A140683 -id:A140683 - cp's OEIS frontend

A140657 Powers of 2 with 3 alternatingly added and subtracted.

4, -1, 7, 5, 19, 29, 67, 125, 259, 509, 1027, 2045, 4099, 8189, 16387, 32765, 65539, 131069, 262147, 524285, 1048579, 2097149, 4194307, 8388605, 16777219, 33554429, 67108867, 134217725, 268435459, 536870909, 1073741827, 2147483645, 4294967299, 8589934589

Offset: 0

Paul Curtz, Jul 10 2008

Jean-François Alcover and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 101 terms from Jean-François Alcover)
Index entries for linear recurrences with constant coefficients, signature (1, 2).

Cf. A000079, A007283, A010734, A033999, A096045, A140683.

[2^n+3*(-1)^n: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011

LinearRecurrence[{1,2},{4,-1},40] (* or *) Total/@Partition[Riffle[ Table[ 2^n, {n,0,40}],{3,-3}],2] (* Harvey P. Dale, Nov 13 2014 *)
CoefficientList[Series[(4 - 5 x) / ((1 + x) (1 - 2 x)), {x, 0, 50}], x] (* Vincenzo Librandi, Jan 14 2015 *)

a(2n) = A000079(2n+1) + 3, a(2n+1) = A000079(2n+2) - 3.
a(n+1) - 2*a(n) = -9*A033999(n) = (-1)^(n+1)*A010734.
a(n) + a(n+1) = 3^*2^n = A007283(n).
a(2n) + a(2n+1) = A096045(n) + 2.
a(-n) = -A140683(n)/2^n.
O.g.f.: (4-5*x)/((1-2*x)(1+x)). - R. J. Mathar, Jul 29 2008
a(n) = 2^n+3*(-1)^n. - R. J. Mathar , Jul 29 2008

Edited and extended by R. J. Mathar, Jul 29 2008

4 inserted as first term and formulas accordingly updated by Jean-François Alcover, Jan 14 2015

A280345 a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2].

Original entry on oeis.org

3, 7, 12, 25, 48, 97, 192, 385, 768, 1537, 3072, 6145, 12288, 24577, 49152, 98305, 196608, 393217, 786432, 1572865, 3145728, 6291457, 12582912, 25165825, 50331648, 100663297, 201326592, 402653185, 805306368, 1610612737, 3221225472, 6442450945, 12884901888

Offset: 0

Paul Curtz, Jan 01 2017

a(n) mod 9 is a periodic sequence of length 2: repeat [3, 7].
From 7, the last digit is of period 4: repeat [7, 2, 5, 8].
(Main sequence for the signature (2,1,-2): 0, 0, 1, 2, 5, 10, 21, 42, ... = 0 followed by A000975(n) = b(n), which first differences are A001045(n) (Paul Barry, Oct 08 2005). Then, 0 followed by b(n) is an autosequence of the first kind. The corresponding autosequence of the second kind is 0, 0, 2, 3, 8, 15, 32, 63, ... . See A277078(n).)
Difference table of a(n):
3,   7, 12, 25, 48,  97, 192, ...
4,   5, 13, 23, 49,  95, 193, ...  = -(-1)^n* A140683(n)
1,   8, 10, 26, 46,  98, 190, ...  = A259713(n)
7,   2, 16, 20, 52,  92, 196, ...
-5, 14,  4, 32, 40, 104, 184, ...
... .

			a(0) = 3, a(1) = 2*3 + 1 = 7, a(2) = 2*7 - 2 = 12, a(3) = 2*12 + 1 = 25.

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

Cf. A005010, A051049, A140657, A140683, A164346, A199116, A259713.

a[0] = 3; a[n_] := a[n] = 2 a[n - 1] + 1 + (-3) Boole[EvenQ@ n]; Table[a@ n, {n, 0, 32}] (* or *)
CoefficientList[Series[(3 + x - 5 x^2)/((1 - x) (1 + x) (1 - 2 x)), {x, 0, 32}], x] (* Michael De Vlieger, Jan 01 2017 *)

Vec((3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Jan 01 2017

a(2n) = 3*4^n, a(2n+1) = 6*4^n + 1.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n>2.
a(n+2) = a(n) + 9*2^n.
a(n) = 2^(n+2) - A051049(n).
From Colin Barker, Jan 01 2017: (Start)
a(n) = 3*2^n for n even.
a(n) = 3*2^n + 1 for n odd.
G.f.: (3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
(End)
Binomial transform of 3, followed by (-1)^n* A140657(n).

