A014551 -id:A014551 - cp's OEIS frontend

A003665 a(n) = 2^(n-1)*( 2^n + (-1)^n ).

1, 1, 10, 28, 136, 496, 2080, 8128, 32896, 130816, 524800, 2096128, 8390656, 33550336, 134225920, 536854528, 2147516416, 8589869056, 34359869440, 137438691328, 549756338176, 2199022206976, 8796095119360, 35184367894528, 140737496743936, 562949936644096, 2251799847239680

Offset: 0

N. J. A. Sloane

Binomial transform of expansion of cosh(3*x), the aerated version of A001019, 1,0,9,0,81,0,729,... - Paul Barry, Apr 05 2003
Alternatively: start with the fraction 1/1, take the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 9 times the bottom to get the new top. The limit of the sequence of fractions used to generate this sequence is sqrt(9). - Cino Hilliard, Sep 25 2005
This sequence also gives the number of ordered pairs of subsets (A, B) of {1, 2, ..., n} such that |A u B| is even. (Here "u" stands for the set-theoretic union.) The special case n = 13 can be found as in CRUX Problem 3257. - Walther Janous (walther.janous(AT)tirol.com), Mar 01 2008
For n > 0, a(n) is term (1,1) in the n-th power of the 2 X 2 matrix [1,3; 3,1]. - Gary W. Adamson, Aug 06 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 9 types of other natural numbers. - Milan Janjic, Aug 13 2010
a(n) = ((1+3)^n+(1-3)^n)/2. In general, if b(0),b(1),... is the k-th binomial transform of the sequence ((3^n+(-3)^n)/2), then b(n) = ((k+3)^n+(k-3)^n)/2. More generally, if b(0),b(1),... is the k-th binomial transform of the sequence ((m^n+(-m)^n)/2), then b(n) = ((k+m)^n+(k-m)^n)/2. See A034494, A081340-A081342, A034659. - Charlie Marion, Jun 25 2011
Pisano period lengths: 1, 1, 1, 1, 4, 1, 6, 1, 1, 4, 5, 1, 12, 6, 4, 1, 8, 1, 9, 4, ... - R. J. Mathar, Aug 10 2012
Starting with offset 1 the sequence is the INVERT transform of (1, 9, 9, 9, ...). - Gary W. Adamson, Aug 06 2016

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.
M. Gardner, Riddles of Sphinx, M.A.A., 1987, p. 145.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Bill Sands, Problem 3257, Crux Math. 33 (2007), No.5, p. 298.
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A001019, A001045, A014551, A078008, A098158.

Cf. A034494, A081340-A081342, A034659.

List([0..30], n-> 2^(n-1)*(2^n +(-1)^n)); # G. C. Greubel, Aug 02 2019

[2^(n-1)*( 2^n + (-1)^n ): n in [0..30]]; // Vincenzo Librandi, Aug 19 2011

A003665:=n->2^(n-1)*( 2^n + (-1)^n ): seq(A003665(n), n=0..30); # Wesley Ivan Hurt, Apr 28 2017

CoefficientList[Series[(1+8x)/(1-2x-8x^2), {x,0,30}], x] (* or *)
LinearRecurrence[{2,8}, {1,1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)

PARI
```
a(n)=2^(n-1)*( 2^n + (-1)^n );
```

[2^(n-1)*(2^n +(-1)^n) for n in (0..30)] # G. C. Greubel, Aug 02 2019

From Paul Barry, Mar 01 2003: (Start)
a(n) = 2*a(n-1) + 8*a(n-2), a(0)=a(1)=1.
a(n) = (4^n + (-2)^n)/2.
G.f.: (1-x)/((1+2*x)*(1-4*x)). (End)
From Paul Barry, Apr 05 2003: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*9^k.
E.g.f. exp(x)*cosh(3*x). (End)
a(n) = (A078008(n) + A001045(n+1))*2^(n-1) = A014551(n)*2^(n-1). - Paul Barry, Feb 12 2004
Given a(0)=1, b(0)=1 then for i=1, 2, ..., a(i)/b(i) = (a(i-1) + 9*b(i-1)) / (a(i-1) + b(i-1)). - Cino Hilliard, Sep 25 2005
a(n) = Sum_{k=0..n} A098158(n,k)*9^(n-k). - Philippe Deléham, Dec 26 2007
a(n) = ((1+sqrt(9))^n + (1-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
If p[1]=1, and p[i]=9, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Apr 29 2010
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(9*k-1)/(x*(9*k+8) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013

Entry revised by N. J. A. Sloane, Nov 22 2006

A166920 a(n) = 2^n - (1 + (-1)^n)/2.

