|
%I #16 Jun 18 2019 13:36:22
%S 1,0,1,-3,7,-15,30,-55,90,-125,125,0,-450,1625,-4250,9500,-18999,
%T 34357,-55454,75735,-70890,-26333,379049,-1352078,3713650,-9000225,
%U 20136806,-42409968,84819937,-161567265,292710630,-501416815,801992970,-1167081365,1453179125
%N First term of n-th difference sequence of (floor[k*r]), r = sqrt(2), k >= 0.
%H Clark Kimberling, <a href="/A325664/b325664.txt">Table of n, a(n) for n = 1..200</a>
%F From _Robert Israel_, Jun 04 2019: (Start)
%F a(n) = Sum_{0<=k<=n} (-1)^(n-k)*binomial(n,k)*A001951(k).
%F G.f.: g(x) = (1+x)^(-1)*h(x/(1+x)) where h is the G.f. of A001951. (End)
%e The sequence (floor(k*r)) for k>=0: 0, 1, 2, 4, 5, 7, 8, 9, 11, 12, ...
%e 1st difference sequence: 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, ...
%e 2nd difference sequence: 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, ...
%e 3rd difference sequence: 1, -2, 2, -2, 1, 1, -2, 2, -2, 1, 1, -2, 2, ...
%e 4th difference sequence: -3, 4, -4, 3, 0, -3, 4, -4, 3, 0, -3, 4, -4, ...
%e 5th difference sequence: 7, -8, 7, -3, -3, 7, -8, 7, -3, -3, 7, -8, 7, ...
%p N:= 50: # for a(1)..a(N)
%p L:= [seq(floor(sqrt(2)*n),n=0..N)]: Res:= NULL:
%p for i from 1 to N do
%p L:= L[2..-1]-L[1..-2];
%p Res:= Res, L[1];
%p od:
%p Res; # _Robert Israel_, Jun 04 2019
%t Table[First[Differences[Table[Floor[Sqrt[2]*n], {n, 0, 50}], n]], {n, 1, 50}]
%Y Cf. A001951.
%Y Guide to related sequences:
%Y A325664, r = sqrt(2)
%Y A325665, r = -sqrt(2)
%Y A325666, r = sqrt(3)
%Y A325667, r = -sqrt(3)
%Y A325668, r = sqrt(5)
%Y A325669, r = -sqrt(5)
%Y A325670, r = sqrt(6)
%Y A325671, r = -sqrt(6)
%Y A325672, r = sqrt(7)
%Y A325673, r = -sqrt(7)
%Y A325674, r = sqrt(8)
%Y A325675, r = -sqrt(8)
%Y A325729, r = sqrt(1/2)
%Y A325730, r = sqrt(1/3)
%Y A325731, r = sqrt(2/3)
%Y A325732, r = sqrt(3/4)
%Y A325733, r = 1/2 + sqrt(2)
%Y A325734, r = e
%Y A325735, r = -e
%Y A325736, r = 2e
%Y A325737, r = 3e
%Y A325738, r = e/2
%Y A325739, r = Pi
%Y A325740, r = 2Pi
%Y A325741, r = Pi/2
%Y A325742, r = Pi/3
%Y A325743, r = Pi/4
%Y A325744, r = Pi/6
%Y A325745, r = tau = golden ratio = (1 + sqrt(5))/2
%Y A325746, r = -tau
%Y A325747, r = tau^2 = 1 + tau
%Y A325748, r = 1/e
%Y A325749, r = e/(e-1)
%Y A325750, r = (1+sqrt(3))/2
%Y A325751, r = log 2
%Y A325752, r = log 3
%K easy,sign
%O 1,4
%A _Clark Kimberling_, May 12 2019
|
