A171449 -id:A171449 - cp's OEIS frontend

A122803 Powers of -2: a(n) = (-2)^n.

1, -2, 4, -8, 16, -32, 64, -128, 256, -512, 1024, -2048, 4096, -8192, 16384, -32768, 65536, -131072, 262144, -524288, 1048576, -2097152, 4194304, -8388608, 16777216, -33554432, 67108864, -134217728, 268435456, -536870912, 1073741824, -2147483648, 4294967296, -8589934592, 17179869184

Offset: 0

Franklin T. Adams-Watters, Sep 11 2006

The number -2 can be used as a base of numeration (see the Weisstein link). - Alonso del Arte, Mar 30 2014
Contribution from M. F. Hasler, Oct 21 2014: (Start)
This is the inverse binomial transform of A033999 = n->(-1)^n, and the binomial transform of A033999*A000244 = n->(-3)^n, see also A141413.
Prefixed with one 0, i.e., (0,1,-2,4,...) = -A033999*A131577, it is the binomial transform of (0, 1, -4, 13, -40, 121,...) = -A033999*A003462, and inverse binomial transform of (0,1,0,1,0,1,...) = A000035.
Prefixed with two 0's, i.e., (0,0,1,-2,4,-8,...), it is the binomial transform of (0,0,1,-5,18,-58,179,-543,...) (cf. A000340) and inverse binomial transform of (0,0,1,1,2,2,3,3,...) = A004526. (End)
Prefixed with three 0's, this is the inverse binomial difference of (0, 0, 0, 1, 2, 4, 6, 9, 12, 16,...) = concat(0, A002620), which has as successive differences (0, 0, 1, 1, 2, 2,...) = A004526, then (0, 1, 0, 1,...) = A000035, then (1, -1, 1, -1,...) = A033999, and then (-2)^k*A033999 with k=1,2,3,... - Paul Curtz, Oct 16 2014, edited by M. F. Hasler, Oct 21 2014
Stirling-Bernoulli transform of triangular numbers: 1, 3, 6, 10, 15, 21, 28, ... - Philippe Deléham, May 25 2015

Franklin T. Adams-Watters, Table of n, (-2)^n for n = 0..1000
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Negabinary.
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A000079 (powers of 2), A004526, A005351, A005352, A011782, A034008, A131577, A171449, A248800.

[(-2)^n: n in [0..60]]; // Vincenzo Librandi, Oct 22 2014

A122803:=n->(-2)^n; seq(A122803(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014

Table[(-2)^n, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

a(n)=(-2)^n \\ Charles R Greathouse IV, Sep 24 2015

def A122803(n): return -(1<Chai Wah Wu, Nov 18 2022

a(n) = (-2)^n = (-1)^n * 2^n.
a(n) = -2*a(n-1), n > 0; a(0) = 1. G.f.: 1/(1+2x). - Philippe Deléham, Nov 19 2008
Sum_{n >= 0} 1/a(n) = 2/3. - Jaume Oliver Lafont, Mar 01 2009
E.g.f.: 1/exp(2*x). - Arkadiusz Wesolowski, Aug 13 2012
a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n, k)*A030195(n+1). - R. J. Mathar, Oct 15 2012
G.f.: 1/(1+2x). A122803 = A033999 * A000079. - M. F. Hasler, Oct 21 2014
a(n) = Sum_{k = 0..n} A163626(n,k)*A000217(k+1). - Philippe Deléham, May 25 2015

A186949 a(n) = 2^n - 2*(binomial(1,n) - binomial(0,n)).

