A083582 -id:A083582 - cp's OEIS frontend

A140966 a(n) = (5 + (-2)^n)/3.

2, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881, 1431655767, -2863311529, 5726623063

Offset: 0

Paul Curtz, Jul 27 2008

Inverse binomial transform of A048573.
This is an example of the case k=-1 of sequences with recurrences a(n) = k*a(n-1) + (k+3)*a(n-2) - (2*k+2)*a(n-3).
The case k=1 is covered, for example, by A097163, A135520, A136326, A136336, or A137208.
Sequences with k=2 are A094554 and A094555.
Sequences with k=3 are A084175, A108924, and A139818.

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

Cf. A048573.
Cf. A097163, A135520, A136326, A136336, A137208.
Cf. A094554, A094555.
Cf. A084175, A108924, A139818.
Cf. A000079, A001045, A078008, A083582, A083584, A163834.

[( 5+(-2)^n)/3: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

(5+(-2)^Range[0,30])/3 (* or *) LinearRecurrence[{-1,2},{2,1},40] (* Harvey P. Dale, Apr 23 2019 *)

a(n)=(5+(-2)^n)/3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = -a(n-1) + 2*a(n-2).
G.f.: (2+3*x)/((1-x)*(1+2*x)).
a(n+1) - a(n) = (-1)^(n+1)*A000079(n).
a(n+3) = (-1)^n*A083582(n).
a(n+1) - 2*a(n) = -a(n+2).
a(n+1) - 3*a(n) = 5*(-1)^(n+1)*A078008(n) = (-1)^(n+1)*A001045(n-1).
a(2n+3) = -A083584(n), a(2n) = A163834(n). - Philippe Deléham, Feb 24 2014
E.g.f.: (5*exp(x) + exp(-2*x))/3. - Stefano Spezia, Jul 27 2024

Definition simplified by R. J. Mathar, Sep 11 2009

A083581 a(n) = 8/3 - 5*(-2)^n/3.

Original entry on oeis.org

1, 6, -4, 16, -24, 56, -104, 216, -424, 856, -1704, 3416, -6824, 13656, -27304, 54616, -109224, 218456, -436904, 873816, -1747624, 3495256, -6990504, 13981016, -27962024, 55924056, -111848104, 223696216, -447392424, 894784856, -1789569704, 3579139416, -7158278824

Offset: 0

Paul Barry, May 01 2003

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

Cf. A083582.

[(8-5*(-2)^n)/3: n in [0..40]]; // Vincenzo Librandi, Aug 23 2014

Table[(8 - 5 (-2)^n)/3, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 7 x)/((1 - x) (1 + 2 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 23 2014 *)

a(n)=8/3-5*(-2)^n/3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (8-5(-2)^n)/3.
G.f.: (1+7x)/((1-x)(1+2x)).
E.g.f.: (8*exp(x)-5*exp(-2*x))/3.

A154570 The main diagonal of the successive differences of A154127.

Original entry on oeis.org

1, 3, -4, 2, -6, -2, -14, -18, -46, -82, -174, -338, -686, -1362, -2734, -5458, -10926, -21842, -43694, -87378, -174766, -349522, -699054, -1398098, -2796206, -5592402, -11184814, -22369618, -44739246, -89478482, -178956974, -357913938, -715827886

Offset: 0

Paul Curtz, Jan 12 2009

sign

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

Cf. A020988, A047849, A078008, A083582, A140966, A154127.

f[n_]:=2/(n+1);x=6;Table[x=f[x];Numerator[x],{n,0,5!}](* Absolute Values *) (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)

a(n) = a(n-1) + 2*a(n-2), n>0.
a(n+2) = 2*(-1)^(n+1)*A140966(n).
a(n+5) = -2*A083582(n).
a(2n+1) = 3 - A078008(2n) = 3 - A047849(n).
a(2n+2) = -4 - A078008(2n+1) = -4 - A020988(n).
G.f.: (1+2*x-9*x^2)/((1+x)*(1-2*x)). - R. J. Mathar, Feb 25 2009

Edited and extended by R. J. Mathar, Feb 25 2009

A171160 a(n) = a(n-1) + 2*a(n-2) with a(0)=3, a(1)=4.

