A083978 -id:A083978 - 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)

A083972 Smallest palindromic prime containing exactly n 1's.

Original entry on oeis.org

313, 11, 18181, 11311, 1311131, 1114111, 110111011, 111181111, 11115151111, 1011117111101, 1110111110111, 1111118111111, 111111212111111, 111111151111111, 11121111111112111, 1011111110111111101, 1111111611161111111, 111110111171111011111, 1111111111111111111, 111111111161111111111, 11111111181118111111111, 1101111111119111111111011

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 21 2003

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A083973, A083974, A083975, A083976, A083977, A083978, A083979, A083980, A083981.

digOfInterest = 1;
primes = {};
While[Length[primes] < 6,
  n = Length[primes] + 1;
  i = 0;
  While[Length[primes] != n,
   i++;
   dig = IntegerDigits[Prime[i]];
   If[
    (dig == Reverse[dig]) &&
     (Count[dig, digOfInterest] == n),
    AppendTo[primes, FromDigits[dig]];
    ];
   ];
  ];
primes
(* Kevin Southwick, Jul 25 2015 *)

More terms from David Wasserman, Dec 06 2004
Corrected and extended by Giovanni Resta, Feb 08 2006
2 more terms from Erich Friedman, Mar 23 2008

A083973 Smallest palindromic prime containing exactly n 2's.

Original entry on oeis.org

2, 12421, 72227, 1221221, 3222223, 122232221, 322222223, 12222122221, 1202222222021, 1222227222221, 9222222222229, 122222232222221, 32222322222322223, 72222222422222227, 1202222222222222021, 1222222226222222221, 125222222222222222521, 722222222202222222227

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 21 2003

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A083972, A083974, A083975, A083976, A083977, A083978, A083979, A083980, A083981.

Table[d = 2; i = 1;
 While[ x = IntegerDigits[Prime[i]];
  x != Reverse[x] || Count[x, d] != n , i++];
FromDigits[x], {n, 1, 6}]  (* Robert Price, Mar 25 2019 *)

Corrected and extended by Giovanni Resta, Feb 08 2006

A083976 Smallest palindromic prime containing exactly n 5's.

Original entry on oeis.org

5, 15451, 15551, 1550551, 154555451, 155535551, 355555553, 35555155553, 75555555557, 1555550555551, 152555555555251, 10555555355555501, 15505555555550551, 35555555755555553, 1555555055505555551, 1555555551555555551, 115555555555555555511

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 21 2003

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A083972, A083973, A083974, A083976, A083977, A083978, A083979, A083980, A083981.

Table[d = 5; i = 1;
 While[ x = IntegerDigits[Prime[i]];
  x != Reverse[x] || Count[x, d] != n , i++];
FromDigits[x], {n, 1, 6}]  (* Robert Price, Mar 25 2019 *)

More terms from Giovanni Resta, Feb 08 2006

A083977 Smallest palindromic prime containing exactly n 6's.

Original entry on oeis.org

10601, 16061, 16661, 1660661, 7666667, 166636661, 10666666601, 36666766663, 1766666666671, 3666662666663, 1666666666661, 366666656666663, 11666666666666611, 36666666766666663, 16666666666666661, 3666666665666666663, 1666666666666666661, 166666666636666666661

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 21 2003

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A083972, A083973, A083974, A083975, A083976, A083978, A083979, A083980, A083981.

Table[d = 6; i = 1;
 While[ x = IntegerDigits[Prime[i]];
  x != Reverse[x] || Count[x, d] != n , i++];
FromDigits[x], {n, 1, 5}]  (* Robert Price, Mar 25 2019 *)

More terms from Giovanni Resta, Feb 08 2006

A083979 Smallest palindromic prime containing exactly n 8's.

Original entry on oeis.org

181, 18181, 78887, 1880881, 9888889, 188868881, 188888881, 18888588881, 1488888888841, 7888886888887, 3888888888883, 188888858888881, 188888888888881, 38888888088888883, 1688888888888888861, 3888888882888888883, 128888888888888888821

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 21 2003

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A083972, A083973, A083974, A083975, A083976, A083977, A083978, A083980, A083981.

