A014113 -id:A014113 - cp's OEIS frontend

A078008 Expansion of (1 - x)/((1 + x)(1 - 2x)).

1, 0, 2, 2, 6, 10, 22, 42, 86, 170, 342, 682, 1366, 2730, 5462, 10922, 21846, 43690, 87382, 174762, 349526, 699050, 1398102, 2796202, 5592406, 11184810, 22369622, 44739242, 89478486, 178956970, 357913942, 715827882, 1431655766, 2863311530, 5726623062

Offset: 0

N. J. A. Sloane, Nov 17 2002

Conjecture: a(n) = the number of fractions in the infinite Farey row of 2^n terms with even denominators. Compare the Salamin & Gosper item in the Beeler et al. link. - Gary W. Adamson, Oct 27 2003
Counts closed walks starting and ending at the same vertex of a triangle. 3*a(n) = P(C_n, 3) chromatic polynomial for 3 colors on cyclic graph C_n. A078008(n) + 2*A001045(n) = 2^n provides decomposition of Pascal's triangle. - Paul Barry, Nov 17 2003
Permutations with one fixed point avoiding 123 and 132.
General form: iterate k -> 2^n - k. See also A001045. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
The inverse g.f. generates sequence 1, 0, -2, -2, -2, -2, ...
a(n) gives the number of oriented (i.e., unreduced for symmetry) meanders on an (n+2) X 3 rectangular grid; see A201145. - Jon Wild, Nov 22 2011
Pisano period lengths: 1, 1, 6, 1, 4, 6, 6, 2, 18, 4, 10, 6, 12, 6, 12, 2, 8, 18, 18, 4, ... - R. J. Mathar, Aug 10 2012
a(n) is the number of length n binary words that end in an odd length run of 0's if we do not include the first letter of the word in our run length count. a(4) =6 because we have 0000, 0010, 0110, 1000, 1010, 1110. - Geoffrey Critzer, Dec 16 2013
a(n) is the top left entry of the n-th power of any of the six 3 X 3 matrices [0, 1, 1; 1, 1, 1; 1, 0, 0], [0, 1, 1; 1, 1, 0; 1, 1, 0], [0, 1, 1; 1, 0, 1; 1, 1, 0], [0, 1, 1; 1, 1, 0; 1, 0, 1], [0, 1, 1; 1, 0, 1; 1, 0, 1] or [0, 1, 1; 1, 0, 0; 1, 1, 1]. - R. J. Mathar, Feb 04 2014
a(n) is the number of compositions of n into parts of two kinds without part 1. - Gregory L. Simay, Jun 04 2018
a(n) is the number of words of length n over a binary alphabet whose position in the lexicographic order is a multiple of three.  a(3) = 2: aba, bab. - Alois P. Heinz, Apr 13 2022
a(n) is the number of words of length n over a ternary alphabet starting with a fixed letter (say, 'a') and ending in a different letter, such that no two adjacent letters are the same. a(4) = 6: abab, abac, abcb, acab, acac, acbc. - Ignat Soroko, Jul 19 2023

			G.f. = 1 + 2*x^2 + 2*x^3 + 6*x^4 + 10*x^5 + 22*x^6 + ... - _Michael Somos_, Mar 18 2022

Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 1.1.10a.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms n=0..300 from T. D. Noe)
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015. See Section 4.6.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Paul Barry and A. Hennessy, Notes on a Family of Riordan Arrays and Associated Integer Hankel Transforms , JIS 12 (2009) 09.5.3.
M. Beeler, R. W. Gosper, and R. C. Schroeppel, R. HAKMEM. MIT AI Memo 239, Feb 29 1972. (Item #54 by Salamin & Gosper)
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, arXiv:1907.06517 [math.CO], 2019.
Ernesto Estevez-Rams, Cristy Azanza Ricardo and Beatriz Aragón Fernández, An alternative expression for counting the number of close-packaged polytypes, Z. Krist. 220 (2005) 592-595, eq (4)
Leonhard Euler, Introductio in analysin infinitorum, (1748), section 216.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 13.
T. Mansour and A. Robertson, Refined Restricted Permutations Avoiding Subsets of Patterns of Length Three, arXiv:math/0204005 [math.CO], 2002.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for linear recurrences with constant coefficients, signature (1,2).
Index entries for sequences related to Chebyshev polynomials.

