A077957 -id:A077957 - cp's OEIS frontend

A264036 Stirling transform of A077957 (aerated powers of 2).

1, 0, 2, 6, 18, 70, 330, 1694, 9202, 53334, 332090, 2212782, 15638370, 116365990, 907975146, 7413080510, 63212284498, 561747543414, 5190343710746, 49752410984526, 493844719701186, 5068209425457862, 53705511911500746, 586862875255860062, 6605213319604075186

Offset: 0

Vladimir Reshetnikov, Nov 01 2015

nonn

a(n) is the inverse binomial transform of A264037 without the leading zero [1, 1, 3, 13, 55, ...].

			G.f. = 1 + 2*x^2 + 6*x^3 + 18*x^4 + 70*x^5 + 330*x^6 + 1694*x^7 + 9202*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..567
Eric Weisstein's MathWorld, Bell Polynomial.

Column k=2 of A357681.

Cf. A065143, A077957, A264037.

Table[(BellB[n, Sqrt[2]] + BellB[n, -Sqrt[2]])/2, {n, 0, 24}]

vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k, 2))) \\ Altug Alkan, Nov 01 2015

a(n) = Sum_{k=0..n} A077957(k)*Stirling2(n,k).
a(n) = Sum_{k=0..floor(n/2)} 2^k*Stirling2(n,2*k).
a(n) = (Bell_n(sqrt(2)) + Bell_n(-sqrt(2)))/2, where Bell_n(x) is n-th Bell polynomial.
Bell_n(sqrt(2)) = a(n) + A264037(n)*sqrt(2).
E.g.f.: cosh(sqrt(2)*(exp(x) - 1)).
a(n) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k) * A264037(k). - Seiichi Manyama, Oct 12 2022

A264037 Stirling transform of A077957 (aerated powers of 2) with 0 prepended [0, 1, 0, 2, 0, 4, 0, 8, ...].

Original entry on oeis.org

0, 1, 1, 3, 13, 55, 241, 1171, 6357, 37567, 236521, 1574331, 11068333, 82110535, 640794337, 5239439011, 44723250501, 397481121295, 3671081354137, 35176098791115, 349120380267421, 3583273413146647, 37975511840454673, 415004245048757299, 4670891190907818165

Offset: 0

Vladimir Reshetnikov, Nov 01 2015

nonn

a(n) without the leading zero [1, 1, 3, 13, 55, ...] is the binomial transform of A264036.

			G.f. = x + x^2 + 3*x^3 + 13*x^4 + 55*x^5 + 241*x^7 + 1171*x^8 + 6357*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..567
Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A077957, A264036.

Cf. A024429, A357572, A357598.

Table[(BellB[n, Sqrt[2]] - BellB[n, -Sqrt[2]])/(2 Sqrt[2]), {n, 0, 24}]

vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k+1, 2))) \\ Altug Alkan, Nov 01 2015

a(n) = Sum_{k=0..floor(n/2)} 2^k*Stirling2(n,2*k+1).
a(n) = (Bell_n(sqrt(2)) - Bell_n(-sqrt(2)))/(2*sqrt(2)), where Bell_n(x) is n-th Bell polynomial.
Bell_n(sqrt(2)) = A264036(n) + a(n)*sqrt(2).
E.g.f.: sinh(sqrt(2)*(exp(x) - 1))/sqrt(2).
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A264036(k). - Seiichi Manyama, Oct 12 2022

A171700 Triangle T : T(n,k)= A007318(n,k)*a(n-k) with a(0)=0 and a(n)= A077957(n-1) for n>0.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 2, 0, 3, 0, 0, 8, 0, 4, 0, 4, 0, 20, 0, 5, 0, 0, 24, 0, 40, 0, 6, 0, 8, 0, 84, 0, 70, 0, 7, 0, 0, 64, 0, 224, 0, 112, 0, 8, 0, 16, 0, 288, 0, 504, 0, 168, 0, 9, 0, 0, 160, 0, 960, 0, 1008, 0, 240, 0, 10, 0

Offset: 0

Philippe Deléham, Dec 15 2009

Diagonal sums : A001353(n+1) alternating with zeros.

			Triangle begins : 0 ; 1,0 ; 0,2,0 ; 2,0,3,0 ; 0,8,0,4,0 ; 4,0,20,0,5,0 ; ...

Sum_{k, 0<=k<=n} T(n,k)*x^k = A077957(n-1), A000129(n), A007070(n-1), A081179(n), A081180(n), A081182(n), A081183(n), A081184(n), A081185(n), A153593(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.

A001333 Pell-Lucas numbers: numerators of continued fraction convergents to sqrt(2).

Original entry on oeis.org

1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119, 19601, 47321, 114243, 275807, 665857, 1607521, 3880899, 9369319, 22619537, 54608393, 131836323, 318281039, 768398401, 1855077841, 4478554083, 10812186007, 26102926097, 63018038201, 152139002499, 367296043199