More terms from Colin Barker, Jan 01 2017

A268741 a(n) = 2*a(n - 2) - a(n - 1) for n>1, a(0) = 4, a(1) = 5.

Original entry on oeis.org

4, 5, 3, 7, -1, 15, -17, 47, -81, 175, -337, 687, -1361, 2735, -5457, 10927, -21841, 43695, -87377, 174767, -349521, 699055, -1398097, 2796207, -5592401, 11184815, -22369617, 44739247, -89478481, 178956975, -357913937, 715827887, -1431655761, 2863311535

Offset: 0

Ilya Gutkovskiy, Feb 12 2016

In general, the ordinary generating function for the recurrence relation b(n) = 2*b(n - 2) - b(n - 1) with n>1 and b(0)=k, b(1)=m, is (k + (k + m)*x)/(1 + x - 2*x^2). This recurrence gives the closed form a(n) = ((-2)^n*(k - m) + 2*k + m).

			a(0) = (5 + 3)/2 = 4  because a(1) = 5, a(2) = 3;
a(1) = (3 + 7)/2 = 5  because a(2) = 3, a(3) = 7;
a(2) = (7 - 1)/2 = 3  because a(3) = 7, a(4) = -1, etc.

Ilya Gutkovskiy, Extended graphical example
Index entries for linear recurrences with constant coefficients, signature (-1,2).

Cf. A084247, A140683, A140966, A154570, A171382.

Cf. A001045, A010709, A077925.

[(13-(-2)^n)/3: n in [0..35]]; // Vincenzo Librandi, Feb 13 2016

Table[(13 - (-2)^n)/3, {n, 0, 33}]
LinearRecurrence[{-1, 2}, {4, 5}, 34]
RecurrenceTable[{a[1] == 4, a[2] == 5, a[n] == 2*a[n-2] - a[n-1]}, a, {n, 50}] (* Vincenzo Librandi, Feb 13 2016 *)

Vec((4 + 9*x)/(1 + x - 2*x^2) + O(x^40)) \\ Michel Marcus, Feb 25 2016

G.f.: (4 + 9*x)/(1 + x - 2*x^2).
a(n) = (13 - (-2)^n)/3.
a(n) = A084247(n) + 3.
a(n) = (-1)^n*A154570(n+1) + 1.
a(n) = (-1)^(n-1)*A171382(n-1) + 2.
Limit_{n -> oo} a(n)/a(n + 1) = -1/2.
a(n) = 4 - (-1)^n *A001045(n). - Paul Curtz, Feb 26 2024

A355668 Array read by upwards antidiagonals T(n,k) = J(k) + n*J(k+1) where J(n) = A001045(n) is the Jacobsthal numbers.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 3, 3, 4, 3, 4, 4, 7, 8, 5, 5, 5, 10, 13, 16, 11, 6, 6, 13, 18, 27, 32, 21, 7, 7, 16, 23, 38, 53, 64, 43, 8, 8, 19, 28, 49, 74, 107, 128, 85, 9, 9, 22, 33, 60, 95, 150, 213, 256, 171, 10, 10, 25, 38, 71, 116, 193, 298, 427, 512, 341

Offset: 0

Paul Curtz, Jul 13 2022

			Row n=0 is A001045(k), then for further rows we successively add A001045(k+1).
       k=0  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9 k=10
  n=0:  0    1    1    3    5   11   21   43   85  171 ... = A001045
  n=1:  1    2    4    8   16   32   64  128  256  512 ... = A000079
  n=2:  2    3    7   13   27   53  107  213  427  853 ... = A048573
  n=3:  3    4   10   18   38   74  150  298  598 1194 ... = A171160
  n=4:  4    5   13   23   49   95  193  383  769 1535 ... = abs(A140683)
  ...

Cf. A001477, A000027, A016777, A016885, A017449, A321373.

Antidiagonal sums give A320933(n+1).

T[n_, k_] := (2^k - (-1)^k + n*(2^(k + 1) + (-1)^k))/3; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 13 2022 *)

T(n, k) = (2^k - (-1)^k + n*(2^(k + 1) + (-1)^k))/3.

G.f.: (x*(y-1) - y)/((x - 1)^2*(y + 1)*(2*y - 1)). - Stefano Spezia, Jul 13 2022

cp's OEIS Frontend

A140657 Powers of 2 with 3 alternatingly added and subtracted.

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A280345 a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2].

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A268741 a(n) = 2*a(n - 2) - a(n - 1) for n>1, a(0) = 4, a(1) = 5.

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A355668 Array read by upwards antidiagonals T(n,k) = J(k) + n*J(k+1) where J(n) = A001045(n) is the Jacobsthal numbers.

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