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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). 3
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified September 19 15:18 EDT 2019. Contains 327198 sequences. (Running on oeis4.)