A094024 -id:A094024 - cp's OEIS frontend

A052542 a(n) = 2*a(n-1) + a(n-2), with a(0) = 1, a(1) = 2, a(2) = 4.

1, 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: 0

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

Apart from the initial 1, this sequence is simply twice the Pell numbers, A000129. - Antonio Alberto Olivares, Dec 31 2003
Image of 1/(1-2x) under the mapping g(x) -> g(x/(1+x^2)). - Paul Barry, Jan 16 2005
The intermediate convergents to 2^(1/2) begin with 4/3, 10/7, 24/17, 58/41; essentially, numerators = A052542 and denominators = A001333. - Clark Kimberling, Aug 26 2008
a(n) is the number of generalized compositions of n+1 when there are 2*i-2 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010
Apart from the initial 1, this is the p-INVERT transform of (1,0,1,0,1,0,...) for p(S) = 1 - 2 S. See A291219. - Clark Kimberling, Sep 02 2017
Conjecture: Apart from the initial 1, a(n) is the number of compositions of two types of n having no even parts. - Gregory L. Simay, Feb 17 2018
For n>0, a(n+1) is the length of tau^n(10) where tau is the morphism: 1 -> 101, 0 -> 1. See Song and Wu. - Michel Marcus, Jul 21 2020
The above conjecture is true, as the g.f. can be written as 1/(1 - (2*x)/(1 - x^2)). - John Tyler Rascoe, Jun 01 2024

Iain Fox, Table of n, a(n) for n = 0..2500 (first 1001 terms from Vincenzo Librandi)
Cyril Banderier and Donatella Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Cecília Pereira de Andrade, José Plínio de Oliveira Santos, Elen Viviani Pereira da Silva, and Kênia Cristina Pereira Silva, Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers, Open Journal of Discrete Mathematics, 2013, 3, 25-32 doi:10.4236/ojdm.2013.31006. - From _N. J. A. Sloane_, Feb 20 2013
Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 4.
Ira M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 477
Sergey Kitaev and Jeffrey Remmel, The 1-box pattern on pattern avoiding permutations, arXiv:1305.6970 [math.CO], 2013.
Haocong Song and Wen Wu, Hankel determinants of a Sturmian sequence, arXiv:2007.09940 [math.CO], 2020. See p.2 and 4.
Index entries for linear recurrences with constant coefficients, signature (2,1).

Cf. A052906. Essentially first differences of A001333.

a:=[2,4];; for n in [3..40] do a[n]:=2*a[n-1]+a[n-2]; od; a; # G. C. Greubel, May 09 2019

a052542 n = a052542_list !! n
a052542_list = 1 : 2 : 4 : tail (zipWith (+)
               (map (* 2) $ tail a052542_list) a052542_list)
-- Reinhard Zumkeller, Feb 24 2015

I:=[2,4]; [n le 2 select I[n] else 2*Self(n-1) +Self(n-2): n in [1..40]]; // G. C. Greubel, May 09 2019

spec := [S,{S=Sequence(Prod(Union(Z,Z),Sequence(Prod(Z,Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
A052542 := proc(n)
    option remember;
    if n <=2 then
        2^n;
    else
        2*procname(n-1)+procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Sep 23 2016
A052542List := proc(m) local A, P, n; A := [1,2]; 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: A052542List(31); # Peter Luschny, Mar 26 2022

Join[{1}, LinearRecurrence[{2, 1}, {2, 4}, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2012 *)

