A010903 -id:A010903

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A010920

Pisot sequence T(3,13), a(n) = floor( a(n-1)^2/a(n-2) ).

+10
4

3, 13, 56, 241, 1037, 4462, 19199, 82609, 355448, 1529413, 6580721, 28315366, 121834667, 524227237, 2255632184, 9705479209, 41760499493, 179686059838, 773148800711, 3326685824041, 14313982718072

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305

D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

FORMULA

Empirical G.f.: (3-2*x)/(1-5*x+3*x^2). - Colin Barker, Feb 21 2012

Empirical: a(n) = 5*a(n-1)-3*a(n-2) with n>1, a(0)=3 and a(1)=13. - Vincenzo Librandi, Apr 17 2012

The empirical g.f. and recurrence above hold for n<=6000. - Bruno Berselli, Sep 03 2013

Note the warning in A010925 from Pab Ter (pabrlos(AT)yahoo.com), May 23 2004: [A010925] and other examples show that it is essential to reject conjectured generating functions for Pisot sequences until a proof or reference is provided. - N. J. A. Sloane, Jul 26 2016

MATHEMATICA

RecurrenceTable[{a[0] == 3, a[1] == 13, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 25}] (* Bruno Berselli, Sep 03 2013 *)

PROG

(Magma) I:=[3, 13]; [n le 2 select I[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..25]]; // Bruno Berselli, Sep 03 2013

CROSSREFS

Cf. A010925, A010903.

KEYWORD

nonn,easy

AUTHOR

Simon Plouffe

STATUS

approved

A095934

Expansion of (1-x)^2/(1-5*x+3*x^2).

+10
2

1, 3, 13, 56, 241, 1037, 4462, 19199, 82609, 355448, 1529413, 6580721, 28315366, 121834667, 524227237, 2255632184, 9705479209, 41760499493, 179686059838, 773148800711, 3326685824041, 14313982718072, 61589856118237, 265007332436969, 1140267093830134

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n) is the number of generalized compositions of n when there are i+2 different types of i, (i=1,2,...). [Milan Janjic, Sep 24 2010]

LINKS

Table of n, a(n) for n=0..24.

P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.

P. J. Cameron, Some sequences of integers, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.

Index entries for linear recurrences with constant coefficients, signature (5,-3).

FORMULA

a(n+2) = 5a(n+1) - 3a(n) (n >= 1); a(0) = 1, a(1) = 3, a(2) = 13.

MATHEMATICA

CoefficientList[Series[(1-x)^2/(1-5x+3x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{5, -3}, {1, 3, 13}, 30] (* Harvey P. Dale, Jun 21 2021 *)

PROG

(PARI) a(n)=polcoeff((1-x)^2/(1-5*x+3*x^2)+x*O(x^n), n)