A094803 -id:A094803 - cp's OEIS frontend

A006012 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - 2a(n-2), n >= 2.

1, 2, 6, 20, 68, 232, 792, 2704, 9232, 31520, 107616, 367424, 1254464, 4283008, 14623104, 49926400, 170459392, 581984768, 1987020288, 6784111616, 23162405888, 79081400320, 270000789504, 921840357376, 3147359850496

Offset: 0

N. J. A. Sloane

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 4, s(2n) = 4. - Herbert Kociemba, Jun 12 2004
a(n-1) counts permutations pi on [n] for which the pairs {i, pi(i)} with i < pi(i), considered as closed intervals [i+1,pi(i)], do not overlap; equivalently, for each i in [n] there is at most one j <= i with pi(j) > i. Counting these permutations by the position of n yields the recurrence relation. - David Callan, Sep 02 2003
a(n) is the sum of (n+1)-th row terms of triangle A140070. - Gary W. Adamson, May 04 2008
The binomial transform is in A083878, the Catalan transform in A084868. - R. J. Mathar, Nov 23 2008
Equals row sums of triangle A152252. - Gary W. Adamson, Nov 30 2008
Counts all paths of length (2*n), n >= 0, starting at the initial node on the path graph P_7, see the second Maple program. - Johannes W. Meijer, May 29 2010
From L. Edson Jeffery, Apr 04 2011: (Start)
Let U_1 and U_3 be the unit-primitive matrices (see [Jeffery])
U_1 = U_(8,1) = [(0,1,0,0); (1,0,1,0); (0,1,0,1); (0,0,2,0)] and
U_3 = U_(8,3) = [(0,0,0,1); (0,0,2,0); (0,2,0,1); (2,0,2,0)]. Then a(n) = (1/4) * Trace(U_1^(2*n)) = (1/2^(n+2)) * Trace(U_3^(2*n)). (See also A084130, A001333.) (End)
Pisano period lengths: 1, 1, 8, 1, 24, 8, 6, 1, 24, 24, 120, 8, 168, 6, 24, 1, 8, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012
a(n) is the first superdiagonal of array A228405. - Richard R. Forberg, Sep 02 2013
Conjecture: With offset 1, a(n) is the number of permutations on [n] with no subsequence abcd such that (i) bc are adjacent in position and (ii) max(a,c) < min(b,d). For example, the 4 permutations of [4] not counted by a(4) are 1324, 1423, 2314, 2413. - David Callan, Aug 27 2014
The conjecture of David Callan above is correct - with offset 1, a(n) is the number of permutations on [n] with no subsequence abcd such that (i) bc are adjacent in position and (ii) max(a,c) < min(b,d). - Yonah Biers-Ariel, Jun 27 2017
From Gary W. Adamson, Jul 22 2016: (Start)
A production matrix for the sequence is M =
  1, 1, 0, 0, 0, 0, ...
  1, 0, 3, 0, 0, 0, ...
  1, 0, 0, 3, 0, 0, ...
  1, 0, 0, 0, 3, 0, ...
  1, 0, 0, 0, 0, 3, ...
  ...
Take powers of M, extracting the upper left terms; getting the sequence starting: (1, 1, 2, 6, 20, 68, ...). (End)
From Gary W. Adamson, Jul 24 2016: (Start)
The sequence is the INVERT transform of the powers of 3 prefaced with a "1": (1, 1, 3, 9, 27, ...) and is N=3 in an infinite of analogous sequences starting:
   N=1 (A000079):  1, 2, 4,  8,  16,   32, ...
   N=2 (A001519):  1, 2, 5, 13,  34,   89, ...
   N=3 (A006012):  1, 2, 6, 20,  68,  232, ...
   N=4 (A052961):  1, 2, 7, 29, 124,  533, ...
   N=5 (A154626):  1, 2, 8, 40, 208, 1088, ...
   N=6:            1, 2, 9, 53, 326, 2017, ...
   ... (End)
Number of permutations of length n > 0 avoiding the partially ordered pattern (POP) {1>2, 1>3, 4>2, 4>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first and fourth elements are larger than the second and third elements. - Sergey Kitaev, Dec 08 2020
a(n-1) is the number of permutations of [n] that can be obtained by placing n points on an X-shape (two crossing lines with slopes 1 and -1), labeling them 1,2,...,n by increasing y-coordinate, and then reading the labels by increasing x-coordinate. - Sergi Elizalde, Sep 27 2021
Consider a stack of pancakes of height n, where the only allowed operation is reversing the top portion of the stack. First, perform a series of reversals of decreasing sizes, followed by a series of reversals of increasing sizes. The number of distinct permutations of the initial stack that can be reached through these operations is a(n). - Thomas Baruchel, May 12 2025
Number of permutations of [n] that are correctly sorted after performing one left-to-right pass and one right-to-left pass of the cocktail sort. - Thomas Baruchel, May 16 2025

