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A088207
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a(n)=sum(k=0,n,floor(k*phi^2)) where phi=(1+sqrt(5))/2.
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1
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0, 2, 7, 14, 24, 37, 52, 70, 90, 113, 139, 167, 198, 232, 268, 307, 348, 392, 439, 488, 540, 594, 651, 711, 773, 838, 906, 976, 1049, 1124, 1202, 1283, 1366, 1452, 1541, 1632, 1726, 1822, 1921, 2023, 2127, 2234, 2343, 2455, 2570, 2687, 2807, 2930, 3055, 3183
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Partial sums of A001950.
A001950 is the upper Beatty sequence for the constant Phi^2, where Phi = (1 + sqrt(5))/2 and the sequence is generated by floor(n*Phi). A054347 = partial sums of the lower Beatty sequence (A000201). Conjecture: a(n)/A054347(n) tends to Phi. Example: a(28)/A054347(28) = 1049/643 = 1.6314...
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FORMULA
| a(n) = Sum(1, n) floor(n*Phi^2)
a(n) = floor( n*(n+1)/2*phi^2- n/2) +0 or +1. - Benoit Cloitre, Sep 27 2003
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EXAMPLE
| A001950(1) = 2, then 5, 7, 10, 13...; partial sums are 2, 7, 14, 24, 37...
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MATHEMATICA
| a[0] = 0; a[n_] := a[n] = (a[n - 1] + Floor[n*(1 + Sqrt[5])^2/4]); Table[ a[n], {n, 1, 50}] (from Robert G. Wilson v)
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CROSSREFS
| Cf. A001950, A054347, A000201.
Sequence in context: A018392 A051640 A119354 * A194111 A102999 A014112
Adjacent sequences: A088204 A088205 A088206 * A088208 A088209 A088210
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KEYWORD
| nonn
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 23 2003
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com) and Benoit Cloitre, Sep 27 2003
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