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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-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

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Last modified September 28 10:48 EDT 2021. Contains 347714 sequences. (Running on oeis4.)