Table[d = 8; i = 1;
While[ x = IntegerDigits[Prime[i]];
  x != Reverse[x] || Count[x, d] != n , i++];
FromDigits[x], {n, 1, 5}]  (* Robert Price, Mar 25 2019 *)
Module[{prs=Select[Prime[Range[105*10^5]],PalindromeQ]},Table[ SelectFirst[ prs,DigitCount[ #,10,8]==n&],{n,6}]] (* The program generates the first 6 terms of the sequence. *) (* Harvey P. Dale, May 11 2022 *)

Corrected and extended by Giovanni Resta, Feb 08 2006

A083980 Smallest palindromic prime containing exactly n 9's.

Original entry on oeis.org

191, 919, 19991, 1993991, 9199919, 199909991, 199999991, 90999599909, 95999999959, 99999199999, 9599999999959, 199999959999991, 959999999999959, 39999999299999993, 94999999999999949, 99999999299999999, 9299999999999999929, 9999999992999999999

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 21 2003

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A083972, A083973, A083974, A083975, A083976, A083977, A083978, A083979, A083981.

Table[d = 9; i = 1;
 While[ x = IntegerDigits[Prime[i]];
  x != Reverse[x] || Count[x, d] != n , i++];
FromDigits[x], {n, 1, 5}]  (* Robert Price, Mar 25 2019 *)

Corrected and extended by Giovanni Resta, Feb 08 2006

A083981 Smallest palindromic prime containing exactly n zeros.

Original entry on oeis.org

101, 10301, 1300031, 1003001, 100404001, 100030001, 10060006001, 10000500001, 1000030300001, 1000008000001, 140000000000041, 300000020000003, 10000003030000001, 10000000500000001, 1000000400040000001, 1000000008000000001, 100000000303000000001

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 21 2003

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A083972, A083973, A083974, A083975, A083976, A083977, A083978, A083979, A083980.

Table[d = 0; i = 1;
 While[ x = IntegerDigits[Prime[i]];
  x != Reverse[x] || Count[x, d] != n , i++];
FromDigits[x], {n, 1, 5}]  (* Robert Price, Mar 25 2019 *)

Corrected and extended by Giovanni Resta, Feb 08 2006

A083974 Smallest palindromic prime containing exactly n 3's.

Original entry on oeis.org

3, 313, 13331, 33533, 1333331, 3331333, 323333323, 13333633331, 31333333313, 1333335333331, 3233333333323, 133333353333331, 332333333333233, 333333313333333, 33233333333333233, 1333333336333333331, 3333333313133333333, 133333333383333333331

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 21 2003

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A083972, A083973, A083975, A083976, A083977, A083978, A083979, A083980, A083981.

Table[d = 3; i = 1;
 While[ x = IntegerDigits[Prime[i]];
  x != Reverse[x] || Count[x, d] != n , i++];
FromDigits[x], {n, 1, 6}]  (* Robert Price, Mar 25 2019 *)

More terms from Giovanni Resta, Feb 08 2006

A083975 Smallest palindromic prime containing exactly n 4's.

Original entry on oeis.org

11411, 14341, 1464641, 1447441, 1444441, 144404441, 14244444241, 34444344443, 74444444447, 1444447444441, 3444444444443, 144444454444441, 14144444444444141, 34444444044444443, 1414444444444444141, 1444444447444444441, 104444444444444444401

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 21 2003

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A083972, A083973, A083975, A083976, A083977, A083978, A083979, A083980, A083981.

Table[d = 4; i = 1;
 While[ x = IntegerDigits[Prime[i]];
  x != Reverse[x] || Count[x, d] != n , i++];
FromDigits[x], {n, 1, 6}]  (* Robert Price, Mar 25 2019 *)

More terms from Ray G. Opao, Sep 20 2004

Corrected and extended by Giovanni Resta, Feb 08 2006

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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A083972 Smallest palindromic prime containing exactly n 1's.

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A083973 Smallest palindromic prime containing exactly n 2's.

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A083976 Smallest palindromic prime containing exactly n 5's.

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A083977 Smallest palindromic prime containing exactly n 6's.

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A083979 Smallest palindromic prime containing exactly n 8's.

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A083980 Smallest palindromic prime containing exactly n 9's.

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A083981 Smallest palindromic prime containing exactly n zeros.

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