Original entry on oeis.org

0, 2, 3, 8, 15, 32, 63, 128, 255, 512, 1023, 2048, 4095, 8192, 16383, 32768, 65535, 131072, 262143, 524288, 1048575, 2097152, 4194303, 8388608, 16777215, 33554432, 67108863, 134217728, 268435455, 536870912, 1073741823, 2147483648, 4294967295

Offset: 0

Paul Curtz, Oct 23 2009

Partial sums of A014551. The inverse binomial transform yields a sequence 0,2,-1,5,-7,17,...: zero followed by a sign alternating A014551.
The table of a(n) plus higher order differences in successive rows shows A131577 on the main diagonal.
a(n) = 2^n when n is odd and 2^n-1 when n is even. - Wesley Ivan Hurt, Nov 15 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A004171, A014551, A024036, A131577, A168361.

a166920 n = a166920_list !! n
a166920_list = scanl (+) 0 a014551_list
-- Reinhard Zumkeller, Jan 02 2013

[2^n -(1+(-1)^n)/2: n in [0..30]]; // Vincenzo Librandi, May 16 2011

A166920:=n->2^n-(1+(-1)^n)/2; seq(A166920(n), n=0..50); # Wesley Ivan Hurt, Nov 15 2013

LinearRecurrence[{2,1,-2},{0,2,3},40] (* Harvey P. Dale, Oct 16 2012 *)

a(n)=2^n-(1+(-1)^n)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(2-x)/((1-x)*(1-2*x)*(1+x)).
a(n) = 2^n - (1+(-1)^n)/2.
a(2*n) = A024036(n); a(2*n+1) = A004171(n).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n+1) - 2*a(n) = A168361(n).
a(n) = A000225(n+1) - A051049(n) = A014551(n) - A168361(n).
E.g.f.: exp(2*x) - cosh(x). - G. C. Greubel, May 28 2016
a(n) = Sum_{k=1..n+1} Sum_{i=0..n+1} C(n-k,i). - Wesley Ivan Hurt, Sep 22 2017
a(n) = 2*A001045(n) + A000975(n-1) for n>0. - Yuchun Ji, Aug 30 2018

Edited and extended by R. J. Mathar, Mar 02 2010

A075117 Table by antidiagonals of generalized Lucas numbers: T(n,k) = T(n,k-1) + n*T(n,k-2) with T(n,0)=2 and T(n,1)=1.

Original entry on oeis.org

2, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 4, 5, 1, 2, 1, 7, 7, 7, 1, 2, 1, 11, 17, 10, 9, 1, 2, 1, 18, 31, 31, 13, 11, 1, 2, 1, 29, 65, 61, 49, 16, 13, 1, 2, 1, 47, 127, 154, 101, 71, 19, 15, 1, 2, 1, 76, 257, 337, 297, 151, 97, 22, 17, 1, 2, 1, 123, 511, 799, 701, 506, 211, 127, 25, 19, 1, 2

Offset: 0

Henry Bottomley, Sep 02 2002

			Array starts as:
  2, 1,  1,  1,  1,   1, ...;
  2, 1,  3,  4,  7,  11, ...;
  2, 1,  5,  7, 17,  31, ...;
  2, 1,  7, 10, 31,  61, ...;
  2, 1,  9, 13, 49, 101, ...;
  2, 1, 11, 16, 71, 151, ...; etc.

G. C. Greubel, Antidiagonals n = 0..100, flattened

Cf. A060959.
Rows include: A054977, A000032, A014551, A075118, A072265.
Columns include: A007395, A000012, A005408, A016777, A056220, A062786.