Offset: 0

N. J. A. Sloane and R. K. Guy

Number of n-step non-selfintersecting paths starting at (0,0) with steps of types (1,0), (-1,0) or (0,1) [Stanley].
Number of n steps one-sided prudent walks with east, west and north steps. - Shanzhen Gao, Apr 26 2011
Number of ternary strings of length n-1 with subwords (0,2) and (2,0) not allowed. - Olivier Gérard, Aug 28 2012
Number of symmetric 2n X 2 or (2n-1) X 2 crossword puzzle grids: all white squares are edge connected; at least 1 white square on every edge of grid; 180-degree rotational symmetry. - Erich Friedman
a(n+1) is the number of ways to put molecules on a 2 X n ladder lattice so that the molecules do not touch each other.
In other words, a(n+1) is the number of independent vertex sets and vertex covers in the n-ladder graph P_2 X P_n. - Eric W. Weisstein, Apr 04 2017
Number of (n-1) X 2 binary arrays with a path of adjacent 1's from top row to bottom row, see A359576. - R. H. Hardin, Mar 16 2002
a(2*n+1) with b(2*n+1) := A000129(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation a^2 - 2*b^2 = -1.
a(2*n) with b(2*n) := A000129(2*n), n >= 1, give all (positive integer) solutions to Pell equation a^2 - 2*b^2 = +1 (see Emerson reference).
Bisection: a(2*n) = T(n,3) = A001541(n), n >= 0 and a(2*n+1) = S(2*n,2*sqrt(2)) = A002315(n), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. See A053120, resp. A049310.
Binomial transform of A077957. - Paul Barry, Feb 25 2003
For n > 0, the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 2. - Herbert Kociemba, Jun 02 2004
For n > 1, a(n) corresponds to the longer side of a near right-angled isosceles triangle, one of the equal sides being A000129(n). - Lekraj Beedassy, Aug 06 2004
Exponents of terms in the series F(x,1), where F is determined by the equation F(x,y) = xy + F(x^2*y,x). - Jonathan Sondow, Dec 18 2004
Number of n-words from the alphabet A={0,1,2} which two neighbors differ by at most 1. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 30 2006
Consider the mapping f(a/b) = (a + 2b)/(a + b). Taking a = b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 3/2, 7/5, 17/12, 41/29, ... converging to 2^(1/2). Sequence contains the numerators. - Amarnath Murthy, Mar 22 2003 [Amended by Paul E. Black (paul.black(AT)nist.gov), Dec 18 2006]
Odd-indexed prime numerators are prime RMS numbers (A140480) and also NSW primes (A088165). - Ctibor O. Zizka, Aug 13 2008
The intermediate convergents to 2^(1/2) begin with 4/3, 10/7, 24/17, 58/41; essentially, numerators=A052542 and denominators here. - Clark Kimberling, Aug 26 2008
Equals right border of triangle A143966. Starting (1, 3, 7, ...) equals INVERT transform of (1, 2, 2, 2, ...) and row sums of triangle A143966. - Gary W. Adamson, Sep 06 2008
Inverse binomial transform of A006012; Hankel transform is := [1, 2, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008
From Charlie Marion, Jan 07 2009: (Start)
In general, denominators, a(k,n) and numerators, b(k,n), of continued fraction convergents to sqrt((k+1)/k) may be found as follows:
let a(k,0) = 1, a(k,1) = 2k; for n>0, a(k,2n) = 2*a(k,2n-1) + a(k,2n-2) and a(k,2n+1) = (2k)*a(k,2n) + a(k,2n-1);
let b(k,0) = 1, b(k,1) = 2k+1; for n>0, b(k,2n) = 2*b(k,2n-1) + b(k,2n-2) and b(k,2n+1) = (2k)*b(k,2n) + b(k,2n-1).
For example, the convergents to sqrt(2/1) start 1/1, 3/2, 7/5, 17/12, 41/29.
In general, if a(k,n) and b(k,n) are the denominators and numerators, respectively, of continued fraction convergents to sqrt((k+1)/k) as defined above, then
k*a(k,2n)^2 - a(k,2n-1)*a(k,2n+1) = k = k*a(k,2n-2)*a(k,2n) - a(k,2n-1)^2 and
b(k,2n-1)*b(k,2n+1) - k*b(k,2n)^2 = k+1 = b(k,2n-1)^2 - k*b(k,2n-2)*b(k,2n);
for example, if k=1 and n=3, then b(1,n)=a(n+1) and
1*a(1,6)^2 - a(1,5)*a(1,7) = 1*169^2 - 70*408 = 1;
1*a(1,4)*a(1,6) - a(1,5)^2 = 1*29*169 - 70^2 = 1;
b(1,5)*b(1,7) - 1*b(1,6)^2 = 99*577 - 1*239^2 = 2;
b(1,5)^2 - 1*b(1,4)*b(1,6) = 99^2 - 1*41*239 = 2.
Cf. A000129, A142238-A142239, A153313-A153318.
(End)
This sequence occurs in the lower bound of the order of the set of equivalent resistances of n equal resistors combined in series and in parallel (A048211). - Sameen Ahmed Khan, Jun 28 2010
Let M = a triangle with the Fibonacci series in each column, but the leftmost column is shifted upwards one row. A001333 = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 27 2010
a(n) is the number of compositions of n when there are 1 type of 1 and 2 types of other natural numbers. - Milan Janjic, Aug 13 2010
Equals the INVERTi transform of A055099. - Gary W. Adamson, Aug 14 2010
From L. Edson Jeffery, Apr 04 2011: (Start)
Let U be the unit-primitive matrix (see [Jeffery])
U = U_(8,2) = (0 0 1 0)
              (0 1 0 1)
              (1 0 2 0)
              (0 2 0 1).
Then a(n) = (1/4)*Trace(U^n). (See also A084130, A006012.)
(End)
For n >= 1, row sums of triangle
  m/k.|..0.....1.....2.....3.....4.....5.....6.....7
  ==================================================
  .0..|..1
  .1..|..1.....2
  .2..|..1.....2.....4
  .3..|..1.....4.....4.....8
  .4..|..1.....4....12.....8....16
  .5..|..1.....6....12....32....16....32
  .6..|..1.....6....24....32....80....32....64
  .7..|..1.....8....24....80....80...192....64...128
which is the triangle for numbers 2^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 12 2012
a(n) is also the number of ways to place k non-attacking wazirs on a 2 X n board, summed over all k >= 0 (a wazir is a leaper [0,1]). - Vaclav Kotesovec, May 08 2012
The sequences a(n) and b(n) := A000129(n) are entries of powers of the special case of the Brahmagupta Matrix - for details see Suryanarayan's paper. Further, as Suryanarayan remark, if we set A = 2*(a(n) + b(n))*b(n), B = a(n)*(a(n) + 2*b(n)), C = a(n)^2 + 2*a(n)*b(n) + 2*b(n)^2 we obtain integral solutions of the Pythagorean relation A^2 + B^2 = C^2, where A and B are consecutive integers. - Roman Witula, Jul 28 2012
Pisano period lengths: 1, 1, 8, 4, 12, 8, 6, 4, 24, 12, 24, 8, 28, 6, 24, 8, 16, 24, 40, 12, .... - R. J. Mathar, Aug 10 2012
This sequence and A000129 give the diagonal numbers described by Theon of Smyrna. - Sture Sjöstedt, Oct 20 2012
a(n) is the top left entry of the n-th power of any of the following six 3 X 3 binary matrices: [1, 1, 1; 1, 1, 1; 1, 0, 0] or [1, 1, 1; 1, 1, 0; 1, 1, 0] or [1, 1, 1; 1, 0, 1; 1, 1, 0] or [1, 1, 1; 1, 1, 0; 1, 0, 1] or [1, 1, 1; 1, 0, 1; 1, 0, 1] or [1, 1, 1; 1, 0, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
If p is prime, a(p) == 1 (mod p) (compare with similar comment for A000032). - Creighton Dement, Oct 11 2005, modified by Davide Colazingari, Jun 26 2016
a(n) = A000129(n) + A000129(n-1), where A000129(n) is the n-th Pell Number; e.g., a(6) = 99 = A000129(6) + A000129(5) = 70 + 29. Hence the sequence of fractions has the form 1 + A000129(n-1)/A000129(n), and the ratio A000129(n-1)/A000129(n)converges to sqrt(2) - 1. - Gregory L. Simay, Nov 30 2018
For n > 0, a(n+1) is the length of tau^n(1) where tau is the morphism: 1 -> 101, 0 -> 1. See Song and Wu. - Michel Marcus, Jul 21 2020
For n > 0, a(n) is the number of nonisomorphic quasitrivial semigroups with n elements, see Devillet, Marichal, Teheux. A292932 is the number of labeled quasitrivial semigroups. - Peter Jipsen, Mar 28 2021
a(n) is the permanent of the n X n tridiagonal matrix defined in A332602. - Stefano Spezia, Apr 12 2022
From Greg Dresden, May 08 2023: (Start)
For n >= 2, 4*a(n) is the number of ways to tile this T-shaped figure of length n-1 with two colors of squares and one color of domino; shown here is the figure of length 5 (corresponding to n=6), and it has 4*a(6) = 396 different tilings.
   _
  || _  
  |||_|||
  |_|
(End)
12*a(n) = number of walks of length n in the cyclic Kautz digraph CK(3,4). - Miquel A. Fiol, Feb 15 2024

			Convergents are 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, 8119/5741, 19601/13860, 47321/33461, 114243/80782, ... = A001333/A000129.
The 15 3 X 2 crossword grids, with white squares represented by an o:
  ooo ooo ooo ooo ooo ooo ooo oo. o.o .oo o.. .o. ..o oo. .oo
  ooo oo. o.o .oo o.. .o. ..o ooo ooo ooo ooo ooo ooo .oo oo.
G.f. = 1 + x + 3*x^2 + 7*x^3 + 17*x^4 + 41*x^5 + 99*x^6 + 239*x^7 + 577*x^8 + ...

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Haocong Song and Wen Wu, Hankel determinants of a Sturmian sequence, arXiv:2007.09940 [math.CO], 2020. See pp. 2, 4.
Claude Soudieux, De l'infini arithmétique, Zurich, 1960. [Annotated scans of selected pages. Contains many sequences including A1333]
E. R. Suryanarayan, The Brahmagupta Polynomials, Fibonacci Quarterly, 34.1 (1996), 30-39.
Wipawee Tangjai, A Non-standard Ternary Representation of Integers, Thai J. Math (2020) Special Issue: Annual Meeting in Mathematics 2019, 269-283.
Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith. [Annotated scanned copy]
G. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes. I, J. Math. Chemistry, 25, 1999, 55-64 (see Eq. (21)).
G. Tasi et al., Quantum algebraic-combinatoric study of the conformational properties of n-alkanes. II, J. Math. Chemistry, 27, 2000, 191-199 (see p. 193).
V. Thebault, Concerning two classes of remarkable perfect square pairs, Amer. Math. Monthly, 56 (1949), 443-448.
Lucyna Trojnar-Spelina and Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
Andrew Vince, The average size of a connected vertex set of a graph - Explicit formulas and open problems, Journal of Graph Theory, Volume 97, Issue 1 pp. 82-103.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Ladder Graph.
Eric Weisstein's World of Mathematics, Pythagoras's Constant.
Eric Weisstein's World of Mathematics, Square Root.
Eric Weisstein's World of Mathematics, Square Triangular Number.
Eric Weisstein's World of Mathematics, Vertex Cover.
Index entries for "core" sequences.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (2,1).

