A097074 -id:A097074 - cp's OEIS frontend

A097073 Expansion of (1-x+2x^2)/((1+x)(1-2*x)).

1, 0, 4, 4, 12, 20, 44, 84, 172, 340, 684, 1364, 2732, 5460, 10924, 21844, 43692, 87380, 174764, 349524, 699052, 1398100, 2796204, 5592404, 11184812, 22369620, 44739244, 89478484, 178956972, 357913940, 715827884, 1431655764, 2863311532

Offset: 0

Paul Barry, Jul 22 2004

Partial sums are A097074.

Pairwise sums are {1, 1, 4, 16, 32, ...} or 2^n -Sum_{k=0..n} binomial(n,k)*(-1)^(n+k)*k.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A046055, A097074.

Cf. A001045, A078008 (form a(n)=2^n-a(n-1)).

[2*2^n/3+4*(-1)^n/3-0^n: n in [0..35]]; // Vincenzo Librandi, Aug 12 2011

k=0;lst={1, k};Do[k=2^n-k;AppendTo[lst, k], {n, 2, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
CoefficientList[Series[(1-x+2x^2)/((1+x)(1-2x)),{x,0,40}],x] (* Harvey P. Dale, Dec 10 2012 *)

a(n)=([0,1; 2,1]^n*[1;0])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

def A097073(n): return ((1<Chai Wah Wu, Apr 18 2025

def A097073(n): return 1 if (n==0) else 2*(2^n +2*(-1)^n)/3
[A097073(n) for n in (0..40)] # G. C. Greubel, Aug 19 2022

a(n) = (2*2^n + 4*(-1)^n)/3 - 0^n.
a(n) = A001045(n+1) + (-1)^n - 0^n.
a(n) = 2*A078008(n) - 0^n.
a(2*n+1) + a(2*n+2) = A000302(n+1). - Paul Curtz, Jun 30 2008
G.f.: 1 - x + x*Q(0), where Q(k) = 1 + 2*x^2 + (4*k+5)*x - x*(4*k+1 + 2*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
E.g.f.: (1/3)*( 2*exp(2*x) + 4*exp(-x) - 3 ). - G. C. Greubel, Aug 19 2022

Obscure variable k in Orlovsky comment replaced with a(n) by R. J. Mathar, Apr 23 2009

A026644 a(n) = a(n-1) + 2*a(n-2) + 2, for n>=3, where a(0)= 1, a(1)= 2, a(2)= 4.

Original entry on oeis.org

1, 2, 4, 10, 20, 42, 84, 170, 340, 682, 1364, 2730, 5460, 10922, 21844, 43690, 87380, 174762, 349524, 699050, 1398100, 2796202, 5592404, 11184810, 22369620, 44739242, 89478484, 178956970, 357913940, 715827882, 1431655764, 2863311530, 5726623060

Offset: 0

Clark Kimberling

Number of moves to solve Chinese rings puzzle.
a(n-1) (with a(0):=0) enumerates all sequences of length m=1,2,...,floor(n/2) with nonzero integer entries n_i satisfying sum |n_i| <= n-m. Rephrasing K. A. Meissner's example p. 6. Example n=4: from length m=1: [1], [2], [3], each in 2 signed versions; from m=2: [1,1] in 2^2 = 4 signed versions. Hence a(3) = a(4-1) = 3*2 + 1*4 = 10.
Also the number of different 3-colorings (out of 4 colors) for the vertices of all triangulated planar polygons on a base with n+1 vertices if the colors of the two base vertices are fixed. - Patrick Labarque, Mar 23 2010
For n > 0, also the total distance that the disks travel from the leftmost peg to the rightmost peg in the Tower of Hanoi puzzle, in the unique solution with 2^n-1 moves (see links). - Sela Fried, Dec 17 2023

Richard I. Hess, Compendium of Over 7000 Wire Puzzles, privately printed, 1991.
Richard I. Hess, Analysis of Ring Puzzles, booklet distributed at 13th International Puzzle Party, Amsterdam, Aug 20 1993.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Thomas Baruchel, Properties of the cumulated deficient binary digit sum, arXiv:1908.02250 [math.NT], 2019.
Sela Fried, Economically solving the Tower of Hanoi puzzle.
Nicolas Gastineau and O. Togni, On S-packing edge-colorings of cubic graphs, arXiv preprint arXiv:1711.10906 [cs.DM], 2017.
Lee Hae-hwang, Illustration of initial terms in terms of rosemary plants
Krzysztof A. Meissner, Black hole entropy in Loop Quantum Gravity, arXiv:gr-qc/0407052, 2004.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Row sums of A026637.
For n >= 1, equals twice A000975, also A001045 - 1.
A167030 is an essentially identical sequence.

