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%I #5 Jul 14 2022 09:35:21
%S 0,1,1,2,3,2,3,4,4,3,4,7,12,7,4,5,8,8,8,8,5,6,11,24,15,24,11,6,7,12,
%T 19,16,16,19,12,7,8,15,28,23,48,23,28,15,8,9,16,16,24,39,39,24,16,16,
%U 9,10,19,48,31,56,51,56,31,48,19,10,11,20,35,32,32,35,35,32,32,35,20,11
%N Square array A(n, k), n, k >= 0, read by antidiagonals; for any number n with runs in binary expansion (r_w, ..., r_0), let p(n) be the polynomial of a single indeterminate x where the coefficient of x^e is r_e for e = 0..w and otherwise 0, and let q be the inverse of p; A(n, k) = q(p(n) + p(k)).
%C In other words, A(n, k) encodes the sum of the polynomials encoded by n and k.
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A(n, k) = A(k, n).
%F A(n, 0) = n.
%F A(n, 1) = A014601(n) for any n > 0.
%F A(n, n) = A001196(n).
%e Array A(n, k) begins:
%e n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12
%e ---+------------------------------------------------------------
%e 0| 0 1 2 3 4 5 6 7 8 9 10 11 12
%e 1| 1 3 4 7 8 11 12 15 16 19 20 23 24
%e 2| 2 4 12 8 24 19 28 16 48 35 44 39 56
%e 3| 3 7 8 15 16 23 24 31 32 39 40 47 48
%e 4| 4 8 24 16 48 39 56 32 96 71 88 79 112
%e 5| 5 11 19 23 39 51 35 47 79 99 76 103 71
%e 6| 6 12 28 24 56 35 60 48 112 67 92 71 120
%e 7| 7 15 16 31 32 47 48 63 64 79 80 95 96
%e 8| 8 16 48 32 96 79 112 64 192 143 176 159 224
%e 9| 9 19 35 39 71 99 67 79 143 195 156 199 135
%e 10| 10 20 44 40 88 76 92 80 176 156 204 152 184
%e 11| 11 23 39 47 79 103 71 95 159 199 152 207 143
%e 12| 12 24 56 48 112 71 120 96 224 135 184 143 240
%o (PARI) toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
%o fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
%o A(n,k) = { fromruns(Vec(Pol(toruns(n)) + Pol(toruns(k)))) }
%Y Cf. A001196, A014601, A101211, A355664.
%K nonn,base,tabl
%O 0,4
%A _Rémy Sigrist_, Jul 13 2022
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