D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms. Birkhäuser, Boston, 3rd edition, 1990, p. 86.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.4.8 Answer to Exer. 8.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Andrei Asinowski and Toufik Mansour, Separable d-Permutations and Guillotine Partitions, arXiv 0803.3414 [math.CO], 2008. Annals of Combinatorics 14 (1) pp.17-43 Springer, 2010; Abstract
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 155
George Balla, Ghislain Fourier, and Kunda Kambaso, PBW filtration and monomial bases for Demazure modules in types A and C, arXiv:2205.01747 [math.RT], 2022.
M. Barnabei, F. Bonetti, and M. Silimbani, Two permutation classes related to the Bubble Sort operator, Electronic Journal of Combinatorics 19(3) (2012), #P25. - From _N. J. A. Sloane_, Dec 25 2012
Yonah Biers-Ariel, A New Quantity Counted by OEIS Sequence A006012, arXiv:1706.07064 [math.CO], 2017.
Yonah Biers-Ariel, The Number of Permutations Avoiding a Set of Generalized Permutation Patterns, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.3.
Rocco Chirivì, Xin Fang, and Ghislain Fourier, Degenerate Schubert varieties in type A, Transformation Groups (2020).
CombOS - Combinatorial Object Server, Generate pattern-avoiding permutations
Sergi Elizalde, The X-class and almost-increasing permutations, arXiv:0710.5168 [math:CO], 2007. Ann. Comb. 15 (2011), 51-68.
S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Luca Ferrari, Enhancing the connections between patterns in permutations and forbidden configurations in restricted elections, arXiv:1906.10553 [math.CO], 2019.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
L. E. Jeffery, Unit-primitive matrices
Donghyun Kim and Lauren Williams, Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations, arXiv:2106.13378 [math.CO], 2021.
Sergey Kitaev and Artem Pyatkin, On permutations avoiding partially ordered patterns defined by bipartite graphs, arXiv:2204.08936 [math.CO], 2022.
Joshua Marsh and Nathan Williams, Nesting Nonpartitions, J. Int. Seq., Vol. 25 (2022), Article 22.8.8.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Joris Nieuwveld, Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding, Master's Thesis, arXiv:2108.11382 [math.NT], 2021.
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
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
Markus Saers, Dekai Wu, and Chris Quirk, On the Expressivity of Linear Transductions.
Yi-dong Sun and Cang-zhi Jia, Counting Dyck paths with strictly increasing peak sequences, J. Math. Res. Expos. 27 (2) (2007) 253, Theorem 3.11.
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A000079, A000129, A001333, A001519, A002203, A003480, A007052, A007070, A024175, A030436, A052961, A056236, A083978, A084130, A084868, A094803, A098158, A140070, A147703, A152252, A154626, A201730, A228405, A265185.

a006012 n = a006012_list !! n
a006012_list = 1 : 2 : zipWith (-) (tail $ map (* 4) a006012_list)
(map (* 2) a006012_list)
-- Reinhard Zumkeller, Oct 03 2011

[n le 2 select n else 4*Self(n-1)- 2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 05 2011

A006012:=-(-1+2*z)/(1-4*z+2*z**2); # Simon Plouffe in his 1992 dissertation
with(GraphTheory): G:=PathGraph(7): A:= AdjacencyMatrix(G): nmax:=24; n2:=2*nmax: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..7); od: seq(a(2*n),n=0..nmax); # Johannes W. Meijer, May 29 2010

