|
|
A325664
|
|
First term of n-th difference sequence of (floor[k*r]), r = sqrt(2), k >= 0.
|
|
37
|
|
|
1, 0, 1, -3, 7, -15, 30, -55, 90, -125, 125, 0, -450, 1625, -4250, 9500, -18999, 34357, -55454, 75735, -70890, -26333, 379049, -1352078, 3713650, -9000225, 20136806, -42409968, 84819937, -161567265, 292710630, -501416815, 801992970, -1167081365, 1453179125
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Sum_{0<=k<=n} (-1)^(n-k)*binomial(n,k)*A001951(k).
G.f.: g(x) = (1+x)^(-1)*h(x/(1+x)) where h is the G.f. of A001951. (End)
|
|
EXAMPLE
|
The sequence (floor(k*r)) for k>=0: 0, 1, 2, 4, 5, 7, 8, 9, 11, 12, ...
1st difference sequence: 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, ...
2nd difference sequence: 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, ...
3rd difference sequence: 1, -2, 2, -2, 1, 1, -2, 2, -2, 1, 1, -2, 2, ...
4th difference sequence: -3, 4, -4, 3, 0, -3, 4, -4, 3, 0, -3, 4, -4, ...
5th difference sequence: 7, -8, 7, -3, -3, 7, -8, 7, -3, -3, 7, -8, 7, ...
|
|
MAPLE
|
N:= 50: # for a(1)..a(N)
L:= [seq(floor(sqrt(2)*n), n=0..N)]: Res:= NULL:
for i from 1 to N do
L:= L[2..-1]-L[1..-2];
Res:= Res, L[1];
od:
|
|
MATHEMATICA
|
Table[First[Differences[Table[Floor[Sqrt[2]*n], {n, 0, 50}], n]], {n, 1, 50}]
|
|
CROSSREFS
|
Guide to related sequences:
A325745, r = tau = golden ratio = (1 + sqrt(5))/2
|
|
KEYWORD
|
easy,sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|