A283940 Interspersion of the signature sequence of sqrt(3).
1, 3, 2, 7, 5, 4, 13, 10, 8, 6, 20, 17, 14, 11, 9, 29, 25, 22, 18, 15, 12, 40, 35, 31, 27, 23, 19, 16, 53, 47, 42, 37, 33, 28, 24, 21, 67, 61, 55, 49, 44, 39, 34, 30, 26, 83, 76, 70, 63, 57, 51, 46, 41, 36, 32, 101, 93, 86, 79, 72, 65, 59, 54, 48, 43, 38
Offset: 1
Examples
Northwest corner: 1 3 7 13 20 29 40 53 2 5 10 17 25 35 47 61 4 8 14 22 31 42 55 70 6 11 18 27 37 49 63 79 9 15 23 33 44 57 72 89 12 19 28 39 51 65 81 99 16 24 34 46 59 74 91 110 21 30 41 54 68 84 102 122
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 = Sqrt[3]; 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}] (* A022778, col 1 of A283940 *) v = Table[s[n], {n, 0, z}] (* A022777, row 1 of A283940*) w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1; Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283940, array *) Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283940, sequence *) -
PARI
r = sqrt(3); 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 21 2017 -
Python
r = 3 ** 0.5 def s(n): return 1 if n<1 else s(n - 1) + 1 + int(n*r) def p(n): return n + 1 + sum([int((n - k)/r) 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 21 2017 -
Python
import numpy as np r = np.sqrt(3) x = np.arange(11) u = np.cumsum(np.ceil(x / r)).astype(int) v = np.cumsum(np.ceil(x * r)).astype(int) print(*[1 + u[k] + v[n-k] + k*(n-k) for n in range(11) for k in range(n+1)], sep=', ') # David Radcliffe, May 10 2025
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