Vec((1-x^2)/(1-2*x-x^2) +O(x^40)) \\ Charles R Greathouse IV, Nov 20 2011

((1-x^2)/(1-2*x-x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

G.f.: (1 - x^2)/(1 - 2*x - x^2).
Recurrence: a(0)=1, a(2)=4, a(1)=2, a(n) + 2*a(n+1) - a(n+2) = 0;
a(n) = Sum_{alpha = RootOf(-1+2*x+x^2)} (1/2)*(1-alpha)*alpha^(-n-1).
a(n) = 2*A001333(n-1) + a(n-1), n > 1. A001333(n)/a(n) converges to sqrt(1/2). - Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003
Binomial transform of A094024. a(n) = 0^n + ((1 + sqrt(2))^n - (1 - sqrt(2))^n)/sqrt(2). - Paul Barry, Apr 22 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1, k)2^(n-2k). - Paul Barry, Jan 16 2005
If p[i] = 2*(i mod 2) 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, May 02 2010
a(n) = round(sqrt(Pell(2n) + Pell(2n-1))). - Richard R. Forberg, Jun 22 2014
a(n) = 2*A000129(n) + A000007(n) - Iain Fox, Nov 30 2017
a(n) = A000129(n) - A000129(n-2). - Gregory L. Simay, Feb 17 2018

A099425 Expansion of (1+x^2)/(1-2*x-x^2).

Original entry on oeis.org

1, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, 39202, 94642, 228486, 551614, 1331714, 3215042, 7761798, 18738638, 45239074, 109216786, 263672646, 636562078, 1536796802, 3710155682, 8957108166, 21624372014, 52205852194, 126036076402, 304278004998

Offset: 0

Paul Barry, Oct 15 2004

Binomial transform of A094024(n+1).
a(n) is the number of matchings of the corona C'(n) of the cycle graph C(n) and the complete graph K(1); in other words, C'(n) is the graph constructed from C(n) to which for each vertex v a new vertex v' and the edge vv' is added. Example: a(3)=14 because in the graph with vertex set {A,B,C,a,b,c} and edge set {AB,AC,BC,Aa,Bb,Cc} we have the following 14 matchings: the empty set, the six singletons containing one of the edges, {Aa,BC}, {Bb,AC}, {Cc,AB}, {Aa,Bb}, {Aa,Cc}, {Bb,Cc} and {Aa,Bb,Cc}. Row sums of A102413. - Emeric Deutsch, Jan 07 2005
Apart from first term, same as A002203. - Peter Shor, May 12 2005
Equals the INVERT transform of integers with repeats. Example: a(4) = 34 = (1, 1, 2, 6, 14) dot (5, 3, 3, 1, 1) = (5 + 3 + 6 + 6 + 14) = 34.

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

Cf. A000129, A001333, A002203, A094024, A102413.

Cf. A014176 (silver mean).

a099425 = sum . a102413_row  -- Reinhard Zumkeller, Apr 15 2014

a:= n-> (<<0|1>, <1|2>>^n. <<2, 2>>)[1, 1]-0^n:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 26 2018

CoefficientList[Series[(1+x^2)/(1-2x-x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,1},{1,2,6},40] (* Harvey P. Dale, Mar 23 2020 *)

a(n) = (1+sqrt(2))^n + (1-sqrt(2))^n - 0^n see silver mean (A014176).
a(n) = Sum_{k=0..n} A000129(n+1-k)*C(1, k/2)*(1+(-1)^k)/2.
a(n) = 2*A001333(n) - 0^n.
a(n) = round((1+sqrt(2))^n). - Bruno Berselli, Feb 04 2013
G.f.: G(0) - 1, where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 30 2013
a(n) = A000129(n+1) + A000129(n-1). - Vladimir Kruchinin, Apr 19 2024

A210552 Triangle of coefficients of polynomials u(n,x) jointly generated with A210553; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 5, 1, 5, 7, 10, 8, 1, 6, 9, 16, 18, 13, 1, 7, 11, 23, 31, 33, 21, 1, 8, 13, 31, 47, 62, 59, 34, 1, 9, 15, 40, 66, 101, 119, 105, 55, 1, 10, 17, 50, 88, 151, 205, 227, 185, 89, 1, 11, 19, 61, 113, 213, 321, 414, 426, 324, 144, 1, 12, 21, 73