For denominators see A000129.
See A040000 for the continued fraction expansion of sqrt(2).
See also A078057 which is the same sequence without the initial 1.
Cf. also A002203, A152113.
Row sums of unsigned Chebyshev T-triangle A053120. a(n)= A054458(n, 0) (first column of convolution triangle).
Row sums of A140750, A160756, A135837.
Equals A034182(n-1) + 2 and A084128(n)/2^n. First differences of A052937. Partial sums of A052542. Pairwise sums of A048624. Bisection of A002965.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Second row of the array in A135597.
Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176500, A176501, A176502.
Cf. A055099.
Cf. A028859, A001906 / A088305, A033303, A000225, A095263, A003945, A006356, A002478, A214260, A001911 and A000217 for other restricted ternary words.
Cf. Triangle A106513 (alternating row sums).
Equals A293004 + 1.
Cf. A033539, A332602, A086395 (subseq. of primes).

a001333 n = a001333_list !! n
a001333_list = 1 : 1 : zipWith (+)
                       a001333_list (map (* 2) $ tail a001333_list)
-- Reinhard Zumkeller, Jul 08 2012

[n le 2 select 1 else 2*Self(n-1)+Self(n-2): n in [1..35]]; // Vincenzo Librandi, Nov 10 2018

A001333 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else 2*procname(n-1)+procname(n-2) fi end;
Digits := 50; A001333 := n-> round((1/2)*(1+sqrt(2))^n);
with(numtheory): cf := cfrac (sqrt(2),1000): [seq(nthnumer(cf,i), i=0..50)];
a:= n-> (M-> M[2, 1]+M[2, 2])(<<2|1>, <1|0>>^n):
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 01 2008
A001333List := proc(m) local A, P, n; A := [1,1]; P := [1,1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(A), P[-2]]);
A := [op(A), P[-1]] od; A end: A001333List(32); # Peter Luschny, Mar 26 2022

Insert[Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[2], n]]], {n, 1, 40}], 1, 1] (* Stefan Steinerberger, Apr 08 2006 *)
Table[((1 - Sqrt[2])^n + (1 + Sqrt[2])^n)/2, {n, 0, 29}] // Simplify (* Robert G. Wilson v, May 02 2006 *)
a[0] = 1; a[1] = 1; a[n_] := a[n] = 2a[n - 1] + a[n - 2]; Table[a@n, {n, 0, 29}] (* Robert G. Wilson v, May 02 2006 *)
Table[ MatrixPower[{{1, 2}, {1, 1}}, n][[1, 1]], {n, 0, 30}] (* Robert G. Wilson v, May 02 2006 *)
a=c=0;t={b=1}; Do[c=a+b+c; AppendTo[t,c]; a=b;b=c,{n,40}]; t (* Vladimir Joseph Stephan Orlovsky, Mar 23 2009 *)
LinearRecurrence[{2, 1}, {1, 1}, 40] (* Vladimir Joseph Stephan Orlovsky, Mar 23 2009 *)
Join[{1}, Numerator[Convergents[Sqrt[2], 30]]] (* Harvey P. Dale, Aug 22 2011 *)
Table[(-I)^n ChebyshevT[n, I], {n, 10}] (* Eric W. Weisstein, Apr 04 2017 *)
CoefficientList[Series[(-1 + x)/(-1 + 2 x + x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)
Table[Sqrt[(ChebyshevT[n, 3] + (-1)^n)/2], {n, 0, 20}] (* Eric W. Weisstein, Apr 17 2018 *)

{a(n) = if( n<0, (-1)^n, 1) * contfracpnqn( vector( abs(n), i, 1 + (i>1))) [1, 1]}; /* Michael Somos, Sep 02 2012 */

{a(n) = polchebyshev(n, 1, I) / I^n}; /* Michael Somos, Sep 02 2012 */

a(n) = real((1 + quadgen(8))^n); \\ Michel Marcus, Mar 16 2021

{ for (n=0, 4000, a=contfracpnqn(vector(n, i, 1+(i>1)))[1, 1]; if (a > 10^(10^3 - 6), break); write("b001333.txt", n, " ", a); ); } \\ Harry J. Smith, Jun 12 2009

from functools import cache
@cache
def a(n): return 1 if n < 2 else 2*a(n-1) + a(n-2)
print([a(n) for n in range(32)]) # Michael S. Branicky, Nov 13 2022

from sage.combinat.sloane_functions import recur_gen2
it = recur_gen2(1,1,2,1)
[next(it) for i in range(30)] ## Zerinvary Lajos, Jun 24 2008

[lucas_number2(n,2,-1)/2 for n in range(0, 30)] # Zerinvary Lajos, Apr 30 2009

a(n) = A055642(A125058(n)). - Reinhard Zumkeller, Feb 02 2007
a(n) = 2a(n-1) + a(n-2);
a(n) = ((1-sqrt(2))^n + (1+sqrt(2))^n)/2.
a(n)+a(n+1) = 2 A000129(n+1). 2*a(n) = A002203(n).
G.f.: (1 - x) / (1 - 2*x - x^2) = 1 / (1 - x / (1 - 2*x / (1 + x))). - Simon Plouffe in his 1992 dissertation.
A000129(2n) = 2*A000129(n)*a(n). - John McNamara, Oct 30 2002
a(n) = (-i)^n * T(n, i), with T(n, x) Chebyshev's polynomials of the first kind A053120 and i^2 = -1.
a(n) = a(n-1) + A052542(n-1), n>1. a(n)/A052542(n) converges to sqrt(1/2). - Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003
E.g.f.: exp(x)cosh(x*sqrt(2)). - Paul Barry, May 08 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)2^k. - Paul Barry, May 13 2003
For n > 0, a(n)^2 - (1 + (-1)^(n))/2 = Sum_{k=0..n-1} ((2k+1)*A001653(n-1-k)); e.g., 17^2 - 1 = 288 = 1*169 + 3*29 + 5*5 + 7*1; 7^2 = 49 = 1*29 + 3*5 + 5*1. - Charlie Marion, Jul 18 2003
a(n+2) = A078343(n+1) + A048654(n). - Creighton Dement, Jan 19 2005
a(n) = A000129(n) + A000129(n-1) = A001109(n)/A000129(n) = sqrt(A001110(n)/A000129(n)^2) = ceiling(sqrt(A001108(n))). - Henry Bottomley, Apr 18 2000
Also the first differences of A000129 (the Pell numbers) because A052937(n) = A000129(n+1) + 1. - Graeme McRae, Aug 03 2006
a(n) = Sum_{k=0..n} A122542(n,k). - Philippe Deléham, Oct 08 2006
For another recurrence see A000129.
a(n) = Sum_{k=0..n} A098158(n,k)*2^(n-k). - Philippe Deléham, Dec 26 2007
a(n) = upper left and lower right terms of [1,1; 2,1]^n. - Gary W. Adamson, Mar 12 2008
If p[1]=1, 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
For n>=2, a(n)=F_n(2)+F_(n+1)(2), where F_n(x) is Fibonacci polynomial (cf. A049310): F_n(x) = Sum_{i=0..floor((n-1)/2)} binomial(n-i-1,i)x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012
a(-n) = (-1)^n * a(n). - Michael Somos, Sep 02 2012
Dirichlet g.f.: (PolyLog(s,1-sqrt(2)) + PolyLog(s,1+sqrt(2)))/2. - Ilya Gutkovskiy, Jun 26 2016
a(n) = A000129(n) - A000129(n-1), where A000129(n) is the n-th Pell Number. Hence the continued fraction is of the form 1-(A000129(n-1)/A000129(n)). - Gregory L. Simay, Nov 09 2018
a(n) = (A000129(n+3) + A000129(n-3))/10, n>=3. - Paul Curtz, Jun 16 2021
a(n) = (A000129(n+6) - A000129(n-6))/140, n>=6. - Paul Curtz, Jun 20 2021
a(n) = round((1/2)*sqrt(Product_{k=1..n} 4*(1 + sin(k*Pi/n)^2))), for n>=1. - Greg Dresden, Dec 28 2021
a(n)^2 + a(n+1)^2 = A075870(n+1) = 2*(b(n)^2 + b(n+1)^2) for all n in Z where b(n) := A000129(n). - Michael Somos, Apr 02 2022
a(n) = 2*A048739(n-2)+1. - R. J. Mathar, Feb 01 2024
Sum_{n>=1} 1/a(n) = 1.5766479516393275911191017828913332473... - R. J. Mathar, Feb 05 2024
From Peter Bala, Jul 06 2025: (Start)
G.f.: Sum_{n >= 1} (-1)^(n+1) * x^(n-1) * Product_{k = 1..n} (1 - k*x)/(1 - 3*x + k*x^2).
The following series telescope:
Sum_{n >= 1} (-1)^(n+1)/(a(2*n) + 1/a(2*n)) = 1/4, since 1/(a(2*n) + 1/a(2*n)) = 1/A077445(n) + 1/A077445(n+1).
Sum_{n >= 1} (-1)^(n+1)/(a(2*n+1) - 1/a(2*n+1)) = 1/8, since. 1/(a(2*n+1) - 1/a(2*n+1)) = 1/(4*Pell(2*n)) + 1/(4*Pell(2*n+2)), where Pell(n) = A000129(n).
Sum_{n >= 1} (-1)^(n+1)/(a(2*n+1) + 9/a(2*n+1)) = 1/10, since 1/(a(2*n+1) + 9/a(2*n+1)) = b(n) + b(n+1), where b(n) = A001109(n)/(2*Pell(2*n-1)*Pell(2*n+1)).
Sum_{n >= 1} (-1)^(n+1)/(a(n)*a(n+1)) = 1 - sqrt(2)/2 = A268682, since (-1)^(n+1)/(a(n)*a(n+1)) = Pell(n)/a(n) - Pell(n+1)/a(n+1). (End)