First differences of A001045.
See A151575 for a signed version.
Bisections: A047849, A020988.
Cf. A151548, A139250, A151555, A153006.

List([0..40], n-> (2^n + 2*(-1)^n)/3); # G. C. Greubel, Jun 28 2019

[(2^n + 2*(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Jun 28 2019

Table[(2^n + 2*(-1)^n)/3, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008; modified by G. C. Greubel, Jun 28 2019 *)
CoefficientList[Series[(1-x)/(1-x-2x^2),{x,0,40}],x]  (* Harvey P. Dale, Mar 30 2011 *)
LinearRecurrence[{1, 2}, {1, 0}, 40] (* Jean-François Alcover, Sep 23 2017 *)

a(n)=(1<Charles R Greathouse IV, Jun 10 2011

def A078008(n): return ((1<Chai Wah Wu, Apr 18 2025

[(2^n + 2*(-1)^n)/3 for n in (0..40)] # G. C. Greubel, Jun 28 2019

Euler expands(1-x)/(1 - x - 2*x^2) into an infinite series and finds that the coefficient of the n-th term is (2^n + (-1)^n 2)/3. Section 226 shows that Euler could have easily found the recursion relation: a(n) = a(n-1) + 2a(n-2) with a(0) = 1 and a(1) = 0. - V. Frederick Rickey (fred-rickey(AT)usma.edu), Feb 10 2006. [Typos corrected by Jaume Oliver Lafont, Jun 01 2009]
a(n) = Sum_{k=0..floor(n/3)} binomial(n, f(n)+3*k) where f(n) = (0, 2, 1, 0, 2, 1, ...) = A080424(n). - Paul Barry, Feb 20 2003
E.g.f. (exp(2*x) + 2*exp(-x))/3. - Paul Barry, Apr 20 2003
a(n) = A001045(n) + (-1)^n = A000079(n) - 2*A001045(n). - Paul Barry, Feb 20 2003
a(n) = (2^n + 2*(-1)^n)/3. - Mario Catalani (mario.catalani(AT)unito.it), Aug 29 2003
a(n) = T(n, i/(2*sqrt(2)))*(-i*sqrt(2))^n - U(n-1, i/(2*sqrt(2)))*(-i*sqrt(2))^(n-1)/2. - Paul Barry, Nov 17 2003
From Paul Barry, Jul 30 2004: (Start)
a(n) = 2*a(n-1) + 2*(-1)^n for n > 0, with a(0)=1.
a(n) = Sum_{k=0..n} (-1)^k*(2^(n-k-1) + 0^(n-k)/2). (End)
a(n) = A014113(n-1) for n > 0; a(n) = A052953(n-1) - 2*(n mod 2) = sum of n-th row of the triangle in A108561. - Reinhard Zumkeller, Jun 10 2005
A137208(n+1) - 2*A137208(n) = a(n) signed. - Paul Curtz, Aug 03 2008
a(n) = A001045(n+1) - A001045(n) - Paul Curtz, Feb 09 2009
If p[1] =0, and p[i]=2, (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
a(n) = 2*(a(n-2) + a(n-3) + a(n-4) ... + a(0)), that is, twice the sum of all the previous terms except the last; with a(0) = 1 and a(1) = 0. - Benoit Jubin, Nov 21 2011
a(n+1) = 2*A001045(n). - Benoit Jubin, Nov 22 2011
G.f.: 1 - x + x*Q(0), where Q(k) = 1 + 2*x^2 + (2*k+3)*x - x*(2*k+1 + 2*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013
G.f.: 1+ x^2*Q(0), where Q(k) = 1 + 1/(1 - x*(4*k+1+2*x)/(x*(4*k+3+2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 01 2014
a(n) = 3*a(n-2) + 2*a(n-3). - David Neil McGrath, Sep 10 2014
a(n) = (-1)^n * A151575(n). - G. C. Greubel, Jun 28 2019
a(n)+a(n+1) = 2^n. - R. J. Mathar, Feb 24 2021
a(n) = -a(2-n) * (-2)^(n-1) = (3/2)*(a(n-1) + a(n-2)) - a(n-3) for all n in Z. - Michael Somos, Mar 18 2022

A163271 Numerators of fractions in a 'zero-transform' approximation of sqrt(2) by means of a(n) = (a(n-1) + c)/(a(n-1) + 1) with c=2 and a(1)=0.

