A083878 -id:A083878 - 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)

A056236 a(n) = (2 + sqrt(2))^n + (2 - sqrt(2))^n.

Original entry on oeis.org

2, 4, 12, 40, 136, 464, 1584, 5408, 18464, 63040, 215232, 734848, 2508928, 8566016, 29246208, 99852800, 340918784, 1163969536, 3974040576, 13568223232, 46324811776, 158162800640, 540001579008, 1843680714752, 6294719700992

Offset: 0

Henry Bottomley, Aug 11 2000

First differences give A060995. - Jeremy Gardiner, Aug 11 2013
Binomial transform of A002203 [Bhadouria].
The binomial transform of this sequence is 2, 6, 22, 90, 386, .. = 2*A083878(n). - R. J. Mathar, Nov 10 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Pooja Bhadouria, Deepika Jhala, and Bijendra Singh, Binomial Transforms of the k-Lucas Sequences and its Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92, sequence B_2.
Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 9.
Takao Komatsu, Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions, arXiv:2105.08277 [math.CO], 2021.
Youngwoo Kwon, Binomial transforms of the modified k-Fibonacci-like sequence, arXiv:1804.08119 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A006012, A007052, A020727.

LinearRecurrence[{4,-2},{2,4},30] (* Harvey P. Dale, Jan 18 2013 *)

PARI
```
a(n) = 2*real((2+quadgen(8))^n);
```

[lucas_number2(n,4,2) for n in range(37)] # Zerinvary Lajos, Jun 25 2008

a(n) = 4*a(n-1) - 2*a(n-2).
a(n) = a(n-2) - a(n-1) + 2*A020727(n-1).
a(n) = 2*A006012(n) = 4*A007052(n-1).
For n>2, a(n) = floor((2+sqrt(2))*a(n-1)).
G.f.: 2*(1-2*x)/(1-4*x+2*x^2).
From L. Edson Jeffery, Apr 08 2011: (Start)
a(n) = 2^(2*n)*(cos(Pi/8)^(2*n) + cos(3*Pi/8)^(2*n)).
a(n) = 3*a(n-1) + Sum_{k=1..(n-2)} a(k), for n>1, with a(0)=2, a(1)=4. (End)
a(n) = [x^n] ( (1 + 4*x + sqrt(1 + 8*x + 8*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

More terms from James Sellers, Aug 25 2000

A164299 a(n) = ((1+4sqrt(2))(3+sqrt(2))^n + (1-4sqrt(2))(3-sqrt(2))^n)/2.

Original entry on oeis.org

1, 11, 59, 277, 1249, 5555, 24587, 108637, 479713, 2117819, 9348923, 41268805, 182170369, 804140579, 3549650891, 15668921293, 69165971521, 305313380075, 1347718479803, 5949117218293, 26260673951137, 115920223178771

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009

nonn

Binomial transform of A164298. Third binomial transform of A164587. Inverse binomial transform of A164300.

This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Sequences in the class a(n, m): A164298 (m=1), this sequence (m=2), A164300 (m=3), A164301 (m=4), A164598 (m=5), A164599 (m=6), A081185 (m=7), A164600 (m=8).

Cf. A081179, A083878, A164587.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+4*r)*(3+r)^n+(1-4*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 17 2009

LinearRecurrence[{6,-7}, {1,11}, 50] (* or *) CoefficientList[Series[(1 + 5*x)/(1 - 6*x + 7*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 12 2017 *)

my(x='x+O('x^50)); Vec((1+5*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Sep 12 2017

[( (1+5*x)/(1-6*x+7*x^2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 1, a(1) = 11.
G.f.: (1+5*x)/(1-6*x+7*x^2).
E.g.f.: (cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(3*x). - G. C. Greubel, Sep 12 2017
From G. C. Greubel, Mar 12 2021: (Start)
a(n) = 2*A083878(n) + 8*A081179(n).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*2^(n-k)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 17 2009

A208345 Triangle of coefficients of polynomials v(n,x) jointly generated with A208344; see the Formula section.

