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A283941 Interspersion of the signature sequence of sqrt(5). 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Row n is the ordered sequence of numbers k such that A023117(k)=n. As a sequence, A283941 is a permutation of the positive integers. This is a transposable interspersion; i.e., every row intersperses all other rows, and every column intersperses all other columns.
LINKS
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
Sequence in context: A095833 A163990 A082156 * A285090 A114577 A285089
KEYWORD
nonn,tabl,easy
AUTHOR
Clark Kimberling, Mar 19 2017
EXTENSIONS
Edited by Clark Kimberling, Feb 27 2018
STATUS
approved

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)