Original entry on oeis.org

0, 2, 4, 10, 24, 58, 140, 338, 816, 1970, 4756, 11482, 27720, 66922, 161564, 390050, 941664, 2273378, 5488420, 13250218, 31988856, 77227930, 186444716, 450117362, 1086679440, 2623476242, 6333631924, 15290740090, 36915112104, 89120964298, 215157040700

Offset: 1

Mark Dols, Jul 24 2009

Twice the Pell numbers; for denominators see A001333 (numerators of the approximation of sqrt(2) for a(1) = 1).
Row sums of the triangle A128966. - Reinhard Zumkeller, Jul 20 2013
Because a(n+1)/A001333(n) approximates sqrt(2) and a(n) = 2*A001333(n) - a(n+1), we get that a(n)/A001333(n) approximates 2 - sqrt(2). - Danny Rorabaugh, Mar 14 2015
Number of weak orderings R on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1 < ... < n and for which {1,...,n} has exactly one minimal and one maximal element for the weak ordering R. - J. Devillet, Sep 28 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA], 2017.
I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5.
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A000129 (Pell numbers), A001333 (denominators), A052542.
Cf. A014113, A189687.
Cf. A293004.

a := [0, 2];; for n in [3..10^2] do a[n] := 2*a[n-1] + a[n-2]; od; A163271:=a; # Muniru A Asiru, Oct 08 2017

a163271 = sum . a128966_row . (subtract 1)
-- Reinhard Zumkeller, Jul 20 2013

A163271:=gfun:-rectoproc({a(n) = 2 * a(n-1) + a(n-2), a(1) = 0, a(2) = 2}, a(n), remember):  map(A163271, [$1..100]);  # Muniru A Asiru, Oct 08 2017

CoefficientList[Series[2*t^2/(1-2*t - t^2), {t,0,50}], t] (* or *) LinearRecurrence[{2,1},{0,2},50] (* G. C. Greubel, Dec 12 2016 *)

concat([0], Vec(2*t^2/(1-2*t - t^2) + O(t^50))) \\ G. C. Greubel, Dec 12 2016

a(n) = A052542(n-1), n > 1.
G.f.: x + x^2/(2*G(0)-x) where G(k) = 1 - (k+1)/(1 - x/(x +(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2012
a(n) = 2*a(n-1) + a(n-2). - Jacob Antony, Jun 07 2013
a(n) = b(n) - b(n-1) = 2b(n-1) - a(n-1) where b(n) = A001333(n). - Danny Rorabaugh, Mar 14 2015
G.f.: 2*t^2/(1 - 2*t - t^2). - G. C. Greubel, Dec 12 2016
a(n) = 2*A000129(n-1) (see the first comment). - J. Devillet, Sep 28 2017

A216800 T(n,k)=Number of permutations of an nXk array with each element moving exactly one horizontally or vertically and without 2-loops.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 6, 0, 6, 0, 0, 10, 8, 8, 10, 0, 0, 22, 0, 88, 0, 22, 0, 0, 42, 32, 292, 292, 32, 42, 0, 0, 86, 0, 1774, 0, 1774, 0, 86, 0, 0, 170, 128, 7676, 10140, 10140, 7676, 128, 170, 0, 0, 342, 0, 39844, 0, 207408, 0, 39844, 0, 342, 0, 0, 682, 512, 186996

Offset: 1

R. H. Hardin Sep 17 2012

Table starts
.0....0....0........0.........0............0..............0.................0
.0....2....2........6........10...........22.............42................86
.0....2....0........8.........0...........32..............0...............128
.0....6....8.......88.......292.........1774...........7676.............39844
.0...10....0......292.........0........10140..............0............361200
.0...22...32.....1774.....10140.......207408........1879040..........27918806
.0...42....0.....7676.........0......1879040..............0.........489300384
.0...86..128....39844....361200.....27918806......489300384.......22902801416
.0..170....0...186996.........0....302667484..............0......539812977524
.0..342..512...927134..12911864...3991662992...129127695440....20296105418276
.0..682....0..4460016.........0..46492284664..............0...550843711073468
.0.1366.2048.21812696.461788640.585384941184.34175930646380.18708387030790218

			Some solutions for n=4 k=4
..1..5..3..7....1..5..3..7....4..0..1..2....1..5..3..7....4..0..1..2
..0..9..2..6....0..4..2..6....8..6..7..3....0..4..2..6....5..6..7..3
..4.13.11.15....9.13.11.15...12..5..9.10...12..8.14.10...12..8..9.10
..8.12.10.14....8.12.10.14...13.14.15.11...13..9.15.11...13.14.15.11