Original entry on oeis.org

1, 0, 3, 0, 1, 7, 0, 1, 3, 17, 0, 1, 3, 10, 41, 0, 1, 3, 11, 30, 99, 0, 1, 3, 12, 35, 87, 239, 0, 1, 3, 13, 40, 108, 245, 577, 0, 1, 3, 14, 45, 130, 322, 676, 1393, 0, 1, 3, 15, 50, 153, 406, 938, 1836, 3363, 0, 1, 3, 16, 55, 177, 497, 1236, 2682, 4925, 8119, 0, 1

Offset: 1

Clark Kimberling, Feb 25 2012

Row sums, u(n,1): (1,2,5,13,...), odd-indexed Fibonacci numbers.
Row sums, v(n,1): (1,3,8,21,...), even-indexed Fibonacci numbers.
As triangle T(n,k) with 0<=k<=n, it is (0, 1/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

			First five rows:
  1
  0   3
  0   1   7
  0   1   3   17
  0   1   3   10   41
First five polynomials u(n,x):
  1, 3*x, x + 7*x^2, x + 3*x^2 + 17*x^3, x + 3*x^2 + 10*x^3 + 41*x^4.

Cf. A208344.

Cf. A084938, A000045, A000244, A001906.

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x];
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[%]    (* A208344 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208345 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]
Table[v[n, x] /. x -> 1, {n, 1, z}]

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 26 2012: (Start)
As triangle T(n,k), 0<=k<=n:
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-2) - 2*T(n-2,k-1) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1+(y-1)*x)/(1-(1+2*y)*x+y*(2-y)*x^2).
Sum_{k=0..n} T(n,k)*x^k = A152167(n), A000007(n), A001906(n+1), A003948(n) for x = -1, 0, 1, 2 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A078057(n), A001906(n+1), A000244(n), A081567(n), A083878(n), A165310(n) for x = 0, 1, 2, 3, 4, 5 respectively. (End)

A083880 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 23a(n-2), n >= 2.

Original entry on oeis.org

1, 5, 27, 155, 929, 5725, 35883, 227155, 1446241, 9237845, 59114907, 378678635, 2427143489, 15561826285, 99793962603, 640017621475, 4104915074881, 26328745454885, 168874407826587, 1083182932803515, 6947717948023649

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A083879.

Inverse binomial transform of A147957. 5th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A083878, A006012.

[ n eq 1 select 1 else n eq 2 select 5 else 10*Self(n-1)-23*Self(n-2): n in [1..21] ]; // Klaus Brockhaus, Dec 16 2008

LinearRecurrence[{10,-23},{1,5},30] (* Harvey P. Dale, May 14 2018 *)

a(n)=if(n<0,0,polsym(23-10*x+x^2,n)[n+1]/2)

G.f.: (1-5x)/(1-10x+23x^2).
E.g.f.: exp(5x)cosh(x*sqrt(2)).
a(n) = ((5-sqrt(2))^n + (5+sqrt(2))^n)/2;
a(n) = Sum_{k=0..n} C(n, 2k)*5^(n-2k)*2^k.
a(n) = (Sum_{k=0..n} A098158(n,k)*5^(2k)*2^(n-k))/5^n. - Philippe Deléham, Nov 30 2008

Typo in definition corrected by Klaus Brockhaus, Dec 16 2008

A083879 a(0)=1, a(1)=4, a(n) = 8a(n-1) - 14a(n-2), n >= 2.

Original entry on oeis.org

1, 4, 18, 88, 452, 2384, 12744, 68576, 370192, 2001472, 10829088, 58612096, 317289536, 1717746944, 9299922048, 50350919168, 272608444672, 1475954689024, 7991119286784, 43265588647936, 234249039168512, 1268274072276992

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A083878.
4th binomial transform of A077957. Inverse binomial transform of A083880. - Philippe Deléham, Nov 30 2008
From L. Edson Jeffery, Apr 26 2011: (Start)
Let G be the Gram matrix
  G =
  (4  1  0  1)
  (1  4  1  0)
  (0  1  4 -1)
  (1  0 -1  4)
of A028997. Then a(n) = (1/4)*Trace(G^n). (End)

Index entries for linear recurrences with constant coefficients, signature (8,-14).

Cf. A028997, A083880.

