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A052101 One of the three sequences associated with the polynomial x^3 - 2. 8
1, 1, 1, 3, 9, 21, 45, 99, 225, 513, 1161, 2619, 5913, 13365, 30213, 68283, 154305, 348705, 788049, 1780947, 4024809, 9095733, 20555613, 46454067, 104982561, 237252321, 536171481, 1211705163, 2738358009, 6188472981, 13985460405 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

If x^3 = 2 and n >= 0, then there are unique integers a, b, c such that (1 + x)^n = a + b*x + c*x^2. The coefficient a is a(n).

REFERENCES

R. Schoof, Catalan's Conjecture, Springer-Verlag, 2008, pp. 17-18.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

A. Kumar Gupta and A. Kumar Mittal, Integer Sequences associated with Integer Monic Polynomial, arXiv:math/0001112 [math.GM], Jan 2000.

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

FORMULA

a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3).

a(n)/a(n-1) tends to 2.259921049... = 1 + 2^(1/3) (a real root to (x - 1)^3 = 2 or x^3 - 3x^2 + 3x - 3 = 0). A 3 X 3 matrix corresponding to the latter polynomial is [0 1 0 / 0 0 1 / 3 -3 3]. Let the matrix = M. Then a(n) = the center term in M^n * [1, 1, 1]. M^[1, 1, 1] = [9, 21, 45], center term = a(4) - Gary W. Adamson, Mar 28 2004

a(n) = Sum_{0..floor(n/3)}, 2^k * binomial(n, 3*k). - Ralf Stephan, Aug 30 2004

From Paul Curtz, Mar 10 2008: (Start)

Equals the first differences of A052102.

Equals the second differences of A052103.

Equals the binomial transform of A077959.

a(n) = 4*a(n-1) - 6*a(n-2) + 6*a(n-3) - 3*a(n-4).

A052103 is binomial transform of c(n)=0, 1, 1, 0, 2, 2, 0, 4, 4, 0, 8, 8, ... b(n+1) - 2*b(n) is essentially 3*b(n). (End)

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

EXAMPLE

From the Schoof reference, pp. 17, 18: Set pi = 1 + sqrt[3]{2}. For every integer k >= 0, there are unique a_k,b_k,c_k in Q such that pi^k = a_k + b_k sqrt[3]{2} + c_k sqrt[3]{4}. The coefficients a_k,b_k,c_k are actually in Z:

     Coefficients a_k, b_k, c_k:

     k      0     1     2     3     4     5      6

     ----------------------------------------------

     a_k    1     1     1     3     9     21     45

     b_k    0     1     2     3     6     15     36

     c_k    0     0     1     3     6     12     27

     ----------------------------------------------

G.f. = 1 + x + x^2 + 3*x^3 + 9*x^4 + 21*x^5 + 45*x^6 + 99*x^7 + 225*x^8 + ...

MAPLE

A052101 := n -> add(2^j*binomial(n, 3*j), j = 0..floor(n/3));

seq(A052101(n), n = 0..40); # G. C. Greubel, Apr 15 2021

MATHEMATICA

LinearRecurrence[{3, -3, 3}, {1, 1, 1}, 31] (* Ray Chandler, Sep 23 2015 *)

PROG

(PARI) {a(n) = polcoeff( lift( Mod(1 + x, x^3 - 2)^n ), 0)} /* Michael Somos, Aug 05 2009 */

(PARI) {a(n) = sum(k=0, n\3, 2^k * binomial(n, 3*k))} /* Michael Somos, Aug 05 2009 */

(PARI) {a(n) = if( n<0, 0, polcoeff( (1 - x)^2 / (1 - 3*x + 3*x^2 - 3*x^3) + x * O(x^n), n))} /* Michael Somos, Aug 05 2009 */

(Magma) [n le 3 select 1 else 3*(Self(n-1) -Self(n-2) +Self(n-3)): n in [1..41]]; // G. C. Greubel, Apr 15 2021

(Sage) [sum(2^j*binomial(n, 3*j) for j in (0..n//3)) for n in (0..40)] # G. C. Greubel, Apr 15 2021

CROSSREFS

Cf. A052102, A052103.

Sequence in context: A107351 A068156 A166452 * A063830 A062444 A141156

Adjacent sequences:  A052098 A052099 A052100 * A052102 A052103 A052104

KEYWORD

nonn,easy

AUTHOR

Ashok K. Gupta and Ashok K. Mittal (akgjkiapt(AT)hotmail.com), Jan 20 2000

STATUS

approved

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Last modified June 22 06:39 EDT 2021. Contains 345373 sequences. (Running on oeis4.)