A062510 -id:A062510 - cp's OEIS frontend

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

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

A253145 Triangular numbers (A000217) omitting the term 1.

Original entry on oeis.org

0, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275

Offset: 0

Paul Curtz, Mar 23 2015

The full triangle of the inverse Akiyama-Tanigawa transform applied to (-1)^n*A062510(n)=3*(-1)^n*A001045(n) yielding a(n) is
0,      3,    6,  10,   15,  21,  28, 36, ...
-3,    -6,  -12, -20,  -30, -42, -56, ...    essentially -A002378
3,     12,   24,  40,   60,  84, ...         essentially  A046092
-9,   -24,  -48, -80, -120, ...              essentially -A033996
15,    48,   96, 160, ...
-33,  -96, -192, ...
63,   192, ...
-129, ...
etc.
First column: (-1)^n*A062510(n).
The following columns are multiples of A122803(n)=(-2)^n. See A007283(n), A091629(n), A020714(n+1), A110286, A175805(n), 4*A005010(n).
An autosequence of the first kind is a sequence whose main diagonal is A000004 = 0's.
b(n) = 0, 0 followed by a(n) is an autosequence of the first kind.
The successive differences of b(n) are
0,   0,  0, 3, 6, 10, 15, 21, ...
0,   0,  3, 3, 4,  5,  6,  7, ...  see A194880(n)
0,   3,  0, 1, 1,  1,  1,  1, ...
3,  -3,  1, 0, 0,  0,  0,  0, ...
-6,  4, -1, 0, 0,  0,  0,  0, ...
10, -5,  1, 0, 0,  0,  0,  0, ...
-15, 6, -1, 0, 0,  0,  0,  0, ...
21, -7,  1, 0, 0,  0,  0,  0, ...
The inverse binomial transform (first column) is the signed sequence. This is general.
Also generalized hexagonal numbers without 1. - Omar E. Pol, Mar 23 2015

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A179865, A001045, A062510, A002378, A046092, A033996, A122803, A007283, A091629, A020714, A110286, A161680, A175805, A005010, A194880, A001840, A022003, A255935.

Concatenation([0],List([1..50],n->(n+1)*(n+2)/2)); # Muniru A Asiru, Oct 31 2018

Prepend[Table[(n + 1) (n + 2)/2, {n, 49}], 0] (* Michael De Vlieger, Mar 23 2015 *)

a(n)=if(n,(n+1)*(n+2)/2,0) \\ Charles R Greathouse IV, Mar 23 2015

Inverse Akiyama-Tanigawa transform of (-1)^n*A062510(n).
a(n) = (n+1)*(n+2)/2 for n > 0. - Charles R Greathouse IV, Mar 23 2015
a(n+1) = 3*A001840(n+1) + A022003(n).
a(n) = A161680(n+2) for n >= 1. - Georg Fischer, Oct 30 2018
From Stefano Spezia, May 28 2025: (Start)
G.f.: x*(3 - 3*x + x^2)/(1 - x)^3.
E.g.f.: exp(x)*(2 + 4*x + x^2)/2 - 1. (End)

A255935 Triangle read by rows: a(n) = Pascal's triangle A007318(n) + A197870(n+1).

Original entry on oeis.org

0, 1, 2, 1, 2, 0, 1, 3, 3, 2, 1, 4, 6, 4, 0, 1, 5, 10, 10, 5, 2, 1, 6, 15, 20, 15, 6, 0, 1, 7, 21, 35, 35, 21, 7, 2, 1, 8, 28, 56, 70, 56, 28, 8, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 2, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0