LinearRecurrence[{4,-2},{1,2},50] (* or *) With[{c=Sqrt[2]}, Simplify[ Table[((2+c)^n+(3+2c)(2-c)^n)/(2(2+c)),{n,50}]]] (* Harvey P. Dale, Aug 29 2011 *)

{a(n) = real(((2 + quadgen(8))^n))}; /* Michael Somos, Feb 12 2004 */

{a(n) = if( n<0, 2^n, 1) * polsym(x^2 - 4*x + 2, abs(n))[abs(n)+1] / 2}; /* Michael Somos, Feb 12 2004 */

Vec((1-2*x)/(1-4*x+2*x^2) + O(x^100)) \\ Altug Alkan, Dec 05 2015

l = [1, 2]
for n in range(2, 101): l.append(4 * l[n - 1] - 2 * l[n - 2])
print(l)  # Indranil Ghosh, Jul 02 2017

A006012=BinaryRecurrenceSequence(4,-2,1,2)
print([A006012(n) for n in range(41)]) # G. C. Greubel, Aug 27 2025

G.f.: (1-2*x)/(1 - 4*x + 2*x^2).
a(n) = 2*A007052(n-1) = A056236(n)/2.
Limit_{n -> oo} a(n)/a(n-1) = 2 + sqrt(2). - Zak Seidov, Oct 12 2002
From Paul Barry, May 08 2003: (Start)
Binomial transform of A001333.
E.g.f.: exp(2*x)*cosh(sqrt(2)*x). (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*2^(n-k) = Sum_{k=0..n} binomial(n, k)*2^(n-k/2)(1+(-1)^k)/2. - Paul Barry, Nov 22 2003 (typo corrected by Manfred Scheucher, Jan 17 2023)
a(n) = ((2+sqrt(2))^n + (2-sqrt(2))^n)/2.
a(n) = [M^n]{2,2}, where M is the 3 X 3 matrix: M = [1,1,1;1,2,1;1,1,1]. - _Simone Severini, Jun 11 2006
a(n) = Sum_{k=0..n} 2^k*A098158(n,k). - Philippe Deléham, Dec 04 2006
a(n) = A007070(n) - 2*A007070(n-1). - R. J. Mathar, Nov 16 2007
a(n) = Sum_{k=0..n} A147703(n,k). - Philippe Deléham, Nov 29 2008
a(n) = Sum_{k=0..n} A201730(n,k). - Philippe Deléham, Dec 05 2011
G.f.: G(0) where G(k)= 1 + 2*x/((1-2*x) - 2*x*(1-2*x)/(2*x + (1-2*x)*2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 10 2012
G.f.: G(0)*(1-2*x)/2, where G(k) = 1 + 1/(1 - 2*x*(4*k+2-x)/( 2*x*(4*k+4-x) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 27 2014
a(-n) = a(n) / 2^n for all n in Z. - Michael Somos, Aug 24 2014
a(n) = A265185(n) / 4, connecting this sequence to the simple Lie algebra B_4. - Tom Copeland, Dec 04 2015
From G. C. Greubel, Aug 27 2025: (Start)
a(n) = 2^((n-2)/2)*( (n+1 mod 2)*A002203(n) + 2*sqrt(2)*(n mod 2)*A000129(n) ).
a(n) = 2^(n/2)*ChebyshevT(n, sqrt(2)). (End)

A030436 Expansion of g.f. (1 + x - 2x^2 - x^3)/(1 - 4x^2 + 2*x^4).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 34, 68, 116, 232, 396, 792, 1352, 2704, 4616, 9232, 15760, 31520, 53808, 107616, 183712, 367424, 627232, 1254464, 2141504, 4283008, 7311552, 14623104, 24963200, 49926400, 85229696, 170459392, 290992384, 581984768, 993510144, 1987020288, 3392055808