Chebyshev comments from Wolfdieter Lang, Jan 10 2003

A035363 Number of partitions of n into even parts.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 42, 0, 56, 0, 77, 0, 101, 0, 135, 0, 176, 0, 231, 0, 297, 0, 385, 0, 490, 0, 627, 0, 792, 0, 1002, 0, 1255, 0, 1575, 0, 1958, 0, 2436, 0, 3010, 0, 3718, 0, 4565, 0, 5604, 0, 6842, 0, 8349, 0, 10143, 0, 12310, 0

Offset: 0

Olivier Gérard

Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009
Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009
Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013
Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016
From Gus Wiseman, May 22 2021: (Start)
The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290.
For n > 1, also the number of integer partitions of n-1 whose only odd part is the smallest. The Heinz numbers of these partitions are given by A341446. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns shown as dots, A..D = 10..13) are:
  1  .  3   .  5    .  7     .  9      .  B       .  D
        21     41      43       63        65         85
               221     61       81        83         A3
                       421      441       A1         C1
                       2221     621       443        643
                                4221      641        661
                                22221     821        841
                                          4421       A21
                                          6221       4441
                                          42221      6421
                                          222221     8221
                                                     44221
                                                     62221
                                                     422221
                                                     2222221
Also the number of integer partitions of n whose greatest part is the sum of all the other parts. The Heinz numbers of these partitions are given by A344415. For example, the a(2) = 1 through a(12) = 11 partitions (empty columns not shown) are:
  (11)  (22)   (33)    (44)     (55)      (66)
        (211)  (321)   (422)    (532)     (633)
               (3111)  (431)    (541)     (642)
                       (4211)   (5221)    (651)
                       (41111)  (5311)    (6222)
                                (52111)   (6321)
                                (511111)  (6411)
                                          (62211)
                                          (63111)
                                          (621111)
                                          (6111111)
Also the number of integer partitions of n of length n/2. The Heinz numbers of these partitions are given by A340387. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns not shown) are:
  (2)  (22)  (222)  (2222)  (22222)  (222222)  (2222222)
       (31)  (321)  (3221)  (32221)  (322221)  (3222221)
             (411)  (3311)  (33211)  (332211)  (3322211)
                    (4211)  (42211)  (333111)  (3332111)
                    (5111)  (43111)  (422211)  (4222211)
                            (52111)  (432111)  (4322111)
                            (61111)  (441111)  (4331111)
                                     (522111)  (4421111)
                                     (531111)  (5222111)
                                     (621111)  (5321111)
                                     (711111)  (5411111)
                                               (6221111)
                                               (6311111)
                                               (7211111)
                                               (8111111)
(End)

			From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(12)=11 symmetric unimodal compositions of 12+2=14 where the maximal part appears twice:
01:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
02:  [ 1 1 1 1 3 3 1 1 1 1 ]
03:  [ 1 1 1 4 4 1 1 1 ]
04:  [ 1 1 2 3 3 2 1 1 ]
05:  [ 1 1 5 5 1 1 ]
06:  [ 1 2 4 4 2 1 ]
07:  [ 1 6 6 1 ]
08:  [ 2 2 3 3 2 2 ]
09:  [ 2 5 5 2 ]
10:  [ 3 4 4 3 ]
11:  [ 7 7 ]
There are a(14)=15 symmetric unimodal compositions of 14 where the maximal part appears an even number of times:
01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
03:  [ 1 1 1 1 3 3 1 1 1 1 ]
04:  [ 1 1 1 2 2 2 2 1 1 1 ]
05:  [ 1 1 1 4 4 1 1 1 ]
06:  [ 1 1 2 3 3 2 1 1 ]
07:  [ 1 1 5 5 1 1 ]
08:  [ 1 2 2 2 2 2 2 1 ]
09:  [ 1 2 4 4 2 1 ]
10:  [ 1 3 3 3 3 1 ]
11:  [ 1 6 6 1 ]
12:  [ 2 2 3 3 2 2 ]
13:  [ 2 5 5 2 ]
14:  [ 3 4 4 3 ]
15:  [ 7 7 ]
(End)
a(8)=5 because we  have [8], [6,2], [4,4], [4,2,2], and [2,2,2,2]. - _Emeric Deutsch_, Jan 27 2016
From _Gus Wiseman_, May 22 2021: (Start)
The a(0) = 1 through a(12) = 11 partitions into even parts are the following (empty columns shown as dots, A = 10, C = 12). The Heinz numbers of these partitions are given by A066207.
  ()  .  (2)  .  (4)   .  (6)    .  (8)     .  (A)      .  (C)
                 (22)     (42)      (44)       (64)        (66)
                          (222)     (62)       (82)        (84)
                                    (422)      (442)       (A2)
                                    (2222)     (622)       (444)
                                               (4222)      (642)
                                               (22222)     (822)
                                                           (4422)
                                                           (6222)
                                                           (42222)
                                                           (222222)
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

Robert Price, Table of n, a(n) for n = 0..2001

Bisection (even part) gives the partition numbers A000041.
Column k=0 of A103919, A264398.
Cf. A036469, A000070.
Cf. A135010, A138121.
Note: A-numbers of ranking sequences are in parentheses below.
The version for odd instead of even parts is A000009 (A066208).
The version for parts divisible by 3 instead of 2 is A035377.
The strict case is A035457.
The Heinz numbers of these partitions are given by A066207.
The ordered version (compositions) is A077957 prepended by (1,0).
This is column k = 2 of A168021.
The multiplicative version (factorizations) is A340785.
A000569 counts graphical partitions (A320922).
A004526 counts partitions of length 2 (A001358).
A025065 counts palindromic partitions (A265640).
A027187 counts partitions with even length/maximum (A028260/A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
The following count partitions of even length:
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A000041, A000290, A087897, A100484, A110618, A209816, A210249, A233771, A339004, A340385, A340387, A340786, A341447.

ZL:= [S, {C = Cycle(B), S = Set(C), E = Set(B), B = Prod(Z,Z)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..69); # Zerinvary Lajos, Mar 26 2008
g := 1/mul(1-x^(2*k), k = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0 .. 78); # Emeric Deutsch, Jan 27 2016
# Using the function EULER from Transforms (see link at the bottom of the page).
[1,op(EULER([0,1,seq(irem(n,2),n=0..66)]))]; # Peter Luschny, Aug 19 2020
# next Maple program:
a:= n-> `if`(n::odd, 0, combinat[numbpart](n/2)):
seq(a(n), n=0..84);  # Alois P. Heinz, Jun 22 2021

nmax = 50; s = Range[2, nmax, 2];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

from sympy import npartitions
def A035363(n): return 0 if n&1 else npartitions(n>>1) # Chai Wah Wu, Sep 23 2023

G.f.: Product_{k even} 1/(1 - x^k).
Convolution with the number of partitions into distinct parts (A000009, which is also number of partitions into odd parts) gives the number of partitions (A000041). - Franklin T. Adams-Watters, Jan 06 2006
If n is even then a(n)=A000041(n/2) otherwise a(n)=0. - Omar E. Pol, Nov 20 2009
G.f.: 1 + x^2*(1 - G(0))/(1-x^2) where G(k) =  1 - 1/(1-x^(2*k+2))/(1-x^2/(x^2-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) = A096441(n) - A000009(n), n >= 1. - Omar E. Pol, Aug 16 2013
G.f.: exp(Sum_{k>=1} x^(2*k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018

A005418 Number of (n-1)-bead black-white reversible strings; also binary grids; also row sums of Losanitsch's triangle A034851; also number of caterpillar graphs on n+2 vertices.