Offset: 0

Paul Curtz, Mar 11 2015

Consider the difference table of a sequence with A000004(n)=0's as main diagonal. (Example: A000045(n).) We call this sequence an autosequence of the first kind.
Based on Pascal's triangle, a(n) =
0,                   T1
1, 2,
1, 2, 0,
1, 3, 3, 2,
etc.
transforms every sequence s(n) in an autosequence of the first kind via the multiplication by the triangle
s0,                  T2
s0, s1,
s0, s1, s2,
s0, s1, s2, s3,
etc.
Examples.
1) s(n) = A198631(n)/A006519(n+1), the second fractional Euler numbers (see A209308). This yields 0*1, 1*1+2*1/2=2, 1*1+2*1/2+0*0=2, 1*1+3*1/2++3*0+2*(-1/4)=2, ... .
The autosequence is 0 followed by 2's or 2*(0,1,1,1,1,1,1,1,... = b(n)).
b(n), the basic autosequence of the first kind, is not in the OEIS (see A140575 and A054977).
2) s(n) = A164555(n)/A027642(n), the second Bernoulli numbers, yields 0,2,2,3,4,5,6,7,... = A254667(n).
Row sums of T1: A062510(n) = 3*A001045(n).
Antidiagonal sums of T1: A111573(n).
With 0's instead of the spaces, every column, i.e.,
0, 1, 1, 1, 1,  1,  1,  1,  1,  1,   1, ...
0, 2, 2, 3, 4,  5,  6,  7,  8,  9,  10, ... = A001477(n) with 0 instead of 1 = A254667(n)
0, 0, 0, 3, 6, 10, 15, 21, 28, 36,  45, ... = A161680(n) with 0 instead of 1
0, 0, 0, 2, 4, 10, 20, 35, 56, 84, 120, ...
etc., is an autosequence of the first kind.
With T(0,0) = 1, it is (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 24 2015

			Triangle starts:
0;
1, 2;
1, 2, 0;
1, 3, 3, 2;
1, 4, 6, 4, 0;
1, 5, 10, 10, 5, 2;
1, 6, 15, 20, 15, 6, 0;
...

Cf. A001045, A001477, A010673, A011973, A054977, A062510, A007318, A197870, A006519, A198631, A140575, A209308, A164555, A027642, A111573, A161680, A254667.

a[n_, k_] := If[k == n, 2*Mod[n, 2], Binomial[n, k]]; Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 23 2015 *)

a(n) = Pascal's triangle A007318(n) with main diagonal A010673(n) (= period 2: repeat 0, 2) instead of 1's=A000012(n).
a(n) = reversal abs(A140575(n)).
a(n) = A007318(n) + A197870(n+1).
T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k>n or if k<0 . - Philippe Deléham, May 24 2015
G.f.: (-1-2*x*y+x^2*y+x^2*y^2)/((x*y+1)*(x*y+x-1)) - 1. - R. J. Mathar, Aug 12 2015

A274073 a(n) = 6^n-(-1)^n.

Original entry on oeis.org

0, 7, 35, 217, 1295, 7777, 46655, 279937, 1679615, 10077697, 60466175, 362797057, 2176782335, 13060694017, 78364164095, 470184984577, 2821109907455, 16926659444737, 101559956668415, 609359740010497, 3656158440062975, 21936950640377857, 131621703842267135

Offset: 0

Colin Barker, Jun 09 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,6).

Cf. A015540.

Sequences of the type k^n-(-1)^n: A062157 (k=0), A010673 (k=1), A062510 (k=2), A105723 (k=3), A247281 (k=4), A274072 (k=5), this sequence (k=6).

concat(0, Vec(7*x/((1+x)*(1-6*x)) + O(x^30)))

O.g.f.: 7*x/((1+x)*(1-6*x)).
E.g.f.: exp(6*x) - exp(-x).
a(n) = 5*a(n-1) + 6*a(n-2) for n>1.
a(n) = 7*A015540(n).

A321483 a(n) = 7*2^n + (-1)^n.

Original entry on oeis.org

8, 13, 29, 55, 113, 223, 449, 895, 1793, 3583, 7169, 14335, 28673, 57343, 114689, 229375, 458753, 917503, 1835009, 3670015, 7340033, 14680063, 29360129, 58720255, 117440513, 234881023, 469762049, 939524095, 1879048193, 3758096383, 7516192769, 15032385535

Offset: 0

Paul Curtz, Nov 11 2018

Difference table:
   8, 13, 29, 55, 113, 223, 449, ...
   5, 16, 26, 58, 110, 226, 446, 898, ...
  11, 10, 32, 52, 116, 220, 452, 892, 1796, ...
  -1, 22, 20, 64, 104, 232, 440, 904, 1784, 3592, ...
      -2, 44, 40, 128, 208, 464, 880, 1808, 3568, 7184, ...
etc.
Every diagonal is a sequence of the form k*2^m.
a(n) is divisible by
.  5 if n is a term of A004767,
. 11 if n is a term of A016885,
. 13 if n is a term of A017533.

