581(n).

G.f.: (1 - 2*x - y + 3*x*y)/((1 - x)^2*(1 - y)*(1 - 2*y)). - Stefano Spezia, Nov 27 2023

Example

This sequence as square array T(n, k):
  n\k  0    1    2    3    4    5    6    7    8    9    10.
  ---------------------------------------------------------.
  0 :  1    2    4    8   16   32   64  128  256  512  1024.
  1 :  0    1    3    7   15   31   63  127  255  511  1023.
  2 : -1    0    2    6   14   30   62  126  254  510  1022.
  3 : -2   -1    1    5   13   29   61  125  253  509  1021.
  4 : -3   -2    0    4   12   28   60  124  252  508  1020.
  5 : -4   -3   -1    3   11   27   59  123  251  507  1019.
  6 : -5   -4   -2    2   10   26   58  122  250  506  1018.
  7 : -6   -5   -3    1    9   25   57  121  249  505  1017.
  8 : -7   -6   -4    0    8   24   56  120  248  504  1016.
  9 : -8   -7   -5   -1    7   23   55  119  247  503  1015.
  10: -9   -8   -6   -2    6   22   54  118  246  502  1014.

Mathematica

Table[2^k-n+k, {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Nov 28 2023 *)

Prog

(PARI) T(n, k) = 2^k-n \\ Thomas Scheuerle, Nov 23 2023

Crossrefs

Cf. A000079, A000225, A000325, A000295.

Cf. A000325, A000918, A084634, A036563.

Cf. A168616, A185346, A246168.

Cf. A002262, A025581.

Sequence in context: A340189 A323873 A365582 * A099020 A179438 A211970

Adjacent sequences: A367556 A367557 A367558 * A367560 A367561 A367562

Keyword

easy,sign,tabl

Author

Paul Curtz, Nov 22 2023

Status

approved