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A355663
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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)).
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1
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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, 19, 16, 16, 19, 12, 7, 8, 15, 28, 23, 48, 23, 28, 15, 8, 9, 16, 16, 24, 39, 39, 24, 16, 16, 9, 10, 19, 48, 31, 56, 51, 56, 31, 48, 19, 10, 11, 20, 35, 32, 32, 35, 35, 32, 32, 35, 20, 11
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OFFSET
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0,4
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COMMENTS
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In other words, A(n, k) encodes the sum of the polynomials encoded by n and k.
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LINKS
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FORMULA
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A(n, k) = A(k, n).
A(n, 0) = n.
A(n, 1) = A014601(n) for any n > 0.
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EXAMPLE
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Array A(n, k) begins:
n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12
---+------------------------------------------------------------
0| 0 1 2 3 4 5 6 7 8 9 10 11 12
1| 1 3 4 7 8 11 12 15 16 19 20 23 24
2| 2 4 12 8 24 19 28 16 48 35 44 39 56
3| 3 7 8 15 16 23 24 31 32 39 40 47 48
4| 4 8 24 16 48 39 56 32 96 71 88 79 112
5| 5 11 19 23 39 51 35 47 79 99 76 103 71
6| 6 12 28 24 56 35 60 48 112 67 92 71 120
7| 7 15 16 31 32 47 48 63 64 79 80 95 96
8| 8 16 48 32 96 79 112 64 192 143 176 159 224
9| 9 19 35 39 71 99 67 79 143 195 156 199 135
10| 10 20 44 40 88 76 92 80 176 156 204 152 184
11| 11 23 39 47 79 103 71 95 159 199 152 207 143
12| 12 24 56 48 112 71 120 96 224 135 184 143 240
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PROG
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(PARI) toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
A(n, k) = { fromruns(Vec(Pol(toruns(n)) + Pol(toruns(k)))) }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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