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

Cf. A000079, A001045, A005029, A010710, A014551, A062092, A062510, A070366, A175805, A199207, A206372.
Cf. A033999, A005009.
Subsequence of A001651, A160543, A160544.

a[n_] := 7*2^n + (-1)^n ; Array[a, 32, 0] (* Amiram Eldar, Nov 12 2018 *)
CoefficientList[Series[E^-x + 7 E^(2 x), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 12 2018 *)
LinearRecurrence[{1,2},{8,13},40] (* Harvey P. Dale, Mar 18 2022 *)

Vec((8 + 5*x) / ((1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Nov 11 2018

O.g.f.: (8 + 5*x) / ((1 + x)*(1 - 2*x)). - Colin Barker, Nov 11 2018
E.g.f.: exp(-x) + 7*exp(2*x). - Stefano Spezia, Nov 12 2018
a(n) = a(n-1) + 2*a(n-2).
a(n) = 2*a(n-1) + 3*(-1)^n for n>0, a(0)=8.
a(2*k) = 7*4^k + 1, a(2*k+1) = 14*4^k - 1.
a(n) = A014551(n) + A014551(n-1) + A014551(n-2).
a(n) = 2^(n+3) - 3*A001045(n).
a(n) mod 9 = A070366(n+3).
a(n) + a(n+1) = 21*2^n.

Two terms corrected, and more terms added by Colin Barker, Nov 11 2018

A340627 a(n) = (112^n - 2(-1)^n)/3 for n >= 0.

Original entry on oeis.org

3, 8, 14, 30, 58, 118, 234, 470, 938, 1878, 3754, 7510, 15018, 30038, 60074, 120150, 240298, 480598, 961194, 1922390, 3844778, 7689558, 15379114, 30758230, 61516458, 123032918, 246065834, 492131670, 984263338, 1968526678, 3937053354, 7874106710, 15748213418, 31496426838

Offset: 0

Paul Curtz, Apr 25 2021

Based on A112387.
Prepended with 0, 1, its difference table is
0,  1,  1,  2,  1,   4,  3,   8, ... = mix A001045(n), 2^n.
1,  0,  1, -1,  3,  -1,  5,  -3, ... = mix A001045(n+1), -A001045(n).
-1, 1, -2,  4, -4,   6, -8,  14, ... = mix -2^n, A084214(n+1).
2, -3,  6, -8, 10, -14, 22, -30, ... = mix 2*A001045(n+2), -a(n).

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

Cf. A000079, A001045, A062510, A078008, A112387.

Cf. also A052997.

LinearRecurrence[{1, 2}, {3, 8}, 35] (* Amiram Eldar, Apr 25 2021 *)

a(n) = (11*2^n - 2*(-1)^n)/3 \\ Felix Fröhlich, Apr 25 2021

a(n) = 2^(n+2) - A078008(n), n>=0.
a(n) = (A062510(n) = 3*A001045(n)) + A001045(n+3), n>=0.
a(0)=3, a(2*n+1) = 2*a(2*n) + 2, a(2*n+2) = 2*a(2*n+1) - 2, n>=0.
a(n) = 4*A052997(n-1) + 2, n>=2. - Hugo Pfoertner, Apr 25 2021
a(n+1) = 11*2^n - a(n) for n>=0.
a(n+3) = 33*2^n - a(n) for n>=0.
a(n+5) = 121*2^n - a(n) for n>=0.
etc.
a(n+2) = a(n) + 11*2^n for n>=0.
a(n+4) = a(n) + 55*2^n for n>=0.
a(n+6) = a(n) + 231*2^n for n>=0.
etc.
G.f.: (3 + 5*x)/(1 - x - 2*x^2). - Stefano Spezia, Apr 26 2021
E.g.f: (11*exp(2*x) - 2*exp(-x))/3. - Jianing Song, Apr 26 2021

More terms from Michel Marcus, Apr 25 2021

New name from Jianing Song, Apr 25 2021

A107663 a(2n) = 2*4^n-1, a(2n+1) = (2^(n+1)+1)^2; interlaces A083420 with A028400.

