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

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

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

Sequence in context: A147400 A002545 A153114 * A290865 A055795 A058695

Adjacent sequences: A325661 A325662 A325663 * A325665 A325666 A325667

Keyword

easy,sign

Author

Clark Kimberling, May 12 2019

Status

approved