[n eq 0 select 1 else (2^(n+2) -3-(-1)^n)/3 : n in [0..40]]; // G. C. Greubel, Jun 28 2024

f:=n-> if n mod 2 = 0 then (2^(n+2)-4)/3 else (2^(n+2)-2)/3; fi;

Join[{1}, Floor[(2^Range[3, 40] - 2)/3]] (* or *) LinearRecurrence[{2,1,-2},{1,2,4,10},40] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
CoefficientList[Series[(1-x^2+2x^3)/((1-x)(1-x-2x^2)),{x,0,1001}],x] (* Vincenzo Librandi, Apr 04 2012 *)

Vec((1-x^2+2*x^3)/(1-x)/(1-x-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Apr 04 2012

def A026644(n): return ((4<Chai Wah Wu, Apr 17 2025

[(2^(n+2)-3-(-1)^n)/3 + int(n==0) for n in range(41)] # G. C. Greubel, Jun 28 2024

a(2*k) = 2*a(2*k-1), a(2*k+1) = 2*a(2*k) + 2. - Peter Shor, Apr 11 2002
For n>0: a(n+1) = a(n) + 2*b(n+1) + 4*b(n), where b(k) = A001045(k). - N. J. A. Sloane, May 16 2003
For n>0: if n mod 2 = 0 then (2^(n+2)-4)/3 else (2^(n+2)-2)/3. - Richard Hess
a(2*n) = 2*n-1 + Sum_{k=0..2*n-1} a(k), n>0; a(2*n+1) = 2*n+1 + Sum_{k=0..n} a(k). - Lee Hae-hwang, Sep 17 2002; corrected by R. J. Mathar, Oct 21 2008
a(n) = 2*n + 2*Sum_{k=1..n-2} a(k), n>0. - Lee Hae-hwang, Sep 19 2002; corrected by R. J. Mathar, Oct 21 2008
From Paul Barry, Oct 24 2007: (Start)
G.f.: (1 - x^2 + 2*x^3)/((1 - x)*(1 - x - 2*x^2)).
a(n) = J(n+2) - 1 + 0^n, where J(n) = A001045(n) (Jacobsthal numbers).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = 0^n + Sum_{k=0..n} (2 - 2*0^(n-k))*J(k+1). (End)
a(n) = A052953(n+1) - 2, n>0. [Moved from A020988, R. J. Mathar, Oct 21 2008]
a(n) = floor(A097074(n+1)/2), n>0. - Gary Detlefs, Dec 19 2010
a(n) = A169969(2*n-1) - 1, n>=2; a(n) = 3*2^(n-1) - 1 - A169969(2*n-7), n>=5 . - Yosu Yurramendi, Jul 05 2016
a(n+3) = 3*2^(n+2) - 2 - a(n), n>=1, a(1)=2, a(2)=4, a(3)=10 . - Yosu Yurramendi, Jul 05 2016
a(n) + A084170(n) = 3*2^n - 2, n>=1. - Yosu Yurramendi, Jul 05 2016
E.g.f: (3 - 4*cosh(x) + 4*cosh(2*x) - 2*sinh(x) + 4*sinh(2*x))/3. - Ilya Gutkovskiy, Jul 05 2016
a(n+3) = 9*2^n + A084170(n), n>=0. - Yosu Yurramendi, Jul 07 2016
a(n) = A000975(n+1) - A000035(n+1), n>0, a(0)=1. - Yuchun Ji, Aug 05 2020

Recurrence in definition line found by Lee Hae-hwang, Apr 03 2002

A020943 a(2n+1) = |a(2n) - a(2n-1)|, a(2n) = a(n) + a(n-1).