Original entry on oeis.org

1, 9, 7, 25, 31, 81, 127, 289, 511, 1089, 2047, 4225, 8191, 16641, 32767, 66049, 131071, 263169, 524287, 1050625, 2097151, 4198401, 8388607, 16785409, 33554431, 67125249, 134217727, 268468225, 536870911, 1073807361, 2147483647

Offset: 0

Creighton Dement, May 19 2005

a(2n) = A085903(2n) = A083420(n).

Floretion Algebra Multiplication Program, FAMP Code: 4tesseq[A*B] with A = + .25'i + .25'j + .25'k + .25i' + .25j' + .25k' + .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' + .25'ki' + .25'kj' + .25e and B = + .5'i + .5i' + 'ii' + e [Factor added to formula by Creighton Dement, Dec 11 2009]

Colin Barker, Table of n, a(n) for n = 0..1000
Henry Bottomley, Illustration of initial terms (A028400)
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
Index entries for linear recurrences with constant coefficients, signature (1,4,-2,-4).

Cf. A083420, A028400, A062510, A085903.

Vec((1 + 8*x - 6*x^2 - 16*x^3) / ((1 + x)*(1 - 2*x)*(1 - 2*x^2)) + O(x^35)) \\ Colin Barker, May 21 2019

G.f.: (-1-8*x+6*x^2+16*x^3) / ((1-2*x)*(x+1)*(2*x^2-1)).
From Colin Barker, May 21 2019: (Start)
a(n) = a(n-1) + 4*a(n-2) - 2*a(n-3) - 4*a(n-4) for n>3.
a(n) = ((-1)^(1+n) + 2^(1+n) + 2^((1+n)/2)*(1+(-1)^(1+n))).
(End)

A216206 a(n) = Product_{i=1..n} ((-2)^i-1).

Original entry on oeis.org

1, -3, -9, 81, 1215, -40095, -2525985, 325852065, 83092276575, -42626337882975, -43606743654283425, 89350217747626737825, 365889141676531491393375, -2997729737755822508985921375, -49111806293653640164716349886625, 1609344780436736134557590069434814625

Offset: 0

R. J. Mathar, Mar 12 2013

Signed partial products of A062510. This implies that all terms from a(1) on are multiples of 3.

Cf. A005329, A015013, A015109, A027871, A062510, A330862.

A216206 := proc(n)
        mul( (-2)^i-1, i=1..n) ;
end proc:

Table[(-1)^n QPochhammer[-2, -2, n], {n, 0, 15}] (* Bruno Berselli, Mar 13 2013 *)
Table[Product[(-2)^k-1,{k,n}],{n,0,20}] (* Harvey P. Dale, Oct 21 2024 *)

A015109(n,k) = a(n)/(a(k)*a(n-k)).
a(n) = (-3)^n*A015013(n) for n>0, a(0)=1. - Bruno Berselli and Alonso del Arte, Mar 13 2013
a(n) ~ (-1)^(floor(n/2)+1) * c * 2^(n*(n+1)/2), where c = Product_{k>=1} (1 - 1/(-2)^k) = 1.21072413030105918013... (A330862). - Amiram Eldar, Aug 10 2025

A274072 a(n) = 5^n-(-1)^n.

Original entry on oeis.org

0, 6, 24, 126, 624, 3126, 15624, 78126, 390624, 1953126, 9765624, 48828126, 244140624, 1220703126, 6103515624, 30517578126, 152587890624, 762939453126, 3814697265624, 19073486328126, 95367431640624, 476837158203126, 2384185791015624, 11920928955078126

Offset: 0

Colin Barker, Jun 09 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A015531.

Sequences of the type k^n-(-1)^n: A062157 (k=0), A010673 (k=1), A062510 (k=2), A105723 (k=3), A247281 (k=4), this sequence (k=5), A274073 (k=6).