LinearRecurrence[{8,-14},{1,4},30] (* Harvey P. Dale, May 08 2013 *)

a(n) = 2^((n-2)/2)*(2*sqrt(2)-1)^n + 2^((n-2)/2)*(2*sqrt(2)+1)^n;
a(n) = Sum_{k=0..n} C(n, 2k)*5^(n-2k)2^k.
G.f.: (1-4x)/(1-8x+14x^2).
E.g.f.: exp(4x)cosh(x*sqrt(2)).
((4+sqrt(2))^n + (4-sqrt(2))^n)/2. Offset=0. a(3)=88. - Al Hakanson (hawkuu(AT)gmail.com), Oct 15 2008
a(n) = Sum_{k=0..n} A098158(n,k)*2^(3*k-n). - Philippe Deléham, Nov 30 2008

A084155 A Pell-related fourth-order recurrence.

Original entry on oeis.org

0, 1, 4, 19, 88, 401, 1804, 8051, 35760, 158401, 700564, 3095731, 13673224, 60375953, 266559388, 1176763859, 5194762080, 22931453953, 101225940772, 446836798675, 1972442421688, 8706804701201, 38433749994028

Offset: 0

Paul Barry, May 16 2003

Binomial transform of A084154.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-18,8,7)

Cf. A001333, A083878.

a:=[0,1,4,19];; for n in [5..25] do a[n]:=8*a[n-1]-18*a[n-2]+8*a[n-3]+7*a[n-4]; od; a; # Muniru A Asiru, Oct 18 2018

I:=[0,1,4,19]; [n le 4 select I[n] else 8*Self(n-1) -18*Self(n-2) +8*Self(n-3) +7*Self(n-4): n in [1..40]]; // G. C. Greubel, Oct 17 2018

seq(coeff(series(x*(1-4*x+5*x^2)/((1-2*x-x^2)*(1-6*x+7*x^2)),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 18 2018

LinearRecurrence[{8,-18,8,7},{0,1,4,19},30] (* Harvey P. Dale, Aug 16 2015 *)

m=40; v=concat([0,1,4,19], vector(m-4)); for(n=5, m, v[n] = 8*v[n-1] -18*v[n-2] +8*v[n-3] +7*v[n-4]); v \\ G. C. Greubel, Oct 17 2018

a(n) = (A083878(n) - A001333(n))/2.
a(n) = 8*a(n-1) - 18*a(n-2) + 8*a(n-3) + 7*a(n-4), a(0)=0, a(1)=1, a(2)=4, a(3)=19.
a(n) = ((3+sqrt(2))^n +(3-sqrt(2))^n -(1+sqrt(2))^n -(1-sqrt(2))^n)/4.
G.f.: x*(1-4*x+5*x^2)/((1-2*x-x^2)*(1-6*x+7*x^2)).
E.g.f.: exp(2*x)*sinh(x)*cosh(sqrt(2)*x).

A161938 a(n) = ((3+sqrt(2))(2+sqrt(2))^n + (3-sqrt(2))(2-sqrt(2))^n)/2.

Original entry on oeis.org

3, 8, 26, 88, 300, 1024, 3496, 11936, 40752, 139136, 475040, 1621888, 5537472, 18906112, 64549504, 220385792, 752444160, 2569005056, 8771131904, 29946517504, 102243806208, 349082189824, 1191841146880, 4069200207872, 13893118537728, 47434073735168

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009, Jun 29 2009

Second binomial transform of A162255.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A162255, A135532, A083878.

a:=[3,8];; for n in [3..25] do a[n]:=4*a[n-1]-2*a[n-2]; od; a; # Muniru A Asiru, Sep 28 2018

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+r)*(2+r)^n+(3-r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009

I:=[3,8]; [n le 2 select I[n] else 4*Self(n-1) - 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 28 2018

seq(coeff(series((3-4*x)/(1-4*x+2*x^2),x,n+1), x, n), n = 0..25); # Muniru A Asiru, Sep 28 2018

