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A325664 First term of n-th difference sequence of (floor[k*r]), r = sqrt(2), k >= 0.

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