[2^(1+k-n)*(&+[Binomial(n-k,2*j)*(1+4*k)^j: j in [0..Floor((n-k)/2)]]): k in [0..n], n in [0..13]]; // G. C. Greubel, Jan 27 2020

seq(seq( 2^(1+k-n)*add( binomial(n-k, 2*j)*(1+4*k)^j, j=0..floor((n-k)/2)), k=0..n), n=0..13); # G. C. Greubel, Jan 27 2020

T[n_, k_]:= ((1 + Sqrt[1+4n])/2)^k + ((1 - Sqrt[1+4n])/2)^k; Table[If[n==0 && k==0, 2, T[k, n-k]]//Simplify, {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 27 2020 *)

def T(n, k): return 2^(1-k)*sum( binomial(k, 2*j)*(1+4*n)^j for j in (0..floor(k/2)) )
[[T(k,n-k) for k in (0..n)] for n in (0..13)] # G. C. Greubel, Jan 27 2020

T(n, k) = ((1+sqrt(4*n+1))/2)^k + ((1-sqrt(4*n+1))/2)^k = 2*A060959(n, k+1) - A060959(n, k).

T(n, k) = 2^(1-k)*Sum_{j=0..floor(k/2)} binomial(k, 2*j)*(1+4*n)^j. - G. C. Greubel, Jan 27 2020

A101622 A Horadam-Jacobsthal sequence.

Original entry on oeis.org

0, 1, 6, 13, 30, 61, 126, 253, 510, 1021, 2046, 4093, 8190, 16381, 32766, 65533, 131070, 262141, 524286, 1048573, 2097150, 4194301, 8388606, 16777213, 33554430, 67108861, 134217726, 268435453, 536870910, 1073741821, 2147483646, 4294967293, 8589934590

Offset: 0

Paul Barry, Dec 10 2004

Companion sequence to A084639.
This is the sequence A(0,1;1,2;5) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
Except for the initial three terms, the decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Mar 27 2017
Named after the Australian mathematician Alwyn Francis Horadam (1923-2016) and the German mathematician Ernst Jacobsthal (1882-1965). - Amiram Eldar, Jun 10 2021

Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart., Vol. 34, No. 1 (1996), pp. 40-54.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [_Wolfdieter Lang_, Oct 18 2010]
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Stephen Wolfram, A New Kind of Science.
Wolfram Research, Wolfram Atlas of Simple Programs.
Index entries for sequences related to cellular automata.
Index to 2D 5-Neighbor Cellular Automata.
Index to Elementary Cellular Automata.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A131953.

[(2^(n+2)+(-1)^n-5)/2: n in [0..35]]; // Vincenzo Librandi, Aug 12 2011

LinearRecurrence[{2,1,-2},{0,1,6},40] (* Harvey P. Dale, Jul 08 2014 *)

concat(0, Vec(x*(1+4*x)/((1-x)*(1+x)*(1-2*x)) + O(x^30))) \\ Colin Barker, Mar 28 2017

a(n) = (2^(n+2) + (-1)^n - 5)/2.
G.f.: x*(1+4*x)/((1-x)*(1+x)*(1-2*x)).
a(n) = (A014551(n+2)-5)/2.
(1, 6, 13, 30, 61, ...) are the row sums of A131953. - Gary W. Adamson, Jul 31 2007
From Paul Curtz, Jan 01 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) + 5.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = A000079(n+1) - A010693(n).
a(n+1) = A141722(n) + 5 = A141722(n) + A010716(n).
a(2n+1) - a(2n) = 1, 7, 31, ... = A083420.
a(2n+1) - 2*a(2n) = 1.
a(2n) = A002446 = 6*A002450, a(2n+1) = A141725. (End)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. - Colin Barker, Mar 28 2017
a(n) = (1/2) * Sum_{k=1..n} binomial(n+1,k) * (2+(-1)^k). - Wesley Ivan Hurt, Sep 23 2017

A048700 Binary palindromes of odd length (written in base 10).

Original entry on oeis.org

1, 5, 7, 17, 21, 27, 31, 65, 73, 85, 93, 99, 107, 119, 127, 257, 273, 297, 313, 325, 341, 365, 381, 387, 403, 427, 443, 455, 471, 495, 511, 1025, 1057, 1105, 1137, 1161, 1193, 1241, 1273, 1285, 1317, 1365, 1397, 1421, 1453, 1501, 1533, 1539, 1571, 1619, 1651

Offset: 1

Antti Karttunen, Mar 07 1999

Note: you get A006995 (all binary palindromes) if you take (after zero) alternatively 2^n (starting from 2^0 = 1) terms from A048700 and as many from A048701 and then each time, twice as many from both.
A178225(a(n)) = 1. - Reinhard Zumkeller, Oct 21 2011
Comment from Altug Alkan, Dec 03 2015: (Start)
a(6*k) is divisible by 9 for k > 0.
a(3*k+(-1)^k-2) is divisible by 3 for k > 1.
The minimum value of a(n+1) - a(n) occurs when n = 2.
A014551(n) appears in this sequence for n > 0. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A048701 (binary palindromes of even length), A002113 (decimal palindromes), A006995 (all binary palindromes).