Offset: 0

N. J. A. Sloane, Dec 11 1999

Also (starting 3, 6, ...) the number of zig-zag paths from top to bottom of a rectangle of width 7 whose color is not that of the top right corner.
From Johannes W. Meijer, May 29 2010: (Start)
The a(n) represent the number of possible chess games, ignoring the fifty-move and the triple repetition rules, after n moves by White in the following position: White Ka1, Nh1, pawns a2, b6, c2, d6, f2, g3 and h2; Black Ka8, Bc8, pawns a3, b7, c3, d7, f3, g4 and h3.
Counts all paths of length n, n>=0, starting at the initial node on the path graph P_7, see the Maple program. (End)
Range of row n of the circular Pascal array of order 8. - Shaun V. Ault, Jun 05 2014.
In general, a(n,m) = (2^n/(m+1))*Sum_{r=1..m} (1-(-1)^r)*cos(Pi*r/(m+1))^n*(1+cos(Pi*r/(m+1))) gives the number of paths of length n starting at the initial node on the path graph P_m. Here we have m=7. - Herbert Kociemba, Sep 17 2020

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 34*x^7 + 68*x^8 + ...

Stefano Spezia, Table of n, a(n) for n = 0..3600
Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, arXiv:1407.2197 [math.CO], 2014.
Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics, Volume 332, October 2014, Pages 45-54.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
Joseph Myers, BMO 2008-2009 Round 1 Problem 1 - Generalisation
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2).

Cf. A006012, A007052, A030435.
a(n) = A094718(7, n).
Cf. A024175, A094803, A000045, A038754, A028495, A061551 and A178381.

with(GraphTheory): P:=7: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=31; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..P); od: seq(a(n),n=0..nmax); # Johannes W. Meijer, May 29 2010
X := j -> (-1)^(j/8) - (-1)^(1-j/8):
a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7])/8:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 17 2020

CoefficientList[Series[(1+x-2x^2-x^3)/(1-4x^2+2x^4),{x,0,40}],x] (* or *) LinearRecurrence[{0,4,0,-2},{1,1,2,3},41] (* Harvey P. Dale, May 11 2011 *)
a[n_,m_]:=2^(n+1)/(m+1) Module[{x=(Pi r)/(m+1)},Sum[Cos[x]^n (1+Cos[x]),{r,1,m,2}]]
Table[a[n,7],{n,0,40}]//Round (* Herbert Kociemba, Sep 17 2020 *)

Vec((1+x-2*x^2-x^3)/(1-4*x^2+2*x^4)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

{a(n) = if( n<0, 0, polsym( x^4 - 4*x^2 + 2, n + n%2)[n + n%2 + 1] / (4 * (n%2 + 1)))}; /* Michael Somos, Feb 08 2015 */

a(0)=a(1)=1, a(2)=2, a(3)=3, a(n)=4*a(n-2)-2*a(n-4). - Harvey P. Dale, May 11 2011
a(n) = (2+sqrt(2+sqrt(2)))/8*(sqrt(2+sqrt(2)))^n + (2-sqrt(2+sqrt(2)))/8*(-sqrt(2+sqrt(2)))^n + (2+sqrt(2-sqrt(2)))/8*(sqrt(2-sqrt(2)))^n + (2-sqrt(2-sqrt(2)))/8*(-sqrt(2-sqrt(2)))^n. - Sergei N. Gladkovskii, Aug 23 2012
a(n) = A030435(n)/2. a(2*n) = A006012(n). a(2*n + 1) = A007052(n). - Michael Somos, Mar 06 2003
a(n) = (2^n/8)*Sum_{r=1..7} (1-(-1)^r)cos(Pi*r/8)^n*(1+cos(Pi*r/8)). - Herbert Kociemba, Sep 17 2020
E.g.f.: (2*cosh(r*x) + 2*cosh(s*x) + r*sinh(r*x) + s*sinh(s*x))/4, where r = sqrt(2 - sqrt(2)) and s = sqrt(2 + sqrt(2)). - Stefano Spezia, Jun 14 2023

Comment and link added and typo in cross-reference corrected by Joseph Myers, Dec 24 2008, May 30 2010

A024175 Expansion of g.f. (x^3 - 6x^2 + 5x - 1)/((2x - 1)(2x^2 - 4x + 1)).