Original entry on oeis.org

3, 4, 10, 18, 38, 74, 150, 298, 598, 1194, 2390, 4778, 9558, 19114, 38230, 76458, 152918, 305834, 611670, 1223338, 2446678, 4893354, 9786710, 19573418, 39146838, 78293674, 156587350, 313174698, 626349398, 1252698794, 2505397590, 5010795178, 10021590358

Offset: 0

Paul Curtz, Dec 04 2009

J. Mulder, Table of n, a(n) for n = 0..2999
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A001045, A078008, A175805.
Cf. A062510, A083582, (-1)^n*A140966.
Cf. A092297, A154879.
Cf. A062092, A083595.

f[n_]:=2/(n+1);x=5;Table[x=f[x];Numerator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
LinearRecurrence[{1,2},{3,4},40] (* Harvey P. Dale, Sep 04 2013 *)

Vec(-(x+3)/((x+1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Feb 10 2015

a(n) = (1/3)*(2*(-1)^n + 7*2^n), with n>=0. - Paolo P. Lava, Dec 14 2009
G.f.: -(x+3) / ((x+1)*(2*x-1)). - Colin Barker, Feb 10 2015
From Paul Curtz, Jun 03 2022: (Start)
a(n) = A078008(n) + A078008(n+1) + A078008(n+2).
a(n) = 2^(n+1) + A078008(n).
a(n) = A001045(n+3) - A001045(n).
(a(n) + a(n+1) = a(n+2) - a(n) = A005009(n).)
a(n) + a(n+3) = A175805(n).
a(n) = A062510(n) + A083582(n-1) with A083582(-1) = 3.
a(n) = A092297(n) + A154879(n). (End)
a(n) = 2*A062092(n-1), for n>0; 2*a(n) = A083595(n+1). - Paul Curtz, Jun 08 2022

Edited by N. J. A. Sloane, Dec 05 2009
More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010
More terms from Max Alekseyev, Apr 24 2010

A338198 Triangle read by rows, T(n,k) = ((k+1)2^(n-k)-(k-2)(-1)^(n-k))/3 for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 2, 3, 2, 1, 6, 5, 4, 3, 1, 10, 11, 8, 5, 4, 1, 22, 21, 16, 11, 6, 5, 1, 42, 43, 32, 21, 14, 7, 6, 1, 86, 85, 64, 43, 26, 17, 8, 7, 1, 170, 171, 128, 85, 54, 31, 20, 9, 8, 1, 342, 341, 256, 171, 106, 65, 36, 23, 10, 9, 1, 682, 683, 512, 341, 214, 127, 76, 41, 26, 11, 10, 1

Offset: 0

Werner Schulte, Oct 15 2020

This triangle is related to the Jacobsthal numbers (A001045).

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :    0     1     2    3    4    5    6   7   8   9
======================================================
  0 :    1
  1 :    0     1
  2 :    2     1     1
  3 :    2     3     2    1
  4 :    6     5     4    3    1
  5 :   10    11     8    5    4    1
  6 :   22    21    16   11    6    5    1
  7 :   42    43    32   21   14    7    6   1
  8 :   86    85    64   43   26   17    8   7   1
  9 :  170   171   128   85   54   31   20   9   8   1
etc.

For columns k = 0 to 8 see A078008, A001045, A000079, A001045, A084214, A014551, A083595, A083582, A259713 respectively.

Table[((k + 1)*2^(n - k) - (k - 2)*(-1)^(n - k))/3, {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 15 2020 *)

T(n,n) = 1 for n >= 0; T(n,n-1) = n-1 for n > 0.
T(n,k) = T(n-1,k) + 2 * T(n-2,k) for 0 <= k <= n-2.
T(n,k) = 2 * T(n-1,k) - (k-2) * (-1)^(n-k) for 0 <= k < n.
T(n,k) = T(n+1-k,1) + (k-1) * T(n-k,1) for 0 <= k < n.
T(n+1,k) * T(n-1,k) - T(n,k+1) * T(n,k-1) = T(n-k,1)^2 for 0 < k < n.
Row sums are A083579(n+1) for n >= 0.
G.f. of column k >= 0: (1+(k-1)*t) * t^k / (1-t-2*t^2).
G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = (1 - (1+x)*t + 2*x*t^2) / ((1 - x*t)^2 * (1 - t - 2*t^2)).
Conjecture: Let M(n,k) be the matrix inverse of T(n,k), seen as a matrix. Then M(i,j) = 0 if j < 0 or j > i, M(n,n) = 1 for n >= 0, M(n,n-1) = 1-n for n > 0, and M(n,k) = (-1)^(n-k) * (k^2-2) * (n-2)! / k! for 0 <= k <= n-2.

cp's OEIS Frontend

A140966 a(n) = (5 + (-2)^n)/3.

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A083581 a(n) = 8/3 - 5*(-2)^n/3.

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A154570 The main diagonal of the successive differences of A154127.

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A171160 a(n) = a(n-1) + 2*a(n-2) with a(0)=3, a(1)=4.

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A338198 Triangle read by rows, T(n,k) = ((k+1)*2^(n-k)-(k-2)*(-1)^(n-k))/3 for 0 <= k <= n.

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A338198 Triangle read by rows, T(n,k) = ((k+1)2^(n-k)-(k-2)(-1)^(n-k))/3 for 0 <= k <= n.