Cf. also A178225.

import Data.Set (singleton, deleteFindMin, insert)
import Data.List (unfoldr)
a048700 n = a048700_list !! (n-1)
a048700_list = f 1 $ singleton 1 where
   f z s = m : f (z+1) (insert (c 0) (insert (c 1) s')) where
     c d = foldl (\v d -> 2 * v + d) 0 $ (reverse b) ++ [d] ++ b
     b = unfoldr
         (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2) z
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Oct 21 2011

bit_i := (x,i) -> `mod`(floor(x/(2^i)),2);
floor_log_2 := proc(n) local nn,i: nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi: nn := floor(nn/2); od: end:

Select[Range@ 1651, # == Reverse@ # && OddQ@ Length@ # &@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Dec 03 2015 *)

{a(n) = local(f); if( n<1, 0, f = length(binary(n)) - 1; 2^f*n + sum(i=1, f, bittest(n,i) * 2^(f-i)))}; /* Michael Somos, Nov 27 2002 */

def A048700(n):
    s = bin(n)[2:]
    return int(s+s[-2::-1],2) # Chai Wah Wu, Feb 26 2021

a(n) = (2^(floor_log_2(n)))*n + sum('(bit_i(n, i)*(2^(floor_log_2(n)-i)))', 'i'=1..floor_log_2(n));

a(A047264(n)) mod 3 = 0, for n > 1. - Altug Alkan, Dec 03 2015

A087451 G.f.: (2-x)/((1+2x)(1-3x)); e.g.f.: exp(3x)+exp(-2x); a(n)=3^n+(-2)^n.

Original entry on oeis.org

2, 1, 13, 19, 97, 211, 793, 2059, 6817, 19171, 60073, 175099, 535537, 1586131, 4799353, 14316139, 43112257, 129009091, 387682633, 1161737179, 3487832977, 10458256051, 31385253913, 94134790219, 282446313697, 847255055011

Offset: 0

Paul Barry, Sep 06 2003

Generalized Lucas-Jacobsthal numbers.

Pisano period lengths: 1, 1, 1, 2, 4, 1, 6, 2, 3, 4, 5, 2, 12, 6, 4, 4, 16, 3, 18, 4,... - R. J. Mathar, Aug 10 2012

Drexel University, Generation X and Y
G. Everest, Y. Puri and T. Ward, Integer sequences counting periodic points, arXiv:math/0204173 [math.NT], 2002.
OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (1,6)

Cf. A014551, A087452, A015441.

for i from 0 to 20 do print(3^i+(-2)^i) od; # Gary Detlefs, Dec 20 2009

a[0] = 2; a[1] = 1; a[n_] := a[n] = a[n - 1] + 6a[n - 2]; a /@ Range[0, 25] (* Robert G. Wilson v, Feb 02 2006 *)

[lucas_number2(n,1,-6) for n in range(0, 26)] # Zerinvary Lajos, Apr 30 2009

a(0) = 2, a(1) = 1, a(n) = a(n-1)+6a(n-2).
a(n) = 3^n + (-2)^n. - Gary Detlefs, Dec 20 2009
The sequence 1, 13, 19... is a(n+1) = 3*3^n-2*(-2)^n.
exp( Sum_{n >= 1} a(n)*x^n/n ) = Sum_{n >= 0} A015441(n+1)*x^n. - Peter Bala, Mar 30 2015
a(n) = 2*A015441(n+1) - A015441(n), a formula given by Paul Curtz for autosequences of the 2nd kind. - Jean-François Alcover, Jun 02 2017

A135440 a(n) = a(n-1) + 2a(n-2).