LinearRecurrence[{4, 5}, {0, 6}, 30] (* Paolo Xausa, Oct 21 2024 *)

concat(0, Vec(6*x/((1+x)*(1-5*x)) + O(x^30)))

O.g.f.: 6*x/((1+x)*(1-5*x)).
E.g.f.: exp(5*x) - exp(-x).
a(n) = 4*a(n-1) + 5*a(n-2) for n>1.
a(n) = 6*A015531(n).

A281166 a(n) = 3a(n-1) - 3a(n-2) + 2*a(n-3) for n>2, a(0)=a(1)=1, a(2)=3.

Original entry on oeis.org

1, 1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, 4096, 8191, 16383, 32768, 65537, 131073, 262144, 524287, 1048575, 2097152, 4194305, 8388609, 16777216, 33554431, 67108863, 134217728, 268435457, 536870913, 1073741824, 2147483647, 4294967295, 8589934592

Offset: 0

Paul Curtz, Jan 16 2017

a(n) is the first sequence on three (with its first and second differences):
1, 1, 3, 8, 17, 33, 64, 127, ...;
0, 2, 5, 9, 16, 31, 63, 128, ..., that is 0 followed by A130752;
2, 3, 4, 7, 15, 32, 65, 129, ..., that is 2 followed by A130755;
1, 1, 3, 8, 17, 33, 64, 127, ..., this sequence.
The main diagonal is 2^n.
The sum of the first three lines is 3*2^n.
Alternated sum and subtraction of a(n) and its inverse binomial transform (period 3: repeat [1, 0, 2]) gives the autosequence of the first kind b(n):
   0,  1,  1,  9, 17, 35, 63, 127, ...
   1,  0,  8,  8, 18, 28, 64, 126, ...
  -1,  8,  0, 10, 10, 36, 62, 134, ...
   9, -8, 10,  0, 26, 26, 72, 118, ... .
The main diagonal is 0's. The first two upper diagonals are A259713.
The sum of the first three lines gives 9*A001045.
a(n) mod 9 gives a periodic sequence of length 6: repeat [1, 1, 3, 8, 8, 6].
a(n) = A130750(n-1) for n > 2. - Georg Fischer, Oct 23 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A000079, A001045, A005010, A007283, A014551 (a diagonal), A057079, A062510 (a diagonal), A128834, A130750, A130752, A130755, A153234, A153237, A259713.

I:=[1,1,3]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018

LinearRecurrence[{3, -3, 2}, {1, 1, 3}, 30] (* Jean-François Alcover, Jan 16 2017 *)

Vec((1 - 2*x + 3*x^2) / ((1 - 2*x)*(1 - x + x^2)) + O(x^40)) \\ Colin Barker, Jan 16 2017

Binomial transform of the sequence of length 3: repeat [1, 0, 2].
a(n+3) = -a(n) + 9*2^n.
a(n) = 2^n - periodic 6: repeat [0, 1, 1, 0, -1, -1, 0].
a(n+6) = a(n) + 63*2^n.
a(n+1) = 2*a(n) - period 6: repeat [1, -1, -2, -1, 1, 2].
a(n) = 2^n - 2*sin(Pi*n/3)/sqrt(3). - Jean-François Alcover and Colin Barker, Jan 16 2017
G.f.: (1 - 2*x + 3*x^2)/((1 - 2*x)*(1 - x + x^2)). - Colin Barker, Jan 16 2017

cp's OEIS Frontend

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

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A253145 Triangular numbers (A000217) omitting the term 1.

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A255935 Triangle read by rows: a(n) = Pascal's triangle A007318(n) + A197870(n+1).

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A274073 a(n) = 6^n-(-1)^n.

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A321483 a(n) = 7*2^n + (-1)^n.

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A340627 a(n) = (11*2^n - 2*(-1)^n)/3 for n >= 0.

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A107663 a(2n) = 2*4^n-1, a(2n+1) = (2^(n+1)+1)^2; interlaces A083420 with A028400.

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A216206 a(n) = Product_{i=1..n} ((-2)^i-1).

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A340627 a(n) = (112^n - 2(-1)^n)/3 for n >= 0.

A281166 a(n) = 3a(n-1) - 3a(n-2) + 2*a(n-3) for n>2, a(0)=a(1)=1, a(2)=3.