Search: a325741 -id:a325741
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A325664
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First term of n-th difference sequence of (floor[k*r]), r = sqrt(2), k >= 0.
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+10
37
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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
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OFFSET
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1,4
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LINKS
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FORMULA
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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)
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EXAMPLE
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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, ...
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MAPLE
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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:
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MATHEMATICA
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Table[First[Differences[Table[Floor[Sqrt[2]*n], {n, 0, 50}], n]], {n, 1, 50}]
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CROSSREFS
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Guide to related sequences:
A325745, r = tau = golden ratio = (1 + sqrt(5))/2
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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