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 132, 428, 1416, 4744, 16016, 54320, 184736, 629280, 2145600, 7319744, 24979584, 85262464, 291057920, 993641216, 3392317952, 11581727232, 39541748736, 135002491904, 460924372992, 1573688313856, 5372896120832, 18344191078400, 62630938517504

Offset: 0

N. J. A. Sloane

Number of (s(0), s(1), ..., s(2*n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1, 2, ..., 2*n, s(0) = 1, s(2*n) = 1. - Herbert Kociemba, Jun 11 2004

Counts all paths of length (2*n), n >= 0, starting and ending at the initial node on the path graph P_7, see the Maple program. - Johannes W. Meijer, May 29 2010

			1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 428*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 27.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
Michael Dairyko, Lara Pudwell, Samantha Tyner, and Casey Wynn, Non-contiguous pattern avoidance in binary trees. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227.
Stefan Felsner and Daniel Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
Matthew Hyatt and Jeffrey Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052, 2012. - From _N. J. A. Sloane_, Dec 24 2012
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (Corollary 3, case k=6, pages 10-11). - From _N. J. A. Sloane_, May 09 2012
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 13.
David Nečas and Ivan Ohlídal, Consolidated series for efficient calculation of the reflection and transmission in rough multilayers, Optics Express, Vol. 22, 2014, No. 4; DOI:10.1364/OE.22.004499.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Lara Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), 2012. - From N. J. A. Sloane, Jan 03 2013
Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (6,-10,4).

Cf. A006012, A030436 and A094803.

Cf. A211216.

with(GraphTheory): G:=PathGraph(7): A:= AdjacencyMatrix(G): nmax:=26; n2:=2*nmax: for n from 0 to n2 do B(n):=A^n; a(n):=B(n)[1,1]; od: seq(a(2*n),n=0..nmax); # Johannes W. Meijer, May 29 2010

CoefficientList[Series[(x^3-6*x^2+5*x-1)/((2*x-1)*(2*x^2-4*x+1)),{x,0,30}],x] (* Vincenzo Librandi, May 10 2012 *)

{a(n) = local(A); A = 1; for( i=1, 6, A = 1 / (1 - x*A)); polcoeff( A + x * O(x^n), n)} /* Michael Somos, May 12 2012 */

From Herbert Kociemba, Jun 11 2004: (Start)
a(n) = (1/4)*Sum_{r=1..7} sin(r*Pi/8)^2*(2*cos(r*Pi/8))^(2n), n >= 1.
a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3), n >= 4. (End)
a(n) = (1/4)*((2 + sqrt(2))^(n - 1) + (2 - sqrt(2))^(n - 1) + 2^n) for n >= 1. - Richard Choulet, Apr 19 2010
a(n) = 2^(n - 2) + A006012(n-1)/2, n > 0. - R. J. Mathar, Mar 14 2011
G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x)))))). - Michael Somos, May 12 2012
E.g.f.: (1 + exp(2*x)*(1 + 2*cosh(sqrt(2)*x) - sqrt(2)*sinh(sqrt(2)*x)))/4. - Stefano Spezia, Jun 14 2023

cp's OEIS Frontend

A006012 a(0) = 1, a(1) = 2, a(n) = 4*a(n-1) - 2*a(n-2), n >= 2.

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A030436 Expansion of g.f. (1 + x - 2*x^2 - x^3)/(1 - 4*x^2 + 2*x^4).

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A024175 Expansion of g.f. (x^3 - 6*x^2 + 5*x - 1)/((2*x - 1)*(2*x^2 - 4*x + 1)).

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A216232 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 3 or if k-n >= 5, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A217257 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 7, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,3) = T(0,4) = T(0,5) = T(0,6) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A006012 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - 2a(n-2), n >= 2.

A030436 Expansion of g.f. (1 + x - 2x^2 - x^3)/(1 - 4x^2 + 2*x^4).

A024175 Expansion of g.f. (x^3 - 6x^2 + 5x - 1)/((2x - 1)(2x^2 - 4x + 1)).