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 36, 72, 136, 272, 528, 1056, 2080, 4160, 8256, 16512, 32896, 65792, 131328, 262656, 524800, 1049600, 2098176, 4196352, 8390656, 16781312, 33558528, 67117056, 134225920, 268451840, 536887296, 1073774592, 2147516416, 4295032832

Offset: 1

N. J. A. Sloane, R. K. Guy

Equivalently, walks on triangle, visiting n+2 vertices, so length n+1, n "corners"; the symmetry group is S3, reversing a walk does not count as different. Walks are not self-avoiding. - Colin Mallows
Slavik V. Jablan observes that this is also the number of rational knots and links with n+2 crossings (cf. A018240). See reference. [Corrected by Andrey Zabolotskiy, Jun 18 2020]
Number of bit strings of length (n-1), not counting strings which are the end-for-end reversal or the 0-for-1 reversal of each other as different. - Carl Witty (cwitty(AT)newtonlabs.com), Oct 27 2001
The formula given in page 1095 of the Balasubramanian reference can be used to derive this sequence. - Parthasarathy Nambi, May 14 2007
Also number of compositions of n up to direction, where a composition is considered equivalent to its reversal, see example. - Franklin T. Adams-Watters, Oct 24 2009
Number of normally non-isomorphic realizations of the associahedron of type I starting with dimension 2 in Ceballos et al. - Tom Copeland, Oct 19 2011
Number of fibonacenes with n+2 hexagons. See the Balaban and the Dobrynin references. - Emeric Deutsch, Apr 21 2013
From the point of view of binary grids, it is a (1,n)-rectangular grid. A225826 to A225834 are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11. - Yosu Yurramendi, May 19 2013
Number of n-vertex difference graphs (bipartite 2K_2-free graphs) [Peled & Sun, Thm. 9]. - Falk Hüffner, Jan 10 2016
The offset should be 0, since the first row of A034851 is row 0. The name would then be: "Number of n bead...". - Daniel Forgues, Jul 26 2018
a(n) is the number of non-isomorphic generalized rigid ladders with n cells. A generalized rigid ladder with n cells is a graph with vertex set is the union of {u_0, u_1, ..., u_n} and {v_0, v_1, ..., v_n}, and for every 0 <= i <= n-1, the edges are of the form {u_i,u_i+1}, {v_i, v_i+1}, {u_i,v_i} and either {u_i,v_i+1} or {u_i+1,v_i}. - Christian Barrientos, Jul 29 2018
Also number of non-isomorphic stairs with n+1 cells. A stair is a snake polyomino allowing only two directions for adjacent cells: east and north. - Christian Barrientos and Sarah Minion, Jul 29 2018
From Robert A. Russell, Oct 28 2018: (Start)
There are two different unoriented row colorings using two colors that give us very similar results here, a difference of one in the offset. In an unoriented row, chiral pairs are counted as one.
a(n) is the number of color patterns (set partitions) of an unoriented row of length n using two or fewer colors (subsets). Two color patterns are equivalent if the colors are permutable.
a(n+1) is the number of ways to color an unoriented row of length n using two noninterchangeable colors (one need not use both colors).
See the examples below of these two different colorings. (End)
Also arises from the enumeration of types of based polyhedra with exactly two triangular faces [Rademacher]. - N. J. A. Sloane, Apr 24 2020
a(n) is the number of (unlabeled) 2-paths with n+4 vertices. (A 2-path with order n at least 4 can be constructed from a 3-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to an existing 2-clique containing an existing 2-leaf.) - Allan Bickle, Apr 05 2022
a(n) is the number of caterpillars with a perfect matching and order 2n+2. - Christian Barrientos, Sep 12 2023
a(n) is also the number of distinct planar embeddings of the (n+2)-centipede graph (up to at least n=8 and likely for all larger n). - Eric W. Weisstein, May 21 2024
a(n) is also the number of distinct planar embeddings of the 2 X (n+2) grid graph i.e., the (n+2)-ladder graph. - Eric W. Weisstein, May 21 2024
Dimension of the homogeneous component of degree n of the free Jordan algebra on two generators (or, in this case, the free special Jordan algebra on two generators). It follows from (Shirshov 1956, Cohn 1959). - Vladimir Dotsenko, Mar 29 2025

			a(5) = 10 because there are 16 compositions of 5 (shown as <vectors>) but only 10 equivalence classes (shown as {sets}): {<5>}, {<4,1>,<1,4>}, {<3,2>,<2,3>}, {<3,1,1>,<1,1,3>}, {<1,3,1>},{<2,2,1>,<1,2,2>}, {<2,1,2>}, {<2,1,1,1>,<1,1,1,2>}, {<1,2,1,1>,<1,1,2,1>}, {<1,1,1,1,1>}. - _Geoffrey Critzer_, Nov 02 2012
G.f. = x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 36*x^7 + 72*x^8 + ... - _Michael Somos_, Jun 24 2018
From _Robert A. Russell_, Oct 28 2018: (Start)
For a(5)=10, the 4 achiral patterns (set partitions) are AAAAA, AABAA, ABABA, and ABBBA. The 6 chiral pairs are AAAAB-ABBBB, AAABA-ABAAA, AAABB-AABBB, AABAB-ABABB, AABBA-ABBAA, and ABAAB-ABBAB. The colors are permutable.
For n=4 and a(n+1)=10, the 4 achiral colorings are AAAA, ABBA, BAAB, and BBBB. The 6 achiral pairs are AAAB-BAAA, AABA-ABAA, AABB-BBAA, ABAB-BABA, ABBB-BBBA, and BABB-BBAB. The colors are not permutable. (End)

