A130755 Binomial transform of periodic sequence (3, 1, 2).

3, 4, 7, 15, 32, 65, 129, 256, 511, 1023, 2048, 4097, 8193, 16384, 32767, 65535, 131072, 262145, 524289, 1048576, 2097151, 4194303, 8388608, 16777217, 33554433, 67108864, 134217727, 268435455, 536870912, 1073741825, 2147483649

Offset

0,1

Comments

The third sequence of "less twisted numbers"; this sequence, A130750 and A130752 form a "suite en trio" (cf. reference, p. 130).

First differences of A130752, second differences of A130750.

Sequence equals its third differences:

3 4 7 15 32 65 129 256 511 1023

1 3 8 17 33 64 127 255 512

2 5 9 16 31 63 128 257

3 4 7 15 32 65 129

References

P. Curtz, Exercise Book, manuscript, 1995.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..300 from Vincenzo Librandi)

Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Formula

G.f.: (3-5*x+4*x^2)/((1-2*x)*(1-x+x^2)).

a(0) = 3; a(1) = 4; a(2) = 7; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).

a(n) = 2^(n+1) + A128834(n+2).

a(0) = 3; for n > 0, a(n) = 2*a(n-1) + A057079(n+3).

Mathematica

CoefficientList[Series[(3-5*x+4*x^2)/((1-2*x)*(1-x+x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{3, -3, 2}, {3, 4, 7}, 30] (* G. C. Greubel, Jan 15 2018 *)

Prog

(Magma) m:=31; S:=[ [3, 1, 2][(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:=[3, 4, 7]; [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]=3; v[2]=4; v[3]=7; 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, 0, -1, -1, 0, 1][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)), A130750 (first differences), A130752 (second differences).

Sequence in context: A329666 A187493 A027020 * A361635 A286348 A116090

Adjacent sequences: A130752 A130753 A130754 * A130756 A130757 A130758

Keyword

nonn,easy

Author

Paul Curtz, Jul 13 2007

Extensions

Edited and extended by Klaus Brockhaus, Aug 03 2007

Status

approved