A176225 A symmetrical triangle sequence: T(n, k) = q^k + q^(n-k) - q^n, with q=3.

1, 1, 1, 1, -3, 1, 1, -15, -15, 1, 1, -51, -63, -51, 1, 1, -159, -207, -207, -159, 1, 1, -483, -639, -675, -639, -483, 1, 1, -1455, -1935, -2079, -2079, -1935, -1455, 1, 1, -4371, -5823, -6291, -6399, -6291, -5823, -4371, 1, 1, -13119, -17487, -18927, -19359, -19359, -18927, -17487, -13119, 1

Offset

0,5

Comments

Row sums are: {1, 2, -1, -28, -163, -730, -2917, -10936, -39367, -137782, -472393, ...}.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n,k) = q^k + q^(n-k) - q^n, with q = 3.

Example

Triangle begins as:
  1;
  1,     1;
  1,    -3,     1;
  1,   -15,   -15,     1;
  1,   -51,   -63,   -51,     1;
  1,  -159,  -207,  -207,  -159,     1;
  1,  -483,  -639,  -675,  -639,  -483,     1;
  1, -1455, -1935, -2079, -2079, -1935, -1455,     1;
  1, -4371, -5823, -6291, -6399, -6291, -5823, -4371, 1;

Maple

q:=3; seq(seq(q^k +q^(n-k) -q^n, k=0..n), n=0..12); # G. C. Greubel, Nov 23 2019

Mathematica

T[n_, k_, q_]:= q^k +q^(n-k) -q^n; Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Nov 23 2019 *)

Prog

(PARI) T(n, k, q) = my(q=3); q^k +q^(n-k) -q^n; \\ G. C. Greubel, Nov 23 2019
(Magma) q:=3; [q^k +q^(n-k) -q^n : k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 23 2019
(Sage) q=3; [[q^k +q^(n-k) -q^n for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 23 2019
(GAP) q:=3;; Flat(List([0..12], n-> List([0..n], k-> q^k +q^(n-k) -q^n ))); # G. C. Greubel, Nov 23 2019

Crossrefs

Cf. A176224 (q=2), this sequence (q=3), A176226 (q=5), A176227 (q=4).

Sequence in context: A110112 A370691 A326800 * A173917 A174410 A156690

Adjacent sequences: A176222 A176223 A176224 * A176226 A176227 A176228

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 12 2010

Extensions

Edited by G. C. Greubel, Nov 23 2019

Status

approved