R. H. Hardin, Table of n, a(n) for n = 1..312

Column 2 is A014113(n-1)

Diagonal is A216678

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

A290732 Number of distinct values of X(3X-1)/2 mod n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 4, 8, 9, 6, 6, 12, 7, 8, 9, 16, 9, 18, 10, 12, 12, 12, 12, 24, 11, 14, 27, 16, 15, 18, 16, 32, 18, 18, 12, 36, 19, 20, 21, 24, 21, 24, 22, 24, 27, 24, 24, 48, 22, 22, 27, 28, 27, 54, 18, 32, 30, 30, 30, 36

Offset: 1

N. J. A. Sloane, Aug 10 2017

			The values taken by (3*X^2-X)/2 mod n for small n are:
   1, [0]
   2, [0, 1]
   3, [0, 1, 2]
   4, [0, 1, 2, 3]
   5, [0, 1, 2]
   6, [0, 1, 2, 3, 4, 5]
   7, [0, 1, 2, 5]
   8, [0, 1, 2, 3, 4, 5, 6, 7]
   9, [0, 1, 2, 3, 4, 5, 6, 7, 8]
  10, [0, 1, 2, 5, 6, 7]
  11, [0, 1, 2, 4, 5, 7]
  12, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], (24-August-2016). See Table 6.

Cf. A000224 (analog for X^2), A014113, A290729, A290730, A290731, A317623.

a:=[]; M:=80;
for n from 1 to M do
q1:={};
for i from 0 to 2*n-1 do q1:={op(q1), i*(3*i-1)/2 mod n}; od;
s1:=sort(convert(q1,list));
a:=[op(a),nops(s1)];
od:
a;

a[n_] := Table[PolynomialMod[X(3X-1)/2, n], {X, 0, 2*n-1}]// Union // Length;
Array[a, 60] (* Jean-François Alcover, Sep 01 2018 *)

a(n)={my(v=vector(n)); for(i=0, 2*n-1, v[i*(3*i-1)/2%n + 1]=1); vecsum(v)} \\ Andrew Howroyd, Oct 27 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); if(p<=3, p^e, 1 + p^(e+1)\(2*p+2)))} \\ Andrew Howroyd, Nov 03 2018

a(3^n) = 3^n. - Hugo Pfoertner, Aug 25 2018
a(n) = A317623(n) * A040001(n). - Andrew Howroyd, Oct 27 2018
Multiplicative with a(2^e) = 2^e, a(3^e) = 3^e, a(p^e) = 1 + floor( p^(e+1)/(2*p+2) ) for prime p >= 5. - Andrew Howroyd, Nov 03 2018

Even terms corrected by Andrew Howroyd, Nov 03 2018

A290604 a(0) = 2, a(1) = 2; for n > 1, a(n) = a(n-1) + 2*a(n-2) + 3.

Original entry on oeis.org

2, 2, 9, 16, 37, 72, 149, 296, 597, 1192, 2389, 4776, 9557, 19112, 38229, 76456, 152917, 305832, 611669, 1223336, 2446677, 4893352, 9786709, 19573416, 39146837, 78293672, 156587349, 313174696, 626349397, 1252698792, 2505397589, 5010795176, 10021590357

Offset: 0

Iain Fox, Oct 11 2017

Ratio of successive terms approaches 2.

			a(0) = 2.
a(1) = 2.
a(2) = 2 + 2*2 + 3 = 9.
a(3) = 9 + 2*2 + 3 = 16.
a(4) = 16 + 9*2 + 3 = 37.
...

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

[(2^(n+2)+2*(-1)^n)/3+2^n-(3-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Oct 20 2017

Table[(2^(n + 2) + 2 (-1)^n) / 3 + 2^n - (3 - (-1)^n) / 2, {n, 0, 40}] (* Vincenzo Librandi, Oct 20 2017 *)
LinearRecurrence[{2,1,-2},{2,2,9},50] (* Harvey P. Dale, Mar 06 2025 *)

Vec((2/(1-x-2*x^2)) + (3*x^2/((1-x)*(1-x-2*x^2))) + O(x^50)) \\ Michel Marcus, Oct 12 2017

first(n) = Vec((2 - 2*x + 3*x^2)/(1 - 2*x - x^2 + 2*x^3) + O(x^n)) \\ Iain Fox, Dec 18 2017

a(n) = (2^(n+2) + 2*(-1)^n)/3 + 2^n - (3-(-1)^n)/2.
a(n) = A014113(n+1) + A141023(n).
G.f.: (2 - 2*x + 3*x^2)/(1 - 2*x - x^2 + 2*x^3).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n > 2. - Iain Fox, Dec 18 2017

A321373 Array T(n,k) read by antidiagonals where the first row is (-1)^k*A140966(k) and each subsequent row is obtained by adding A001045(k) to the preceding one.

