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First term of n-th difference sequence of (floor[k*r]), r = sqrt(2), k >= 0.
37

%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