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

Author

Paul Curtz, Jul 13 2007

Keywords

• nonn
• easy

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).

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).

Programs

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

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 *)

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

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).

Extensions

Edited and extended by Klaus Brockhaus, Aug 03 2007