Original entry on oeis.org

2, 2, -1, 2, 0, 3, 2, 1, 4, 1, 2, 2, 5, 4, 7, 2, 3, 6, 7, 12, 9, 2, 4, 7, 10, 17, 20, 23, 2, 5, 8, 13, 22, 31, 44, 41, 2, 6, 9, 16, 27, 42, 65, 84, 87, 2, 7, 10, 19, 32, 53, 86, 127, 172, 169, 2, 8, 11, 22, 37, 64, 107, 170, 257, 340, 343

Offset: 0

Paul Curtz, Nov 08 2018

Array:
2, -1,  3,  1,  7,  9,  23,  41,  87, ... = (-1)^n*A140966(n)
2,  0,  4,  4, 12, 20,  44,  84, 172, ... = abs(A084247(n+1))
2,  1,  5,  7, 17, 31,  65, 127, 257, ... = A014551(n)
2,  2,  6, 10, 22, 42,  86, 170, 342, ... = A078008(n+2) = A014113(n+1)
2,  3,  7, 13, 27, 53, 107, 213, 427, ... = A048573(n)
2,  4,  8, 16, 32, 64, 128, 256, 512, ... = A000079(n+1)
2,  5,  9, 19, 37, 75, 149, 299, 597, ... = A062092(n)
2,  6, 10, 22, 42, 86, 170, 342, 682, ... = A078008(n+3) = A014113(n+2).
T(n+1,k) = (-1)^k*A140966(k) + (n+1)*A001045(k).
Every row T(n+1,k) has the signature (1,2).
T(0,k) = 2, -2, 2, -2, ... = (-1)^n*2.
T(n+1,k) - T(0,k) = (n+1)*A001045(n).
5*A001045(n) is not in the OEIS.

			Triangle a(n):
  2;
  2, -1;
  2,  0,  3;
  2,  1,  4,  1;
  2,  2,  5,  4,  7;
  2,  3,  6,  7, 12,  9;
  2,  4,  7, 10, 17, 20, 23;
  etc.
Row sums: 2, 1, 5, 8, 20, 39, 83, 166, 338, 677, 1361, 2724, ... = b(n+2).
With b(0) = 2 and b(1) = 0, b(n) = b(n-1) + 2*b(n-2)  + n - 4, n > 1.
b(n) = A001045(n) - A097065(n-1).
b(n) = b(n-2) + A000225(n-2).

Cf. A000079, A001045, A014113, A014551, A048573, A062092, A078008, A084247, A092297, A097073, A140360, A140966.

Cf. A000225, A097065.

T[_, 0] = 2;
T[0, k_] := (2^k + 5(-1)^k)/3;
T[n_ /; n>0, k_ /; k>0] := T[n, k] = T[n-1, k] + (2^k + (-1)^(k+1))/3;
T[, ] = 0;
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

cp's OEIS Frontend

A078008 Expansion of (1 - x)/((1 + x)*(1 - 2*x)).

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A163271 Numerators of fractions in a 'zero-transform' approximation of sqrt(2) by means of a(n) = (a(n-1) + c)/(a(n-1) + 1) with c=2 and a(1)=0.

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A216800 T(n,k)=Number of permutations of an nXk array with each element moving exactly one horizontally or vertically and without 2-loops.

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

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A290732 Number of distinct values of X*(3*X-1)/2 mod n.

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A290604 a(0) = 2, a(1) = 2; for n > 1, a(n) = a(n-1) + 2*a(n-2) + 3.

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A321373 Array T(n,k) read by antidiagonals where the first row is (-1)^k*A140966(k) and each subsequent row is obtained by adding A001045(k) to the preceding one.

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A078008 Expansion of (1 - x)/((1 + x)(1 - 2x)).

A290732 Number of distinct values of X(3X-1)/2 mod n.