K. Balasubramanian, "Combinatorial Enumeration of Chemical Isomers", Indian J. Chem., (1978) vol. 16B, pp. 1094-1096. See page 1095.
Wayne M. Dymacek, Steinhaus graphs. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 399--412, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561065 (81f:05120)
Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World Scientific Press, 2007.
Joseph S. Madachy: Madachy's Mathematical Recreations. New York: Dover Publications, Inc., 1979, p. 46 (first publ. by Charles Scribner's Sons, New York, 1966, under the title: Mathematics on Vacation)
M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]
C. A. Pickover, Keys to Infinity, Wiley 1995, p. 75.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
M. Archibald, A. Blecher, A. Knopfmacher, and M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 151 and 733.
Andrei Asinowski and Alon Regev, Triangulations with Few Ears: Symmetry Classes and Disjointness, Integers 16 (2016), #A5.
A. T. Balaban, Chemical graphs. Part 50. Symmetry and enumeration of fibonacenes (unbranched catacondensed benzenoids isoarithmic with helicenes and zigzag catafusenes), MATCH: Commun. Math. Comput. Chem., 24 (1989) 29-38.
Allan Bickle, How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
A. P. Burger, M. Van Der Merwe, and J. H. Van Vuuren, An asymptotic analysis of the evolutionary spatial prisoner's dilemma on a path, Discrete Appl. Math. 160, No. 15, 2075-2088 (2012), Table 3.1.
C. Ceballos, F. Santos, and G. Ziegler, Many Non-equivalent Realizations of the Associahedron, arXiv:1109.5544 [math.MG], 2011-2013; pp. 15 and 26.
P. M. Cohn, Two embedding theorems for Jordan algebras, Proceedings of the London Mathematical Society, Volume s3-9, Issue 4, October 1959, pp. 503-524.
Jacob Crabtree, Another Enumeration of Caterpillar Trees, arXiv:1810.11744 [math.CO], 2018.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, E. Brendsdal, Zhang Fuji, Guo Xiaofeng, and R. Tosic, Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
Miroslav Marinov Dimitrov, Designing Boolean Functions and Digital Sequences for Cryptology and Communications, Ph. D. Dissertation, Bulgarian Acad. Sci. (Sofia, Bulgaria 2023).
A. A. Dobrynin, On the Wiener index of fibonacenes, MATCH: Commun. Math. Comput. Chem, 64 (2010), 707-726.
Vladimir Dotsenko and Irvin Roy Hentzel, On the conjecture of Kashuba and Mathieu about free Jordan algebras, arXiv:2507.00437 [math.RA], 2025. See p. 14.
J. Eckhoff, Extremal interval graphs, J. Graph Theory 17 1 (1993), 117-127.
Sahir Gill, Bounds for Region Containing All Zeros of a Complex Polynomial, International Journal of Mathematical Analysis (2018), Vol. 12, No. 7, 325-333.
T. A. Gittings, Minimum braids: a complete invariant of knots and links, arXiv:math/0401051 [math.GT], 2004. - _N. J. A. Sloane_, Jan 18 2013
R. K. Guy, Letter to N. J. A. Sloane, Nov 1978.
Frank Harary and Allen J. Schwenk, The number of caterpillars, Discrete Mathematics, Volume 6, Issue 4, 1973, 359-365.
N. Hoffman, Binary grids and a related counting problem, Two-Year College Math. J. 9 (1978), 267-272.
S. V. Jablan, Geometry of Links, XII Yugoslav Geometric Seminar (Novi Sad, 1998), Novi Sad J. Math. 29 (1999), no. 3, 121-139.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Isaac B. Michael and M. R. Sepanski, Net regular signed trees, Australasian Journal of Combinatorics, Volume 66(2) (2016), 192-204.
U. N. Peled and F. Sun, Enumeration of difference graphs, Discrete Appl. Math., 60 (1995), 311-318.
Paulo Renato da Costa Pereira, Lilian Markenzon and Oswaldo Vernet, A clique-difference encoding scheme for labelled k-path graphs, Discrete Appl. Math. 156 (2008), 3216-3222.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Hans Rademacher, On the number of certain types of polyhedra, Illinois Journal of Mathematics 9.3 (1965): 361-380. Reprinted in Coll. Papers, Vol II, MIT Press, 1974, pp. 544-564. See Theorem 8, Eq. 14.3.
A. Regev, Remarks on two-eared triangulations, arXiv preprint arXiv:1309.0743 [math.CO], 2013-2014.
Suthee Ruangwises, The Landscape of Computing Symmetric n-Variable Functions with 2n Cards, arXiv:2306.13551 [cs.CR], 2023.
A. I. Shirshov, On special J-rings, Mat. Sb. (N.S.), 38(80), 1956, pp. 149-166.
N. J. A. Sloane, Classic Sequences
R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), Article #00.1.1.
Eric Weisstein's World of Mathematics, Barker Code.
Eric Weisstein's World of Mathematics, Bishops Problem.
Eric Weisstein's World of Mathematics, Caterpillar Graph.
Eric Weisstein's World of Mathematics, Centipede Graph.
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Ladder Graph.
Eric Weisstein's World of Mathematics, Losanitsch's Triangle.
Eric Weisstein's World of Mathematics, Planar Embedding.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 87 (2014), 1260-1264; see Tables 1 and 2 (and text). - _N. J. A. Sloane_, Mar 26 2015
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Column 2 of A320750 (set partitions).
Cf. A131577 (oriented), A122746(n-3) (chiral), A016116 (achiral), for set partitions with up to two subsets.
Column 2 of A277504, offset by one (colors not permutable).
Cf. A000079 (oriented), A122746(n-2) (chiral), and A060546 (achiral), for a(n+1).
Cf. A001998, A001444, A051436, A051437, A007582, A001445, A032085.

a005418 n = sum $ a034851_row (n - 1) -- Reinhard Zumkeller, Jan 14 2012

A005418 := n->2^(n-2)+2^(floor(n/2)-1): seq(A005418(n), n=1..34);

LinearRecurrence[{2,2,-4}, {1,2,3}, 40] (* or *) Table[2^(n-2)+2^(Floor[n/2]-1), {n,40}] (* Harvey P. Dale, Jan 18 2012 *)

A005418(n)= 2^(n-2) + 2^(n\2-1); \\ Joerg Arndt, Sep 16 2013

def A005418(n): return 1 if n == 1 else 2**((m:= n//2)-1)*(2**(n-m-1)+1) # Chai Wah Wu, Feb 03 2022

a(n) = 2^(n-2) + 2^(floor(n/2) - 1).
G.f.: -x*(-1 + 3*x^2) / ( (2*x - 1)*(2*x^2 - 1) ). - Simon Plouffe in his 1992 dissertation
G.f.: x*(1+2*x)*(1-3*x^2)/((1-4*x^2)*(1-2*x^2)), not reduced. - Wolfdieter Lang, May 08 2001
a(n) = 6*a(n - 2) - 8*a(n - 4). a(2*n) = A063376(n - 1) = 2*a(2*n - 1); a(2*n + 1) = A007582(n). - Henry Bottomley, Jul 14 2001
a(n+2) = 2*a(n+1) - A077957(n) with a(1) = 1, a(2) = 2. - Yosu Yurramendi, Oct 24 2008
a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3). - Jaume Oliver Lafont, Dec 05 2008
Union of A007582 and A161168. Union of A007582 and A063376. - Jaroslav Krizek, Aug 14 2009
G.f.: G(0); G(k) = 1 + 2*x/(1 - x*(1+2^(k+1))/(x*(1+2^(k+1)) + (1+2^k)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 12 2011
a(2*n) = 2*a(2*n-1) and a(2*n+1) = a(2*n) + 4^(n-1) with a(1) = 1. - Johannes W. Meijer, Aug 26 2013
From Robert A. Russell, Oct 28 2018: (Start)
a(n) = (A131577(n) + A016116(n)) / 2 = A131577(n) - A122746(n-3) = A122746(n-3) + A016116(n), for set partitions with up to two subsets.
a(n+1) = (A000079(n) + A060546(n)) / 2 = A000079(n) - A122746(n-2) = A122746(n-2) + A060546(n), for two colors that do not permute.
a(n) = Sum_{j=0..k} (S2(n,j) + Ach(n,j)) / 2, where k=2 is the maximum number of colors, S2(n,k) is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(n+1) = (k^n + k^ceiling(n/2)) / 2, where k=2 is number of colors we can use. (End)
E.g.f.: (cosh(2*x) + 2*cosh(sqrt(2)*x) + sinh(2*x) + sqrt(2)*sinh(sqrt(2)*x) - 3)/4. - Stefano Spezia, Jun 01 2022

A168561 Riordan array (1/(1-x^2), x/(1-x^2)). Unsigned version of A049310.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 1, 0, 3, 0, 4, 0, 1, 1, 0, 6, 0, 5, 0, 1, 0, 4, 0, 10, 0, 6, 0, 1, 1, 0, 10, 0, 15, 0, 7, 0, 1, 0, 5, 0, 20, 0, 21, 0, 8, 0, 1, 1, 0, 15, 0, 35, 0, 28, 0, 9, 0, 1, 0, 6, 0, 35, 0, 56, 0, 36, 0, 10, 0, 1, 1, 0, 21, 0, 70, 0, 84, 0, 45, 0, 11, 0, 1

Offset: 0

Philippe Deléham, Nov 29 2009

Row sums: A000045(n+1), Fibonacci numbers.
A168561*A007318 = A037027, as lower triangular matrices. Diagonal sums : A077957. - Philippe Deléham, Dec 02 2009
T(n,k) is the number of compositions of n+1 into k+1 odd parts. Example: T(4,2)=3 because we have 5 = 1+1+3 = 1+3+1 = 3+1+1.
Coefficients of monic Fibonacci polynomials (rising powers of x). Ftilde(n, x) = x*Ftilde(n-1, x) + Ftilde(n-2, x), n >=0, Ftilde(-1,x) = 0, Ftilde(0, x) = 1. G.f.: 1/(1 - x*z - z^2). Compare with Chebyshev S-polynomials (A049310). - Wolfdieter Lang, Jul 29 2014

			The triangle T(n,k) begins:
n\k 0  1   2   3   4    5    6    7    8    9  10  11  12  13 14 15 ...
0:  1
1:  0  1
2:  1  0   1
3:  0  2   0   1
4:  1  0   3   0   1
5:  0  3   0   4   0    1
6:  1  0   6   0   5    0    1
7:  0  4   0  10   0    6    0    1
8:  1  0  10   0  15    0    7    0    1
9:  0  5   0  20   0   21    0    8    0    1
10: 1  0  15   0  35    0   28    0    9    0   1
11: 0  6   0  35   0   56    0   36    0   10   0   1
12: 1  0  21   0  70    0   84    0   45    0  11   0   1
13: 0  7   0  56   0  126    0  120    0   55   0  12   0   1
14: 1  0  28   0 126    0  210    0  165    0  66   0  13   0  1
15: 0  8   0  84   0  252    0  330    0  220   0  78   0  14  0  1
... reformatted by _Wolfdieter Lang_, Jul 29 2014.
------------------------------------------------------------------------

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From _N. J. A. Sloane_, May 10 2012
Tom Copeland, Addendum to Elliptic Lie Triad
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A162515 (rows reversed), A112552, A102426 (deflated).