Original entry on oeis.org

-1, 4, 2, 10, 14, 34, 62, 130, 254, 514, 1022, 2050, 4094, 8194, 16382, 32770, 65534, 131074, 262142, 524290, 1048574, 2097154, 4194302, 8388610, 16777214, 33554434, 67108862, 134217730, 268435454, 536870914, 1073741822, 2147483650, 4294967294, 8589934594, 17179869182, 34359738370

Offset: 0

Paul Curtz, Feb 18 2008

First differences of A014551. - Reinhard Zumkeller, Jan 02 2013

It can be noticed that, once deprived of its first term, this is an "autosequence" of the second kind, whose companion of the first kind is A014113. - Jean-François Alcover, Aug 19 2022

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (1,2).

a135440 n = a135440_list !! n
a135440_list = zipWith (-) (tail a014551_list) a014551_list
-- Reinhard Zumkeller, Jan 02 2013

f[n_]:=2/(n+1);x=4;Table[x=f[x];Numerator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
LinearRecurrence[{1,2}, {-1,4}, 25] (* or *) Table[2^n - 2*(-1)^n, {n,0,25}] (* G. C. Greubel, Oct 14 2016 *)

From R. J. Mathar, Feb 19 2008: (Start)
O.g.f.: -1/(2*x-1) - 2/(1+x).
a(n) = 2^n - 2*(-1)^n. (End)
a(n) = 2*A014551(n-1), n>0. - Paul Curtz, Jun 01 2011
E.g.f.: exp(2*x) - 2*exp(-x). - G. C. Greubel, Oct 14 2016

More terms from R. J. Mathar, Feb 19 2008

A140253 a(2n) = 2(24^(n-1)-1) and a(2n-1) = 2*4^(n-1)-1.

Original entry on oeis.org

-1, 1, 2, 7, 14, 31, 62, 127, 254, 511, 1022, 2047, 4094, 8191, 16382, 32767, 65534, 131071, 262142, 524287, 1048574, 2097151, 4194302, 8388607, 16777214, 33554431, 67108862, 134217727, 268435454, 536870911

Offset: 0

Paul Curtz, Jun 23 2008

The inverse binomial transform is 1, 1, 4, -2, 10, -14, 34, -62 which leads to (-1)^(n+1)*A135440(n).
For n > 0: A266161(a(n)) = n and A266161(m) < n for m < a(n). - Reinhard Zumkeller, Dec 22 2015
Also, the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 23 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2, 1, -2).

Cf. A000034, A000069, A000079, A002450, A010674, A014551, A083420, A266161.

import Data.List (transpose)
a140253 n = a140253_list !! n
a140253_list = -1 : concat
                    (transpose [a083420_list, map (* 2) a083420_list])
-- Reinhard Zumkeller, Dec 22 2015

A140253:=proc(n): if type(n, odd) then 2*4^(((n+1)/2)-1)-1 else 2*(2*4^((n/2)-1)-1) fi: end: seq(A140253(n),n=0..29); # Johannes W. Meijer, Jun 24 2011

Table[(2^(n+1) - 3 + (-1)^(n+1))/2, {n, 0, 30}] (* Jean-François Alcover, Jun 05 2017 *)

a(2*n) = 2*A083420(n-1) and a(2*n+1) = A083420(n)
a(n+1) - a(n) = A014551(n); Jacobsthal-Lucas numbers.
a(2*n) + a(2*n+1) = 9*A002450(n)
a(n+1) - 2*a(n) = A010674(n+1); repeat 3, 0.
a(n) + A000034(n+1) = A000079(n); powers of 2.
a(n)= a(n-1) + 2*a(n-2) + 3. - Gary Detlefs, Jun 22 2010
a(n+1) = A000069(2^n); odious numbers. - Johannes W. Meijer, Jun 24 2011
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2, a(0) = -1, a(1) = 1, a(2) = 2. - Philippe Deléham, Feb 25 2012
G.f.: (x^2+3*x-1)/((1-2*x)*(1-x)*(1+x)). - Philippe Deléham, Feb 25 2012

Edited, corrected and information added by Johannes W. Meijer, Jun 24 2011

A166956 a(n) = 2^n +(-1)^n - 2.

Original entry on oeis.org

0, -1, 3, 5, 15, 29, 63, 125, 255, 509, 1023, 2045, 4095, 8189, 16383, 32765, 65535, 131069, 262143, 524285, 1048575, 2097149, 4194303, 8388605, 16777215, 33554429, 67108863, 134217725, 268435455, 536870909, 1073741823, 2147483645, 4294967295, 8589934589

Offset: 0

Paul Curtz, Oct 25 2009

The inverse binomial transform yields 0,-1,5,-7,17,-31,..., a sign alternating variant of A014551.
In a table of a(n) and higher-order differences in successive rows, the main diagonal contains 0, 4, 8, 16, ... (zero followed by A020707).
Similar to the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero, which begins with 1,3,5,15,29,63,125. - Robert Price, Aug 08 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..240
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,1,-2)

Cf. A166920, A166956, A290660, A290661, A290662.

