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A084639
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Expansion of x*(1+2*x)/((1+x)*(1-x)*(1-2*x)).
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9
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0, 1, 4, 9, 20, 41, 84, 169, 340, 681, 1364, 2729, 5460, 10921, 21844, 43689, 87380, 174761, 349524, 699049, 1398100, 2796201, 5592404, 11184809, 22369620, 44739241, 89478484, 178956969, 357913940, 715827881, 1431655764, 2863311529, 5726623060, 11453246121
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OFFSET
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0,3
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COMMENTS
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Original name was: Generalized Jacobsthal numbers.
This is the sequence A(0,1;1,2;3) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. [Wolfdieter Lang, Oct 18 2010]
Entries correspond to value bound adjustment for an N-bit string having M bits set and a(n+1) bit transitions. Wolfram Alpha can easily generate an entry. a(5)=41 stems from input as 1111110 base 2 - 1010101 base 2. The subtraction pattern alternates (begins at 1), and bit count is ptr+2 both terms, with the lead term having only its LSB clear. [Bill McEachen, Jul 15 2011]
Also a(n) = 2*A000975(n) if n even, a(n) = 2*A000975(n)- 1 if n odd. - Michel Lagneau, Jan 11 2012
In the above comment by Bill McEachen the binary pattern (in an obvious notation) is for even n 1^(n+1)0 - (10)^((n+2)/2) and for odd n 1^(n+1)0 - (10)^((n+1)/2)1. That is for even n a(n) = sum(2^k, k=1..(n+1)) - sum(2^(2*k-1), k=1..(n+2)/2) = (2^(n+2) - 4)/3, and for odd n a(n) = sum(2^k , k=1..(n+1)) - sum(2^(2*k), k=0..(n+1)/2) = (2^(n+2) - 5)/3. This checks with the formula a(n) = (2^(n+3) + (-1)^n - 9)/6 given below. After a correspondence with Bill McEachen. - Wolfdieter Lang, Jan 24 2014
Michel Lagneau's comment above is equal to the fact that a(n) = A000975(n)-1, or in other words, this sequence gives the partial sums of Jacobsthal sequence, starting from its second 1, A001045(2). From this also follows that this sequence gives the positions of repunits in "Jacobsthal greedy base", A265747. - Antti Karttunen, Dec 17 2015
From Kensuke Matsuoka, Aug 11 2020: (Start)
This sequence is the sum of diagonally arranged powers of 2 repeated in an L shape. For example, a(1)=1, a(2) = 4, a(3)=9, a(4)= 20, a(5)=41, a(6)=84 are obtained from the figure below.
32
16 8
8 4 2
4 2 1 2
2 1 2 4 8
1 2 4 8 16 32
From this figure, a(n) = a(n-2) + 2^n is obtained. (End)
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart. 34, 40-54, 1996.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).
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FORMULA
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G.f.: x*(1+2*x)/((1+x)*(1-x)*(1-2*x)).
E.g.f.: 4*exp(2*x)/3-3*exp(x)/2+exp(-x)/6.
a(n) = a(n-1)+2*a(n-2)+3, a(0)=0, a(1)=1.
a(n) = 2^(n+2)/3+(-1)^n/6-3/2.
a(n) = A001045(n+2) - A000034(n).
a(n) = 5*a(n-2)-4*a(n-4). Cf. A084640, A101622. - Paul Curtz, Apr 03 2008
a(n) = 2*a(n-1) + a(n-2) -2*a(n-3). - R. J. Mathar, Jun 28 2010
a(n) = a(n-1)+2*a(n-2)+3, n>1. - Gary Detlefs, Dec 19 2010
a(n) = 3*a(n-1)-2*a(n-2) +(-1)^n, n>1. - Gary Detlefs, Dec 19 2010
a(n) = a(n-2) + 2^n for n >= 2. - Kensuke Matsuoka, Aug 11 2020
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MAPLE
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a:=proc(n) (2^(n+3) + (-1)^n - 9)/6 end proc: [seq(a(n), n=0..33)]; # Wolfdieter Lang, Jan 24 2014
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MATHEMATICA
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a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n - 1] + 2 a[n - 2] + 3; Array[a, 32, 0] (* Or *)
a[0] = 0; a[1] = 1; a[n_] := a[n] = 3 a[n - 1] - 2 a[n - 2] + (-1)^n; Array[a, 32, 0]
CoefficientList[Series[x*(1+2*x)/((1+x)*(1-x)*(1-2*x)), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 1, -2}, {0, 1, 4}, 40] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)
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PROG
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(MAGMA) [2^(n+2)/3+(-1)^n/6-3/2: n in [0..35]]; // Vincenzo Librandi, Aug 08 2011
(PARI) a(n)=2^(n+2)/3-if(n%2, 5, 4)/3 \\ Charles R Greathouse IV, Aug 08 2011
(PARI) concat(0, Vec(x*(1+2*x)/((1+x)*(1-x)*(1-2*x)) + O(x^100))) \\ Altug Alkan, Dec 17 2015
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CROSSREFS
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Cf. A000975, A000225, A001045.
Cf. A265747.
Sequence in context: A023607 A117074 A072934 * A272268 A284736 A051136
Adjacent sequences: A084636 A084637 A084638 * A084640 A084641 A084642
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, Jun 06 2003
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EXTENSIONS
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Replaced duplicate of a formula by another recurrence - R. J. Mathar, Jun 28 2010
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STATUS
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approved
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