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A094373 Expansion of (1-x-x^2)/((1-x)*(1-2*x)). 25
1, 2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297, 8589934593 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Partial sum of 1,1,1,2,4,8,...

Binomial transform of abs(A073097).

Binomial transform is A094374.

Partial sums are in A006127. - Paul Barry, Aug 05 2004

An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 2, 8, 32 and 128, lead to this sequence. For the central square these vectors lead to the companion sequence A011782. [Johannes W. Meijer, Aug 15 2010]

This sequence has a(0) = 1 and for all n > 0, a(n) = 2^(n-1)+1. Consequently 2*a(n) >= a(n+1) for all n > 0 and the sequence is complete. [Frank M Jackson, Jan 29 2012]

Row lengths of the triangle in A198069. - Reinhard Zumkeller, May 26 2013

LINKS

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

Eric Weisstein, Complete Sequence.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

FORMULA

a(n) = (2^n-0^n)/2 + 1.

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

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

Row sums of triangle A135225. - Gary W. Adamson, Nov 23 2007

a(n) = A131577(n) + 1. - Paul Curtz, Aug 07 2008

a(n) = 2*a(n-1)-1 for n>1, a(0)=1, a(1)=2. [Philippe Deléham, Sep 25 2009]

E.g.f.: exp(x)*(1 + sinh(x)). - Arkadiusz Wesolowski, Aug 13 2012

G.f.: G(0), where G(k)= 1 + 2^k*x/(1 - x/(x + 2^k*x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2013

a(n) = 2^(n-1)+1 = A000051(n-1) for n>0. - M. F. Hasler, Sep 22 2013

EXAMPLE

G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 9*x^4 + 17*x^5 + 33*x^6 + 65*x^7 + ...

MATHEMATICA

CoefficientList[Series[(1 - x - x^2)/((1 - x)*(1 - 2*x)), {x, 0, 40}], x] (* or *) Join[{1}, LinearRecurrence[{3, -2}, {2, 3}, 40]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2012 *)

a[ n_] := If[ n < 0, 0, 1 + Quotient[ 2^n, 2]]; (* Michael Somos, May 26 2014 *)

a[ n_] := SeriesCoefficient[ (1 - x - x^2) / ((1 - x) (1 - 2 x)), {x, 0, n}]; (* Michael Somos, May 26 2014 *)

LinearRecurrence[{3, -2}, {1, 2, 3}, 40] (* Harvey P. Dale, Aug 09 2015 *)

PROG

(MAGMA) [(2^n-0^n)/2+1: n in [0..40]]; // Vincenzo Librandi, Jun 10 2011

(PARI) a(n)=2^n\2+1 \\ Charles R Greathouse IV, Apr 05 2013

(PARI) Vec((1-x-x^2)/((1-x)*(1-2*x))+O(x^99)) \\ Charles R Greathouse IV, Apr 05 2013

CROSSREFS

Apart from the initial 1, identical to A000051.

Cf. A135225.

Column k=1 of A152977.

Row n=2 of A238016.

Sequence in context: A109740 A248155 A000051 * A213705 A061902 A166286

Adjacent sequences:  A094370 A094371 A094372 * A094374 A094375 A094376

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Apr 28 2004

STATUS

approved

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Last modified June 29 06:39 EDT 2017. Contains 288859 sequences.