A283944 Interspersion of the signature sequence of Pi (rectangular array by antidiagonals).
1, 5, 2, 12, 7, 3, 22, 15, 9, 4, 35, 26, 18, 11, 6, 51, 40, 30, 21, 14, 8, 70, 57, 45, 34, 25, 17, 10, 92, 77, 63, 50, 39, 29, 20, 13, 118, 100, 84, 69, 56, 44, 33, 24, 16, 147, 127, 108, 91, 76, 62, 49, 38, 28, 19, 179, 157, 136, 116, 99, 83, 68, 55, 43, 32
Offset: 1
Examples
Northwest corner: 1 5 12 22 35 51 70 92 118 2 7 15 26 40 57 77 100 127 3 9 18 30 45 63 84 108 136 4 11 21 34 50 69 91 115 145 6 14 25 39 56 76 99 125 155 8 17 29 44 62 83 107 134 165
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 = Pi; 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}] (* A022796, col 1 of A283944 *) v = Table[s[n], {n, 0, z}] (* A022795, row 1 of A283944 *) w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1; Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283944, array*) Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283944, sequence *) -
PARI
\\ Produces the triangle when the array is read by antidiagonals r = Pi; 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*pi)) def p(n): return n + 1 + sum([int(math.floor((n - k)/pi)) 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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