OFFSET
1,4
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..200
FORMULA
From Robert Israel, Jun 04 2019: (Start)
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:
Res; # Robert Israel, Jun 04 2019
MATHEMATICA
Table[First[Differences[Table[Floor[Sqrt[2]*n], {n, 0, 50}], n]], {n, 1, 50}]
CROSSREFS
Cf. A001951.
Guide to related sequences:
A325664, r = sqrt(2)
A325665, r = -sqrt(2)
A325666, r = sqrt(3)
A325667, r = -sqrt(3)
A325668, r = sqrt(5)
A325669, r = -sqrt(5)
A325670, r = sqrt(6)
A325671, r = -sqrt(6)
A325672, r = sqrt(7)
A325673, r = -sqrt(7)
A325674, r = sqrt(8)
A325675, r = -sqrt(8)
A325729, r = sqrt(1/2)
A325730, r = sqrt(1/3)
A325731, r = sqrt(2/3)
A325732, r = sqrt(3/4)
A325733, r = 1/2 + sqrt(2)
A325734, r = e
A325735, r = -e
A325736, r = 2e
A325737, r = 3e
A325738, r = e/2
A325739, r = Pi
A325740, r = 2Pi
A325741, r = Pi/2
A325742, r = Pi/3
A325743, r = Pi/4
A325744, r = Pi/6
A325745, r = tau = golden ratio = (1 + sqrt(5))/2
A325746, r = -tau
A325747, r = tau^2 = 1 + tau
A325748, r = 1/e
A325749, r = e/(e-1)
A325750, r = (1+sqrt(3))/2
A325751, r = log 2
A325752, r = log 3
KEYWORD
easy,sign
AUTHOR
Clark Kimberling, May 12 2019
STATUS
approved