A168561:=proc(n,k) if n-k mod 2 = 0 then binomial((n+k)/2,k) else 0 fi end proc:
seq(seq(A168561(n,k),k=0..n),n=0..12) ; # yields sequence in triangular form

Table[If[EvenQ[n + k], Binomial[(n + k)/2, k], 0], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 16 2017 *)

T(n,k) = if ((n+k) % 2, 0, binomial((n+k)/2,k));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print();); \\ Michel Marcus, Oct 09 2016

Sum_{k=0..n} T(n,k)*x^k = A059841(n), A000045(n+1), A000129(n+1), A006190(n+1), A001076(n+1), A052918(n), A005668(n+1), A054413(n), A041025(n), A099371(n+1), A041041(n), A049666(n+1), A041061(n), A140455(n+1), A041085(n), A154597(n+1), A041113(n) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 respectively. - Philippe Deléham, Dec 02 2009
T(2n,2k) = A085478(n,k). T(2n+1,2k+1) = A078812(n,k). Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000045(n+1), A006131(n), A015445(n), A168579(n), A122999(n) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Dec 02 2009
T(n,k) = binomial((n+k)/2,k) if (n+k) is even; otherwise T(n,k)=0.
G.f.: (1-z^2)/(1-t*z-z^2) if offset is 1.
T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = 1, T(0,1) = 0. - Philippe Deléham, Feb 09 2012
Sum_{k=0..n} T(n,k)^2 = A051286(n). - Philippe Deléham, Feb 09 2012
From R. J. Mathar, Feb 04 2022: (Start)
Sum_{k=0..n} T(n,k)*k = A001629(n+1).
Sum_{k=0..n} T(n,k)*k^2 = 0,1,4,11,... = 2*A055243(n)-A099920(n+1).
Sum_{k=0..n} T(n,k)*k^3 = 0,1,8,29,88,236,... = 12*A055243(n) -6*A001629(n+2) +A001629(n+1)-6*(A001872(n)-2*A001872(n-1)). (End)

Typo in name corrected (1(1-x^2) changed to 1/(1-x^2)) by Wolfdieter Lang, Nov 20 2010

A059905 Index of first half of decomposition of integers into pairs based on A000695.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 2, 3, 0, 1, 0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 4, 5, 4, 5, 6, 7, 6, 7, 0, 1, 0, 1, 2, 3, 2, 3, 0, 1, 0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 12, 13, 12, 13, 14

Offset: 0

Marc LeBrun, Feb 07 2001

One coordinate of a recursive non-self-intersecting walk on the square lattice Z^2.

			A000695(a(14)) + 2*A000695(A059906(14)) = A000695(2) + 2*A000695(3) = 4 + 2*5 = 14.
If n=27, then b_0=1, b_1=1, b_2=0, b_3=1, b_4=1. Therefore a(27) = b_0 + b_2*2 + b_4*2^2 = 5. - _Vladimir Shevelev_, Nov 13 2008

Peter Kagey, Table of n, a(n) for n = 0..8192
G. M. Morton, A Computer Oriented Geodetic Data Base; and a New Technique in File Sequencing, IBM, 1966, with a(n) being section 5.1 step (c).
Index entries for sequences related to coordinates of 2D curves

Cf. A000695, A030308, A059906, A057300, A077957.

f:= proc(n) local L; L:= convert(n,base,2); add(L[2*i+1]*2^i,i=0..floor((nops(L)-1)/2)) end;
map(f, [$0..256]); # Robert Israel, Aug 12 2015

a[n_] := Module[{P}, (P = Partition[IntegerDigits[2n, 2]//Reverse, 2][[All, 2]]).(2^(Range[Length[P]]-1))]; Array[a, 100, 0] (* Jean-François Alcover, Apr 24 2019 *)

A059905(n) = { my(t=1,s=0); while(n>0, s += (n%2)*t; n \= 4; t *= 2); (s); }; \\ Antti Karttunen, Apr 14 2018

def a(n): return sum([(n>>2*i&1)<Indranil Ghosh, Jun 25 2017, after Ruby code by Peter Kagey

Python

def A059905(n): return int(bin(n)[:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

Ruby

def a(n) (0..n.bit_length/2).to_a.map { |i| (n >> 2 * i & 1) << i}.reduce(:+) end # Peter Kagey, Aug 12 2015

Formula

n = A000695(a(n)) + 2*A000695(A059906(n)).

To get a(n), write n as Sum b_j*2^j, then a(n) = Sum b_(2j)*2^j. - Vladimir Shevelev, Nov 13 2008

a(n) = Sum_{k>=0} A030308(n,k)*A077957(k). - Philippe Deléham, Oct 18 2011

G.f.: (1-x)^(-1) * Sum_{j>=0} 2^j*x^(2^j)/(1+x^(2^j)). - Robert Israel, Aug 12 2015

a(n) = A059906(2*n). - Velin Yanev, Dec 01 2016

A059906 Index of second half of decomposition of integers into pairs based on A000695.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 5, 5, 4, 4, 5, 5, 6, 6, 7, 7, 6, 6, 7, 7, 4, 4, 5, 5, 4, 4, 5, 5, 6, 6, 7, 7, 6, 6, 7, 7, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 5, 5, 4, 4, 5, 5, 6

Offset: 0

Marc LeBrun, Feb 07 2001

One coordinate of a recursive non-self-intersecting walk on the square lattice Z^2.

			A000695(A059905(14)) + 2*A000695(a(14)) = A000695(2) + 2*A000695(3) = 4 + 2*5 = 14.
If n=27, then b_0=1, b_1=1, b_2=0, b_3=1, b_4=1. Therefore a(27) = b_1 + b_3*2 = 3. - _Vladimir Shevelev_, Nov 13 2008

Rémy Sigrist, Table of n, a(n) for n = 0..8192
G. M. Morton, A Computer Oriented Geodetic Data Base; and a New Technique in File Sequencing, IBM, 1966, with a(n) being section 5.1 step (b).
Index entries for sequences related to coordinates of 2D curves

Cf. A000695, A057300, A059905.

a[n_] := Module[{P}, (P = Partition[IntegerDigits[n, 2]//Reverse, 2][[All, 2]]).(2^(Range[Length[P]]-1))]; Array[a, 105, 0] (* Jean-François Alcover, Apr 24 2019 *)

A059906(n) = { my(t=1,s=0); while(n>0, s += ((n%4)>=2)*t; n \= 4; t *= 2); (s); }; \\ Antti Karttunen, Apr 14 2018

def a(n):
    x=[int(t) for t in list(bin(n)[2:])[::-1]]
    return sum(x[2*i + 1]*2**i for i in range(int(len(x)//2)))
print([a(n) for n in range(105)]) # Indranil Ghosh, Jun 25 2017

def A059906(n): return 0 if n < 2 else int(bin(n)[-2:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

n = A000695(A059905(n)) + 2*A000695(a(n))
To get a(n), write n as Sum b_j*2^j, then a(n) = Sum b_(2j+1)*2^j. - Vladimir Shevelev, Nov 13 2008
a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=0 and b(k)=A077957(k-1) for k>0. - Philippe Deléham, Oct 18 2011
Conjecture: a(n) = n - (1/2)*Sum_{k=1..n} (sqrt(2)^A007814(k) + (-sqrt(2))^A007814(k)) = -Sum_{k=1..n} (-1)^k * 2^floor(k/2) * floor(n/2^k). - Velin Yanev, Dec 01 2016

A052551 Expansion of 1/((1 - x)(1 - 2x^2)).