[2^n-2+(-1)^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

LinearRecurrence[{2,1,-2},{0,-1,3},20] (* G. C. Greubel, May 29 2016 *)

a(n) = A000079(n) - A010684(n).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
G.f.: x*(5*x -1)/((1-x)*(1-2*x)*(1+x)).
E.g.f.: exp(2*x) - 2*exp(x) + exp(-x). - G. C. Greubel, May 29 2016

Edited and extended by R. J. Mathar, Mar 02 2010

A075118 Variant on Lucas numbers: a(n) = a(n-1) + 3*a(n-2) with a(0)=2 and a(1)=1.

Original entry on oeis.org

2, 1, 7, 10, 31, 61, 154, 337, 799, 1810, 4207, 9637, 22258, 51169, 117943, 271450, 625279, 1439629, 3315466, 7634353, 17580751, 40483810, 93226063, 214677493, 494355682, 1138388161, 2621455207, 6036619690, 13900985311, 32010844381, 73713800314, 169746333457

Offset: 0

Henry Bottomley, Sep 02 2002

The sequence 4,1,7,.. = 2*0^n+A075118(n) is given by trace(A^n) where A=[1,1,1,1;1,0,0,0;1,0,0,0;1,0,0,0]. - Paul Barry, Oct 01 2004

For n>2, a(n) is the numerator of the value of the continued fraction 1+3/(1+3/(1+...+3/7)) where there are n-2 1's. - Alexander Mark, Aug 16 2020

			a(4) = a(3)+3*a(2) = 10+3*7 = 31.

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001, p. 471.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (1,3).

Cf. A000032, A006130, A014551, A072265, A075117, A274977.

a:=[2,1];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, Jan 15 2020

I:=[2,1]; [n le 2 select I[n] else Self(n-1)+3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jul 20 2013

R:=PowerSeriesRing(Integers(), 33); Coefficients(R!((2-x)/(1-x-3*x^2))); // Marius A. Burtea, Jan 15 2020

a:= n-> (Matrix([[1,2]]). Matrix([[1,1], [3,0]])^n)[1,2]:
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 15 2008

a[0]=2; a[1]=1; a[n_]:= a[n]= a[n-1] +3a[n-2]; Table[a[n], {n, 0, 30}]
CoefficientList[Series[(2-x)/(1-x-3x^2), {x,0,40}], x] (* Vincenzo Librandi, Jul 20 2013 *)
LinearRecurrence[{1,3},{2,1},40] (* Harvey P. Dale, Jun 18 2017 *)
Table[Round[Sqrt[3]^n*LucasL[n, 1/Sqrt[3]]], {n,0,40}] (* G. C. Greubel, Jan 15 2020 *)

my(x='x+O('x^30)); Vec((2-x)/(1-x-3*x^2)) \\ G. C. Greubel, Dec 21 2017

polsym(x^2-x-3, 44) \\ Joerg Arndt, Jan 22 2023

[lucas_number2(n,1,-3) for n in range(0, 30)] # Zerinvary Lajos, Apr 30 2009

a(n) = ((1+sqrt(13))/2)^n + ((1-sqrt(13))/2)^n.
a(n) = 2*A006130(n) - A006130(n-1) = A075117(3, n).
G.f.: (2-x)/(1-x-3*x^2). - Philippe Deléham, Nov 15 2008
a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x + 13*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n) = 3^(n/2) * Lucas(n, 1/sqrt(3)). - G. C. Greubel, Jan 15 2020

cp's OEIS Frontend

A003665 a(n) = 2^(n-1)*( 2^n + (-1)^n ).

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A166920 a(n) = 2^n - (1 + (-1)^n)/2.

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A075117 Table by antidiagonals of generalized Lucas numbers: T(n,k) = T(n,k-1) + n*T(n,k-2) with T(n,0)=2 and T(n,1)=1.

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A101622 A Horadam-Jacobsthal sequence.

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A048700 Binary palindromes of odd length (written in base 10).

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A087451 G.f.: (2-x)/((1+2x)(1-3x)); e.g.f.: exp(3x)+exp(-2x); a(n)=3^n+(-2)^n.

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A140253 a(2n) = 2(24^(n-1)-1) and a(2n-1) = 2*4^(n-1)-1.