Original entry on oeis.org

1, 0, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824

Offset: 0

Paul Barry, Mar 01 2011

Binomial transform is A186948.
Second binomial transform is A186947.
Inverse binomial transform is (-1)^n * A168277(n).
Essentially the same as A000079, A151821, A155559, A171449, and A171559.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Concatenation([1,0], List([2..30], n-> 2^n )); # G. C. Greubel, Dec 01 2019

[n lt 2 select 1-n else 2^n: n in [0..30]]; // G. C. Greubel, Dec 01 2019

seq( `if`(n<2, 1-n, 2^n), n=0..30); # G. C. Greubel, Dec 01 2019

Table[If[n<2, 1-n, 2^n], {n, 0, 30}] (* G. C. Greubel, Dec 01 2019 *)

vector(31, n, if(n<3, 2-n, 2^(n-1))) \\ G. C. Greubel, Dec 01 2019

[1,0]+[2^n for n in (2..30)] # G. C. Greubel, Dec 01 2019

G.f.: (1 - 2*x + 4*x^2)/(1-2*x).
a(n) = Sum_{k=0..n} binomial(n,k)*(3^k - 2*k).
E.g.f.: exp(2*x) - 2*x. - G. C. Greubel, Dec 01 2019

A173078 a(n) = (52^n - 2(-1)^n - 9)/3.

Original entry on oeis.org

1, 3, 11, 23, 51, 103, 211, 423, 851, 1703, 3411, 6823, 13651, 27303, 54611, 109223, 218451, 436903, 873811, 1747623, 3495251, 6990503, 13981011, 27962023, 55924051, 111848103, 223696211, 447392423, 894784851, 1789569703, 3579139411

Offset: 1

Paul Curtz, Feb 09 2010

The sequence and higher-order differences in subsequent rows are
    1,  3,  11,  23, 51, 103, 211, 423, 851, 1703, 3411, 6823, 13651
    2,  8,  12,  28, 52, 108, 212, 428, 852, 1708, 3412, 6828, 13652
    6,  4,  16,  24, 56, 104, 216, 424, 856, 1704, 3416, 6824, 13656
   -2,  12,  8,  32, 48, 112, 208, 432, 848, 1712, 3408, 6832, 13648
   14,  -4, 24,  16, 64,  96, 224, 416, 864, 1696, 3424, 6816, 13664
  -18,  28, -8,  48, 32, 128, 192, 448, 832, 1728, 3392, 6848,  1363
   46, -36, 56, -16, 96,  64, 256, 384, 896, 1664, 3456, 6784,  1369
The main diagonal 1,8,16,... is essentially A000079.
A subdiagonal is 2, 4, 8, 16, ... A155559.
Other diagonals are 3, 12, 24, 48, ... = 3*A151821, 6, 12, 24, ... = A082505 and -2, -4, -8, -16, ..., a negated variant of A171449.

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

List([1..40], n-> (5*2^n - 2*(-1)^n - 9)/3); # G. C. Greubel, Dec 01 2019

[5*2^n/3-2*(-1)^n/3-3: n in [1..40]]; // Vincenzo Librandi, Aug 05 2011

seq( (5*2^n -2*(-1)^n -9)/3, n=1..40); # G. C. Greubel, Dec 01 2019

LinearRecurrence[{2,1,-2},{1,3,11},40] (* Harvey P. Dale, Oct 01 2018 *)

vector(40, n, (5*2^n - 2*(-1)^n - 9)/3) \\ G. C. Greubel, Dec 01 2019

[(5*2^n - 2*(-1)^n - 9)/3 for n in (1..40)] # G. C. Greubel, Dec 01 2019

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n+1) - 2*a(n) = A010686(n-1).
a(n) = A084214(n+1) - 3.
G.f.: x*(1 + x + 4*x^2) / ( (1-x)*(1-2*x)*(1+x) ).
a(2n+3) - a(2n+1) = 10*A000302(n).
E.g.f.: (-2*exp(-x) + 6 - 9*exp(x) + 5*exp(2*x))/3. - G. C. Greubel, Dec 01 2019

A171501 Inverse binomial transform of A084640.