Offset: 1

Clark Kimberling, Mar 22 2012

Let T(n,k) denote the term in row n, column k.
T(n,n):  A000045 (Fibonacci numbers)
T(n,n-1): A010049 (second-order Fibonacci numbers)
T(n,1): 1,1,1,1,1,1,1,1,1,1,1,,...
T(n,2): 2,3,4,5,6,7,8,9,10,11,...
T(n,3): 3,5,7,9,11,13,15,17,19,...
T(n,4): A052905
Row sums: A000225
Alternating row sums: A094024 (signed)
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
1...3...3
1...4...5...5
1...5...7...10...8
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 3x^2.

Cf. A210553, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210552 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210553 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)

u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A210553 Triangle of coefficients of polynomials v(n,x) jointly generated with A210552; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 3, 5, 3, 5, 4, 9, 8, 5, 6, 5, 14, 15, 15, 8, 7, 6, 20, 24, 31, 26, 13, 8, 7, 27, 35, 54, 57, 46, 21, 9, 8, 35, 48, 85, 104, 108, 80, 34, 10, 9, 44, 63, 125, 170, 209, 199, 139, 55, 11, 10, 54, 80, 175, 258, 360, 404, 366, 240, 89, 12, 11, 65, 99

Offset: 1

Clark Kimberling, Mar 22 2012

Let T(n,k) denote the term in row n, column k.
T(n,n):  A000045 (Fibonacci numbers)
T(n,n-1): A006367
T(n,n-2): A105423
T(n,1): 1,2,3,4,5,6,7,8,9,...
T(n,2): 1,2,3,4,5,6,7,8,9,...
T(n,3): A000096
T(n,4): A005563
T(n,5): A055831
T(n,6): A111694
Row sums: A000225
Alternating row sums: A052551
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2...1
3...2...2
4...3...5...3
5...4...9...8...5
First three polynomials v(n,x): 1, 2 + x , 3 + 2x + 2x^2.

Cf. A210552, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210552 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210553 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)

u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A098656 Expansion of x(1-4x)/((1-2x)(1-8x^2)).

Original entry on oeis.org

0, 1, -2, 4, -24, 16, -224, 64, -1920, 256, -15872, 1024, -129024, 4096, -1040384, 16384, -8355840, 65536, -66977792, 262144, -536346624, 1048576, -4292870144, 4194304, -34351349760, 16777216, -274844352512, 67108864, -2198889037824, 268435456, -17591649173504, 1073741824

Offset: 0

Paul Barry, Sep 19 2004

Let A=[1,2,1;2,0,-2;1,-2,1] the 3 X 3 symmetric Krawtchouk matrix. Than a(n) is the 1,3 element of A^n.

P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum walks, Contemporary Mathematics, 287 2001, pp. 83-96.

Index entries for linear recurrences with constant coefficients, signature (2,8,-16).

Cf. A098655, A098657.

CoefficientList[Series[x (1-4x)/((1-2x)(1-8x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,8,-16},{0,1,-2},40] (* Harvey P. Dale, Jun 30 2011 *)

a(n)=2^(n-1)-2^(3(n-1)/2)(1+(-1)^n)/sqrt(2); a(n)=2a(n-1)+8a(n-2)-16a(n-3).

a(n) = (-2)^(n-1)*A094024(n-1). - R. J. Mathar, Mar 08 2021

cp's OEIS Frontend

A052542 a(n) = 2*a(n-1) + a(n-2), with a(0) = 1, a(1) = 2, a(2) = 4.

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A099425 Expansion of (1+x^2)/(1-2*x-x^2).

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A210552 Triangle of coefficients of polynomials u(n,x) jointly generated with A210553; see the Formula section.

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A210553 Triangle of coefficients of polynomials v(n,x) jointly generated with A210552; see the Formula section.

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A098656 Expansion of x(1-4x)/((1-2x)(1-8x^2)).

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