A323346 Square array read by ascending antidiagonals: T(p,q) is the number of bases e such that e^2 = 1 (including e = 1) in Clifford algebra Cl(p,q)(R).

1, 2, 1, 3, 3, 1, 4, 6, 4, 2, 6, 10, 10, 6, 6, 12, 16, 20, 16, 12, 16, 28, 28, 36, 36, 28, 28, 36, 64, 56, 64, 72, 64, 56, 64, 72, 136, 120, 120, 136, 136, 120, 120, 136, 136, 272, 256, 240, 256, 272, 256, 240, 256, 272, 256, 528, 528, 496, 496, 528, 528, 496, 496, 528, 528, 496

Offset

0,2

Comments

See A323100 for a introduction of Clifford algebras.

Links

Jianing Song, Antidiagonals n = 0..99, flattened

Wikipedia, Clifford algebras

Formula

T(p,q) = Sum_{i=0..p} Sum_{j=0..q} binomial(p, i)*binomial(q, j)*(1 - (binomial(i - j, 2) mod 2)).

T(p,q) = 2^(p+q) - A323100(p,q).

Example

Table begins
p\q|  0   1   2    3    4    5  ...
---+-------------------------------
0  |  1,  1,  1,   2,   6,  16, ...
1  |  2,  3,  4,   6,  12,  28, ...
2  |  3,  6, 10,  16,  28,  56, ...
3  |  4, 10, 20,  36,  64, 120, ...
4  |  6, 16, 36,  72, 136, 256, ...
5  | 12, 28, 64, 136, 272, 528, ...
...
See A323100 for an example that shows T(1,3) = 6.

Maple

s := sqrt(2): h := n -> [ 0, -s, -2, -s, 0, s, 2,  s][1 + modp(n+1, 8)]:
T := proc(n, k) option remember;
if n = 0 then return 2^k*(1 - 1/2) - 2^((k - 3)/2)*h(k + 2) fi;
if k = 0 then return 2^n*(1 - 1/2) - 2^((n - 3)/2)*h(n) fi;
T(n, k-1) + T(n-1, k) end:
for n from 0 to 9 do seq(T(n, k), k=0..9) od; # Peter Luschny, Jan 12 2019

Mathematica

T[n_, k_] := 2^(n + k) - Sum[Binomial[n, i] Binomial[k, j] Mod[Binomial[i - j, 2], 2], {i, 0, n}, {j, 0, k}];
Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 19 2019 *)

Prog

(PARI) T(p, q) = sum(i=0, p, sum(j=0, q, binomial(p, i)*binomial(q, j)*!(binomial(i-j, 2)%2)))

Crossrefs

Cf. A038503((n+1) (first row), A038504(n+1) (first column), A007582 (main diagonal).

A323100 is the complement sequence.

Sequence in context: A239986 A285548 A130305 * A143328 A192001 A122176

Adjacent sequences: A323343 A323344 A323345 * A323347 A323348 A323349

Keyword

nonn,tabl

Author

Jianing Song, Jan 12 2019

Status

approved