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A048739 Expansion of 1/((1-x)*(1-2*x-x^2)). 60
1, 3, 8, 20, 49, 119, 288, 696, 1681, 4059, 9800, 23660, 57121, 137903, 332928, 803760, 1940449, 4684659, 11309768, 27304196, 65918161, 159140519, 384199200, 927538920, 2239277041, 5406093003, 13051463048, 31509019100, 76069501249 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Partial sums of Pell numbers A000129.

W(n){1,3;2,-1,1} = Sum[ i=1 to n ]W(i){1,2;2,-1,0}; where W(n){a,b; p,q,r} implies x(n)=p*x(n-1) - q*x(n-2) + r; x(0)= a, x(1)= b.

Number of 2 X (n+1) binary arrays with path of adjacent 1's from upper left to lower right corner. - R. H. Hardin, Mar 16 2002

Binomial transform of A029744. - Paul Barry, Apr 23 2004

Number of (s(0), s(1), ..., s(n+2)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n+2, s(0) = 1, s(n+2) = 3. - Herbert Kociemba, Jun 16 2004

Equals row sums of triangle A153346. - Gary W. Adamson, Dec 24 2008

Equals the sum of the terms of the antidiagonals of A142978. - J. M. Bergot, Nov 13 2012

a(p-2) == 0 mod p where p is an odd prime, see A270342. - Altug Alkan, Mar 15 2016

Also, the lexicographically earliest sequence of positive integers such that for n>3, {sqrt(2)*a(n)} is located strictly between {sqrt(2)*a(n-1)} and {sqrt(2)*a(n-2)} where {} denotes the fractional part. - Ivan Neretin, May 02 2017

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

M. Bicknell, A Primer on the Pell Sequence and related sequences, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.

M. Bicknell-Johnson and G. E. Bergum, The Generalized Fibonacci Numbers {C(n)}, C(n)=C(n-1)+C(n-2)+K, Applications of Fibonacci Numbers, 1986, pp. 193-205.

A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1065

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1)

FORMULA

a(n) = 2*a(n-1)+a(n-2)+1 with n>1, a(0)=1, a(1)=3.

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

a(0)=1, a(n+1) = ceil(x*a(n)) for n>0, where x=1+sqrt(2). - Paul D. Hanna, Apr 22 2003

a(n) = 3*a(n-1)-a(n-2)-a(n-3). With two leading zeros, e.g.f. is exp(x)(cosh(sqrt(2)x)-1)/2. a(n)=sum{k=0..floor((n+2)/2), binomial(n+2, 2k+2)2^k }. - Paul Barry, Aug 16 2003

-a(-3-n) = A077921(n). - N. J. A. Sloane, Sep 13 2003

E.g.f.: exp(x)(cosh(x/sqrt(2))+sqrt(2)sinh(x/sqrt(2)))^2. - N. J. A. Sloane, Sep 13 2003

a(n) = floor((1+sqrt(2))^(n+2)/4). - Bruno Berselli, Feb 06 2013

a(n) = (((1-sqrt(2))^(n+2) + (1+sqrt(2))^(n+2) - 2) / 4). - Altug Alkan, Mar 16 2016

MAPLE

a:=n->sum(fibonacci(i, 2), i=0..n): seq(a(n), n=1..29); # Zerinvary Lajos, Mar 20 2008

MATHEMATICA

Join[{a=1, b=3}, Table[c=2*b+a+1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)

CoefficientList[Series[1/(1-3x+x^2+x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, -1, -1}, {1, 3, 8}, 30] (* Harvey P. Dale, Jun 13 2011 *)

PROG

(PARI) a(n)=local(w=quadgen(8)); -1/2+(3/4+1/2*w)*(1+w)^n+(3/4-1/2*w)*(1-w)^n

(PARI) vector(100, n, n--; floor((1+sqrt(2))^(n+2)/4)) \\ Altug Alkan, Oct 07 2015

(PARI) Vec(1/((1-x)*(1-2*x-x^2)) + O(x^40)) \\ Michel Marcus, May 06 2017

CROSSREFS

First row of table A083087.

With a different offset, a(4n)=A008843(n), a(4n-2)=8*A001110(n), a(2n-1)=A001652(n).

Cf. A001333, A048654, A048655, A083044, A083047, A083050, A153346.

Sequence in context: A038746 A126876 A090757 * A054192 A124523 A054185

Adjacent sequences:  A048736 A048737 A048738 * A048740 A048741 A048742

KEYWORD

easy,nice,nonn

AUTHOR

Barry E. Williams

EXTENSIONS

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 11 2002

STATUS

approved

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Last modified May 23 19:46 EDT 2017. Contains 286926 sequences.