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 2, 2, 0, 1, 1, 2, 1, 4, 3, 4, 1, 2, 1, 1, 0, 2, 2, 3, 1, 3, 2, 5, 3, 7, 4, 7, 3, 5, 2, 3, 1, 3, 2, 2, 0, 1, 1, 2, 1, 4, 3, 5, 2, 4, 2, 4, 2, 5, 3, 7, 4, 8, 4, 10, 6, 11, 5, 11, 6, 10, 4, 8, 4, 7, 3, 5, 2, 4, 2, 4, 2, 5, 3, 4, 1, 2, 1, 1, 0, 2, 2, 3, 1, 3, 2, 5, 3, 7, 4, 8, 4, 7, 3, 6, 3, 6, 3

Offset: 1

Clark Kimberling

Presumably, the positions n such that a(n)=0 are the terms of A097074. - Ivan Neretin, Jul 06 2015

Ivan Neretin, Table of n, a(n) for n = 1..5461

Cf. A020947, A020948, A020949, A020950.

a = {0, 1, 1}; Do[AppendTo[a, a[[n]] + a[[n - 1]]]; AppendTo[a, Abs[a[[-1]] - a[[-2]]]], {n, 2, 51}]; a (* Ivan Neretin, Jul 06 2015 *)

More terms from Henry Bottomley, May 16 2001

A173114 a(0)=a(1)=1, a(n) = 2*a(n-1)- A010686(n), n>1.

Original entry on oeis.org

1, 1, 1, -3, -7, -19, -39, -83, -167, -339, -679, -1363, -2727, -5459, -10919, -21843, -43687, -87379, -174759, -349523, -699047, -1398099, -2796199, -5592403, -11184807, -22369619, -44739239, -89478483, -178956967, -357913939, -715827879, -1431655763

Offset: 0

Paul Curtz, Feb 10 2010

The sequence in the first row and successive differences in followup rows defines the array
  1, 1, 1, -3, -7, -19, -39, -83, -167, -339, ..
  0, 0, -4, -4, -12, -20, -44, -84, -172, -340, ..
  0, -4, 0, -8, -8, -24, -40, -88, -168, -344, ..
 -4, 4, -8, 0, -16, -16, -48, -80, -176, -336, ..
  8, -12, 8, -16, 0, -32, -32, -96, -160, -352, ..
The first two subdiagonals show essentially the powers of 2.

Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

a(n) = 3 + 2*( (-1)^n-2^n )/3 = 3-A078008(n+1), n>0. - R. J. Mathar, Jun 30 2010
a(n+2)-a(n)= A154589(n+2) = -2^(n+1), n>0.
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3), n>3.
G.f.: (-x-2*x^2-4*x^3+1)/( (1-x)*(1-2*x)*(1+x) ).
a(n) + A173078(n) = 2^n.
a(n) - a(n-1) = -4*A001045(n-2) = -A097074(n-1), n>1.

Edited and extended by R. J. Mathar, Jun 30 2010

A322783 a(n) = 1 - n + (2^(n+2) - (-1)^n)/3.

Original entry on oeis.org

2, 3, 4, 9, 18, 39, 80, 165, 334, 675, 1356, 2721, 5450, 10911, 21832, 43677, 87366, 174747, 349508, 699033, 1398082, 2796183, 5592384, 11184789, 22369598, 44739219, 89478460, 178956945, 357913914, 715827855, 1431655736

Offset: 0

Paul Curtz, Dec 26 2018

a(n) mod 10 = period 20: repeat [2, 3, 4, 9, 8, 9, 0, 5, 4, 5, 6, 1, 0, 1, 2, 7, 6, 7, 8, 3] = disordered [0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9].

Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A001045, A004442(n-2), A097073, A097074.

Vec((2 - 3*x - 3*x^2 + 6*x^3) / ((1 - x)^2*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 26 2018

a(n+1) - 2*(n) = -1, -2, 1, 0, 3, 2, 5, 4, ..., n >= 0.
a(n+1) - a(n) = A097074(n).
a(n+2) - 2*a(n+1) + a(n) = A097073(n+1).
From Colin Barker, Dec 26 2018: (Start)
G.f.: (2 - 3*x - 3*x^2 + 6*x^3) / ((1 - x)^2*(1 + x)*(1 - 2*x)).
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4) for n > 3.
(End)

cp's OEIS Frontend

A097073 Expansion of (1-x+2*x^2)/((1+x)*(1-2*x)).

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A026644 a(n) = a(n-1) + 2*a(n-2) + 2, for n>=3, where a(0)= 1, a(1)= 2, a(2)= 4.

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A020943 a(2n+1) = |a(2n) - a(2n-1)|, a(2n) = a(n) + a(n-1).

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A173114 a(0)=a(1)=1, a(n) = 2*a(n-1)- A010686(n), n>1.

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A322783 a(n) = 1 - n + (2^(n+2) - (-1)^n)/3.

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A097073 Expansion of (1-x+2x^2)/((1+x)(1-2*x)).