Original entry on oeis.org

1, 1, 3, 3, 7, 7, 15, 15, 31, 31, 63, 63, 127, 127, 255, 255, 511, 511, 1023, 1023, 2047, 2047, 4095, 4095, 8191, 8191, 16383, 16383, 32767, 32767, 65535, 65535, 131071, 131071, 262143, 262143, 524287, 524287, 1048575, 1048575, 2097151, 2097151

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Equals row sums of triangle A137865. - Gary W. Adamson, Feb 18 2008
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 566", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 05 2017
Number of nonempty subsets of {1,2,...,n+1} that contain only odd numbers. a(0) = a(1) = 1: {1}; a(6) = a(7) = 15: {1}, {3}, {5}, {7}, {1,3}, {1,5}, {1,7}, {3,5}, {3,7}, {5,7}, {1,3,5}, {1,3,7}, {1,5,7}, {3,5,7}, {1,3,5,7}. - Enrique Navarrete, Mar 16 2018
Number of nonempty subsets of {1,2,...,n+2} that contain only even numbers. a(0) = a(1) = 1: {2}; a(4) = a(5) = 7: {2}, {4}, {6}, {2,4}, {2,6}, {4,6}, {2,4,6}. - Enrique Navarrete, Mar 26 2018
Doubling of A000225(n+1), n >= 0 entries. First differences give A077957. - Wolfdieter Lang, Apr 08 2018
a(n-2) is the number of achiral rows or cycles of length n partitioned into two sets or the number of color patterns using exactly 2 colors. An achiral row or cycle is equivalent to its reverse. Two color patterns are equivalent if the colors are permuted. For n = 4, the a(n-2) = 3 row patterns are AABB, ABAB, and ABBA; the cycle patterns are AAAB, AABB, and ABAB. For n = 5, the a(n-2) = 3 patterns for both rows and cycles are AABAA, ABABA, and ABBBA. For n = 6, the a(n-2) = 7 patterns for rows are AAABBB, AABABB, AABBAA, ABAABA, ABABAB, ABBAAB, and ABBBBA; the cycle patterns are AAAAAB, AAAABB, AAABAB, AAABBB, AABAAB, AABABB, and ABABAB. - Robert A. Russell, Oct 15 2018
For integers m > 1, the expansion of 1/((1 - x)*(1 - m*x^2)) generates a(n) = (sqrt(m)^(n + 1)*((-1)^n*(sqrt(m) - 1) + sqrt(m) + 1) - 2)/(2*(m - 1)). It appears, for integer values of n >= 0 and m > 1, that it could be simplified in the integral domain a(n) = (m^(1 + floor(n/2)) - 1)/(m - 1). - Federico Provvedi, Nov 23 2018
From Werner Schulte, Mar 04 2019: (Start)
More generally: For some fixed integers q and r > 0 the expansion of A(q,r; x) = 1/((1-x)*(1-q*x^r)) generates coefficients a(q,r; n) = (q^(1+floor(n/r))-1)/(q-1) for n >= 0; the special case q = 1 leads to a(1,r; n) = 1 + floor(n/r).
The a(q,r; n) satisfy for n > r a linear recurrence equation with constant coefficients. The signature vector is given by the sum of two vectors v and w where v has terms 1 followed by r zeros, i.e., (1,0,0,...,0), and w has r-1 leading zeros followed by q and -q, i.e., (0,0,...,0,q,-q).
Let a_i(q,r; n) be the convolution inverse of a(q,r; n). The terms are given by the sum a_i(q,r; n) = b(n) + c(n) for n >= 0 where b(n) has terms 1 and -1 followed by infinitely zeros, i.e., (1,-1,0,0,0,...), and c(n) has r leading zeros followed by -q, q and infinitely zeros, i.e., (0,0,...,0,-q,q,0,0,0,...).
Here is an example for q = 3 and r = 5: The expansion of A(3,5; x) = 1/((1-x)*(1-3*x^5)) = Sum_{n>=0} a(3,5; n)*x^n generates the sequence of coefficients (a(3,5; n)) = (1,1,1,1,1,4,4,4,4,4,13,13,13,13,13,40,...) where r = 5 controls the repetition and q = 3 the different values.
The a(3,5; n) satisfy for n > 5 the linear recurrence equation with constant coefficients and signature (1,0,0,0,0,0) + (0,0,0,0,3,-3) = (1,0,0,0,3,-3).
The convolution inverse a_i(3,5; n) has terms (1,-1,0,0,0,0,0,0,0,...) + (0,0,0,0,0,-3,3,0,0,...) = (1,-1,0,0,0,-3,3,0,0,...).
For further examples and informations see A014983 (q,r = -3,1), A077925 (q,r = -2,1), A000035 (q,r = -1,1), A000012 (q,r = 0,1), A000027 (q,r = 1,1), A000225 (q,r = 2,1), A003462 (q,r = 3,1), A077953 (q,r = -2,2), A133872 (q,r = -1,2), A004526 (q,r = 1,2), A052551 (this sequence with q,r = 2,2), A077886 (q,r = -2,3), A088911 (q,r = -1,3), A002264 (q,r = 1,3) and A077885 (q,r = 2,3). The offsets might be different.
(End)
a(n) is the number of palindromes of length n over the alphabet {1,2} containing the letter 1. More generally, the number of palindromes of length n over the alphabet {1,2,...,k} containing the letter 1 is given by k^ceiling(n/2)-(k-1)^ceiling(n/2). - Sela Fried, Dec 10 2024

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

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 488
Donatella Merlini and Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.
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 (1,2,-2).

Cf. A000225, A077957, A136865, A289404, A289405, A032085, A050686.
Column 2 (offset by two) of A304972.
Cf. A000225 (oriented), A056326 (unoriented), and A122746(n-2) (chiral) for rows.
Cf. A056295 (oriented), A056357 (unoriented), and A059053 (chiral) for cycles.

Flat(List([1..21],n->[2^n-1,2^n-1])); # Muniru A Asiru, Oct 16 2018

[2^Floor(n/2)-1: n in [2..50]]; // Vincenzo Librandi, Aug 16 2011

spec := [S,{S=Prod(Sequence(Prod(Z,Union(Z,Z))),Sequence(Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[StirlingS2[Floor[n/2] + 2, 2], {n, 0, 50}] (* Robert A. Russell, Dec 20 2017 *)
Drop[LinearRecurrence[{1, 2, -2}, {0, 1, 1}, 50], 1] (* Robert A. Russell, Oct 14 2018 *)
CoefficientList[Series[1/((1-x)*(1-2*x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 16 2018 *)
2^(1+Floor[(Range[0,50])/2])-1 (* Federico Provvedi, Nov 22 2018 *)
((-1)^#(Sqrt[2]-1)+Sqrt[2]+1)2^((#-1)/2)-1&@Range[0, 50] (* Federico Provvedi, Nov 23 2018 *)

x='x+O('x^50); Vec(1/((1-x)*(1-2*x^2))) \\ Altug Alkan, Mar 19 2018

[2^(floor(n/2)) -1 for n in (2..50)] # G. C. Greubel, Mar 04 2019

G.f.: 1/((1 - x)*(1 - 2*x^2)).
Recurrence: a(1) = 1, a(0) = 1, -2*a(n) - 1 + a(n+2) = 0.
a(n) = -1 + Sum((1/2)*(1 + 2*alpha)*alpha^(-1 - n)) where the sum is over alpha = the two roots of -1 + 2*x^2.
a(n) = A016116(n+2) - 1. - R. J. Mathar, Jun 15 2009
a(n) = A060546(n+1) - 1. - Filip Zaludek, Dec 10 2016
From Robert A. Russell, Oct 15 2018: (Start)
a(n-2) = S2(floor(n/2)+1,2), where S2 is the Stirling subset number A008277.
a(n-2) = 2*A056326(n) - A000225(n) = A000225(n) - 2*A122746(n-2) = A056326(n) - A122746(n-2).
a(n-2) = 2*A056357(n) - A056295(n) = A056295(n) - 2*A059053(n) = A056357(n) - A059053(n). (End)
From Federico Provvedi, Nov 22 2018: (Start)
a(n) = 2^( 1 + floor(n/2) ) - 1.
a(n) = ( (-1)^n*(sqrt(2)-1) + sqrt(2) + 1 ) * 2^( (n - 1)/2 ) - 1.  (End)
E.g.f.: 2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x) - cosh(x) - sinh(x). - Franck Maminirina Ramaharo, Nov 23 2018

More terms from James Sellers, Jun 06 2000

cp's OEIS Frontend

A264036 Stirling transform of A077957 (aerated powers of 2).

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A264037 Stirling transform of A077957 (aerated powers of 2) with 0 prepended [0, 1, 0, 2, 0, 4, 0, 8, ...].

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A171700 Triangle T : T(n,k)= A007318(n,k)*a(n-k) with a(0)=0 and a(n)= A077957(n-1) for n>0.

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A001333 Pell-Lucas numbers: numerators of continued fraction convergents to sqrt(2).

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A035363 Number of partitions of n into even parts.

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A005418 Number of (n-1)-bead black-white reversible strings; also binary grids; also row sums of Losanitsch's triangle A034851; also number of caterpillar graphs on n+2 vertices.

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A052551 Expansion of 1/((1 - x)(1 - 2x^2)).