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A306773
Triangle read by rows, 0 <= k < n, n >= 2: T(n,k) is the eventual period of the modified Fibonacci sequence x(j) (or 0 if x(j) never enters a cycle) defined as follows: x(0) = 0, x(1) = 1, and for j > 1 x(j) is obtained from x(j-1) + x(j-2) by deleting all occurrences of the digit k in base n.
8
1, 1, 3, 1, 3, 16, 1, 3, 24, 6, 1, 3, 13, 6, 100, 1, 3, 8, 252, 120, 24, 1, 3, 8, 42, 119, 96, 576, 1, 3, 60, 588, 378, 36, 624, 1932, 1, 3, 126, 600, 144, 381, 200, 936, 912, 1, 3, 480, 51, 9, 2760, 6220, 540, 1800, 5700, 1, 3, 170, 750, 2480, 14880, 10990, 300, 1440, 3660, 840, 1, 3, 13800, 5880, 432, 48096, 60528, 456, 17640, 8496, 10560
OFFSET
2,3
COMMENTS
For k = 1 x(j) = 0 for all j > 1, so T(n,1) = 1 for n >= 2.
For k = 2 x(j) is the periodic sequence 0, 1, 1, 0, 1, 1, ... (A011655), so T(n,2) = 3 for n >= 3.
LINKS
Pontus von Brömssen, Rows n = 2..24, flattened
EXAMPLE
Triangle begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 11
---------------------------------------------------------------------------
2: 1 1
3: 3 1 3
4: 16 1 3 24
5: 6 1 3 13 6
6: 100 1 3 8 252 120
7: 24 1 3 8 42 119 96
8: 576 1 3 60 588 378 36 624
9: 1932 1 3 126 600 144 381 200 936
10: 912 1 3 480 51 9 2760 6220 540 1800
11: 5700 1 3 170 750 2480 14880 10990 300 1440 3660
12: 840 1 3 13800 5880 432 48096 60528 456 17640 8496 10560
T(10,0) = 912 because A243063 eventually enters a cycle of length 912.
PROG
(PARI) isok(v) = {for (n=1, #v-2, if ((v[#v] == v[#v - n]) && (v[#v-1] == v[#v - n - 1]), return (n)); ); 0; }
f(x, y, n, k) = {my(z=x+y, d = digits(z, n)); fromdigits(select(t->(t!=k), d), n); }
T(n, k) = {my(v = [0, 1], len = 2); while (! (per = isok(v)), v = concat(v, f(v[len-1], v[len], n, k)); len++; ); per; } \\ Michel Marcus, May 01 2019
(Python)
# Note: the function hangs if the sequence never enters a cycle.
import functools, sympy
def drop(x, n, k): return functools.reduce(lambda x, j:n*x+j if j!=k else x, sympy.ntheory.factor_.digits(x, n)[1:], 0) # Drop all digits k from x in base n.
def A306773(n, k): return next(sympy.cycle_length(lambda x:(x[1], drop(x[0]+x[1], n, k)), (0, 1)))[0]
# Pontus von Brömssen, May 09 2019
CROSSREFS
Sequence in context: A179658 A112811 A197272 * A276639 A361840 A361839
KEYWORD
nonn,base,tabf
AUTHOR
STATUS
approved