A283943 Interspersion of the signature sequence of e (a rectangular array, by antidiagonals).
1, 4, 2, 10, 6, 3, 19, 13, 8, 5, 30, 23, 16, 11, 7, 44, 35, 27, 20, 14, 9, 61, 50, 40, 32, 24, 17, 12, 81, 68, 56, 46, 37, 28, 21, 15, 103, 89, 75, 63, 52, 42, 33, 25, 18, 128, 112, 97, 83, 70, 58, 48, 38, 29, 22, 156, 138, 121, 106, 91, 77, 65, 54, 43, 34
Offset: 1
Examples
Northwest corner: 1 4 10 19 30 44 61 81 103 2 6 13 23 35 50 68 89 112 3 8 16 27 40 56 75 97 121 5 11 20 32 46 63 83 106 131 7 14 24 37 52 70 91 115 141 9 17 28 42 58 77 99 124 151
Links
- Clark Kimberling, Antidiagonals n = 1..60, flattened
- Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Programs
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Mathematica
r = E; z = 100; s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r]; u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022786, col 1 of A283943 *) v = Table[s[n], {n, 0, z}] (* A022785, row 1 of A283943 *) w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1; Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283943, array *) Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283943, sequence *) -
PARI
\\ Produces the triangle when the array is read by antidiagonals r = exp(1); z = 100; s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r)); p(n) = n + 1 + sum(k=0, n, floor((n - k)/r)); u = v = vector(z + 1); for(n=1, 101, (v[n] = s(n - 1))); for(n=1, 101, (u[n] = p(n - 1))); w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1; tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); ); print(); ); }; tabl(10) \\ Indranil Ghosh, Mar 26 2017 -
Python
# Produces the triangle when the array is read by antidiagonals import math from mpmath import * mp.dps = 100 def s(n): return 1 if n<1 else s(n - 1) + 1 + int(math.floor(n*e)) def p(n): return n + 1 + sum([int(math.floor((n - k)/e)) for k in range(0, n+1)]) v=[s(n) for n in range(0, 101)] u=[p(n) for n in range(0, 101)] def w(i, j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1 for n in range(1, 11): print([w(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017
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