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A130750
Binomial transform of A010882.
8
1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, 4096, 8191, 16383, 32768, 65537, 131073, 262144, 524287, 1048575, 2097152, 4194305, 8388609, 16777216, 33554431, 67108863, 134217728, 268435457, 536870913, 1073741824, 2147483647
OFFSET
0,2
COMMENTS
The first sequence of "less twisted numbers"; this sequence, A130752 and A130755 form a "suite en trio" (cf. reference, p. 130).
First differences of A130755, second differences of A130752.
Sequence equals its third differences:
1 3 8 17 33 64 127 255 512 1025
2 5 9 16 31 63 128 257 513
3 4 7 15 32 65 129 256
1 3 8 17 33 64 127
REFERENCES
P. Curtz, Exercise Book, manuscript, 1995.
FORMULA
G.f.: (1+2*x^2)/((1-2*x)*(1-x+x^2)).
a(0) = 1; a(1) = 3; a(2) = 8; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
a(n) = 2^(n+1) + A128834(n+4).
a(0) = 1; for n > 0, a(n) = 2*a(n-1) + A057079(n-1).
MATHEMATICA
CoefficientList[Series[(1+2*x^2)/((1-2*x)*(1-x+x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{3, -3, 2}, {1, 3, 8}, 30] (* G. C. Greubel, Jan 15 2018 *)
PROG
(Magma) m:=31; S:=[ [1, 2, 3][(n-1) mod 3 +1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; // Klaus Brockhaus, Aug 03 2007
(Magma) I:=[1, 3, 8]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018
(PARI) {m=31; v=vector(m); v[1]=1; v[2]=3; v[3]=8; for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} \\ Klaus Brockhaus, Aug 03 2007
(PARI) {for(n=0, 30, print1(2^(n+1)+[ -1, -1, 0, 1, 1, 0][n%6+1], ", "))} \\ Klaus Brockhaus, Aug 03 2007
CROSSREFS
Cf. A010882 (periodic (1, 2, 3)), A128834 (periodic (0, 1, 1, 0, -1, -1)), A057079 (periodic (1, 2, 1, -1, -2, -1)), A130752 (first differences), A130755 (second differences).
Sequence in context: A360848 A002625 A027181 * A281166 A165273 A002626
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Jul 13 2007
EXTENSIONS
Edited and extended by Klaus Brockhaus, Aug 03 2007
STATUS
approved