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A130755 Binomial transform of periodic sequence (3, 1, 2). 9
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 (list; graph; refs; listen; history; text; internal format)
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 * A286348 A116090 A287741

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

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Last modified May 14 04:40 EDT 2021. Contains 343872 sequences. (Running on oeis4.)