Original entry on oeis.org

0, 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

Offset: 0

Paul Curtz, Dec 10 2009

a(n) and differences are
0,     1,   3,  -1,    7,   -9,
1,     2,  -4,   8,  -16,   32,  =(-1)^(n+1) * A171449(n),
1,    -6,  12, -24,   48,  -96,
-7,   18, -36,  72, -144,  288,
25,  -54, 108, -216, 432, -864,
Vertical: 1) 0 followed with A168589(n).
          2) (-1 followed with A008776(n) ) signed. See A025192(n).
Main diagonal: see A167747(1+n). - Paul Curtz, Jun 16 2011

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

I:=[0, 1, 3]; [n le 3 select I[n] else -Self(n-1) + 2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 18 2012

CoefficientList[Series[x*(1 + 4*x)/((1 + 2*x)*(1 - x)), {x, 0, 30}], x]
LinearRecurrence[{-1,2},{0,1,3},40] (* Harvey P. Dale, Jan 14 2020 *)

a(n) = A140966(n), n>0.
G.f.: x*(1+4*x) / ( (1+2*x)*(1-x) ). - R. J. Mathar, Jun 14 2011
a(1+n)= (-1)^(1+n) * A001045(1+n) + 2. - Paul Curtz, Jun 16 2011

Mathematica program by Olivier Gérard, Jul 06 2011

A171559 Powers of 2 (cf. A000079) with 1 replaced by 3.

Original entry on oeis.org

3, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Paul Curtz, Dec 11 2009

First differences of A171449.

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

a(n+1)-a(n) = A171449(n).
a(n) + A171449(n) = A000079(n+1) = 2*A000079(n).
G.f.: (3-4*x)/(1-2*x).

Edited by R. J. Mathar and N. J. A. Sloane, Dec 15 2009

A173197 a(0)=1, a(n)= 2+2^n/6+4*(-1)^n/3, n>0.

Original entry on oeis.org

1, 1, 4, 2, 6, 6, 14, 22, 46, 86, 174, 342, 686, 1366, 2734, 5462, 10926, 21846, 43694, 87382, 174766, 349526, 699054, 1398102, 2796206, 5592406, 11184814, 22369622, 44739246, 89478486, 178956974, 357913942, 715827886, 1431655766, 2863311534, 5726623062, 11453246126, 22906492246, 45812984494, 91625968982, 183251937966, 366503875926, 733007751854

Offset: 0

Paul Curtz, Feb 12 2010

Linked to Jacobsthal numbers (expansion of tan(x), a.k.a. Zag numbers) A000182=1,2,16,272,...: a(n+1)-2a(n) = -(-1)^n*(A000182(n) mod 10) = (-1,2,-6,2,-6,2,-6,...).
Cf. A173114, related to Euler (or secant, or Zig) numbers, A000364. a(n+1)+A010684=A001045.
First differences: 0,3,-2,4,0,8,8,24,... = 0,A154879 (third differences of A001045).
Main diagonal: A003945; first upper diagonal: -A171449; second: 4*A011782.

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

a(n) = A093380(n+4), n>3.
a(n) = +2*a(n-1) +a(n-2) -2*a(n-3), n>3.
G.f.: 1-x*(-1-2*x+7*x^2)/((x-1)*(2*x-1)*(1+x)).
a(2n+2)+a(2n+3)=6*A047689.
a(2n)-a(2n-2) = 3,1,2,4,8,16,... = 3,A000079.

cp's OEIS Frontend

A122803 Powers of -2: a(n) = (-2)^n.

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A186949 a(n) = 2^n - 2*(binomial(1,n) - binomial(0,n)).

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A173078 a(n) = (5*2^n - 2*(-1)^n - 9)/3.

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A171501 Inverse binomial transform of A084640.

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A171559 Powers of 2 (cf. A000079) with 1 replaced by 3.

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A173197 a(0)=1, a(n)= 2+2^n/6+4*(-1)^n/3, n>0.

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A173078 a(n) = (52^n - 2(-1)^n - 9)/3.