|
|
A194458
|
|
Total number of entries in rows 0,1,...,n of Pascal's triangle not divisible by 5.
|
|
2
|
|
|
1, 3, 6, 10, 15, 17, 21, 27, 35, 45, 48, 54, 63, 75, 90, 94, 102, 114, 130, 150, 155, 165, 180, 200, 225, 227, 231, 237, 245, 255, 259, 267, 279, 295, 315, 321, 333, 351, 375, 405, 413, 429, 453, 485, 525, 535, 555, 585, 625, 675, 678, 684, 693, 705, 720, 726
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
The number of zeros in the first n rows is binomial(n+1,2) - a(n).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = ((C(d0+1,2)*15^0*(d1+1) + C(d1+1,2)*15^1)*(d1+1) + C(d1+1,2)*15^1)*(d2+1) + C(d2+1,2)*15^2 ..., where d_k...d_1d_0 is the base 5 expansion of n+1 and 15 = binomial(5+1,2). The formula generalizes to other prime bases p.
|
|
EXAMPLE
|
n = 38: n+1 = 39 = 124_5, thus a(38) = (C(5,2)*15^0*3 + C(3,2)*15^1)*2 + C(2,2)*15^2 = (10*1*3 + 3*15)*2 + 1*225 = 375.
|
|
MAPLE
|
a:= proc(n) local l, m, h, j;
m:= n+1;
l:= [];
while m>0 do l:= [l[], irem (m, 5, 'm')+1] od;
h:= 0;
for j to nops(l) do h:= h*l[j] +binomial (l[j], 2) *15^(j-1) od:
h
end:
seq (a(n), n=0..100);
|
|
MATHEMATICA
|
a[n_] := Module[{l, m, r, h, j}, m = n+1; l = {}; While[m>0, l = Append[l, {m, r} = QuotientRemainder[m, 5]; r+1]]; h = 0; For[j = 1, j <= Length[l], j++, h = h*l[[j]] + Binomial [l[[j]], 2] *15^(j-1)]; h]; Table [a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 26 2017, translated from Maple *)
|
|
CROSSREFS
|
a(n+1) = a(n) + A194459(n+1) for p=5.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|