CoefficientList[Series[(3-4*x)/(1-4*x+2*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 28 2018 *)

my(x='x+O('x^50)); Vec((3-4*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Sep 28 2018

a(n) = 4*a(n-1) - 2*a(n-2) for n>1; a(0) = 3; a(1) = 8.
G.f.: (3-4*x)/(1-4*x+2*x^2).
From G. C. Greubel, Sep 28 2018: (Start)
a(2*n) = 2^(n-1) * (Q(2*n +1) + 2*Q(2*n)), Q(m) = Pell-Lucas numbers = A002203(m).
a(2*n+1) = 2^(n-1) * (P(2*n+2) + 2*P(2*n+1)), P(m) = Pell numbers = A000129(m). (End)

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 01 2009

A171753 Expansion of g.f. 1/(1-3x-x^2/(1-3x-x^2/(1-3*x))).

Original entry on oeis.org

1, 3, 10, 36, 137, 543, 2218, 9264, 39329, 168939, 731770, 3188364, 13948745, 61196775, 269007994, 1184076216, 5216618369, 22996827795, 101421591466, 447422614068, 1974197123657, 8712062181999, 38449506441994, 169702143024768, 749034931995041, 3306200447618043

Offset: 0

Paul Barry, Dec 17 2009

3rd binomial transform of 1,0,1,0,2,0,4,0,8,0,...

Stefano Spezia, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (9,-25,21).

Cf. A083878.

LinearRecurrence[{9,-25,21},{1,3,10},26] (* Stefano Spezia, May 11 2024 *)

G.f.: (1-6x+8x^2)/(1-9x+25x^2-21x^3) = -(4*x-1)*(2*x-1)/((3*x-1)*(7*x^2-6*x+1)).
a(n) = (3-sqrt(2))^n/4 + (3+sqrt(2))^n/4 + 3^n/2.
a(n) = (3^n+A083878(n))/2. - R. J. Mathar, Oct 08 2016
E.g.f.: exp(3*x)*cosh(x/sqrt(2))^2. - Stefano Spezia, May 11 2024

A147688 a(n) = ((6 + sqrt(8))^n + (6 - sqrt(8))^n)/2.

Original entry on oeis.org

1, 6, 44, 360, 3088, 26976, 237248, 2091648, 18456832, 162915840, 1438198784, 12696741888, 112091336704, 989587267584, 8736489783296, 77129433907200, 680931492954112, 6011553766047744, 53072563389857792, 468547255228956672

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 10 2008

Index entries for linear recurrences with constant coefficients, signature (12, -28).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-8); S:=[ ((6+r8)^n+(6-r8)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 13 2008

LinearRecurrence[{12,-28},{1,6},30]  (* Harvey P. Dale, Apr 23 2011 *)

From Philippe Deléham, Nov 13 2008: (Start)
a(n) = 12*a(n-1) - 28*a(n-2), a(0)=1, a(1)=6.
G.f.: (1-6x)/(1-12x+28x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2k)*8^(n-k))/6^n. (End)
a(n) = 2^n*A083878(n). - R. J. Mathar, Feb 04 2021

Extended beyond a(6) by Klaus Brockhaus, Nov 13 2008

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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A056236 a(n) = (2 + sqrt(2))^n + (2 - sqrt(2))^n.

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A164299 a(n) = ((1+4*sqrt(2))*(3+sqrt(2))^n + (1-4*sqrt(2))*(3-sqrt(2))^n)/2.

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A208345 Triangle of coefficients of polynomials v(n,x) jointly generated with A208344; see the Formula section.

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A083880 a(0)=1, a(1)=5, a(n) = 10*a(n-1) - 23*a(n-2), n >= 2.

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A083879 a(0)=1, a(1)=4, a(n) = 8*a(n-1) - 14*a(n-2), n >= 2.

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

A164299 a(n) = ((1+4sqrt(2))(3+sqrt(2))^n + (1-4sqrt(2))(3-sqrt(2))^n)/2.

A083880 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 23a(n-2), n >= 2.

A083879 a(0)=1, a(1)=4, a(n) = 8a(n-1) - 14a(n-2), n >= 2.

A161938 a(n) = ((3+sqrt(2))(2+sqrt(2))^n + (3-sqrt(2))(2-sqrt(2))^n)/2.

A171753 Expansion of g.f. 1/(1-3x-x^2/(1-3x-x^2/(1-3*x))).