A283941 Interspersion of the signature sequence of sqrt(5).
1, 4, 2, 9, 6, 3, 16, 12, 8, 5, 25, 20, 15, 11, 7, 37, 30, 24, 19, 14, 10, 51, 43, 35, 29, 23, 18, 13, 67, 58, 49, 41, 34, 28, 22, 17, 85, 75, 65, 56, 47, 40, 33, 27, 21, 106, 94, 83, 73, 63, 54, 46, 39, 32, 26, 129, 116, 103, 92, 81, 71, 61, 53, 45, 38, 31
Offset: 1
Examples
Northwest corner: 1 4 9 16 25 37 51 67 2 6 12 20 30 43 58 76 3 8 15 24 35 49 65 83 5 11 19 29 41 56 73 92 7 14 23 34 47 63 81 101 10 18 28 40 54 71 90 111
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[5]; 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}] (* A022780 , col 1 of A283941 *) v = Table[s[n], {n, 0, z}] (* A022779, row 1 of A283941*) w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1; Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283941, array *) Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283941, sequence *) -
PARI
r = sqrt(5); 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(20) \\ Indranil Ghosh, Mar 21 2017 -
Python
from sympy import sqrt import math def s(n): return 1 if n<1 else s(n - 1) + 1 + int(math.floor(n*sqrt(5))) def p(n): return n + 1 + sum([int(math.floor((n - k)/sqrt(5))) 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
Extensions
Edited by Clark Kimberling, Feb 27 2018
Comments