A086348 -id:A086348 - cp's OEIS frontend

A057087 Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence.

1, 4, 20, 96, 464, 2240, 10816, 52224, 252160, 1217536, 5878784, 28385280, 137056256, 661766144, 3195289600, 15428222976, 74494050304, 359689093120, 1736732573696, 8385686667264, 40489676963840, 195501454524416

Offset: 0

Wolfdieter Lang, Aug 11 2000

a(n) gives the length of the word obtained after n steps with the substitution rule 0->1111, 1->11110, starting from 0. The number of 1's and 0's of this word is 4*a(n-1) and 4*a(n-2), respectively.
Inverse binomial transform of odd Pell bisection A001653. With a leading zero, inverse binomial transform of even Pell bisection A001542, divided by 2. - Paul Barry, May 16 2003
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 4's along the main diagonal, and 2's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 19 2011
Pisano period lengths: 1, 1, 8, 1, 3, 8, 6, 1, 24, 3, 120, 8, 21, 6, 24, 1, 16, 24, 360, 3, ... . - R. J. Mathar, Aug 10 2012
Exponential convolution of Pell numbers (A000129) and companion Pell numbers (A002203), divided by 2 and leading zero dropped. - Vladimir Reshetnikov, Oct 07 2016

Indranil Ghosh, Table of n, a(n) for n = 0..1459
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=4, q=4.
Tanya Khovanova, Recursive Sequences.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419; Eqs.(39) and (45),rhs, m=4.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,4)

Pairwise sums are in A086347.

Appears in A086346, A086347 and A086348. - Johannes W. Meijer, Aug 01 2010

I:=[1,4]; [n le 2 select I[n] else 4*Self(n-1) + 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

A057087 := n -> `if`(n=0, 1, 4^n*hypergeom([1/2-n/2, -n/2], [-n], -1)):
seq(simplify(A057087(n)), n=0..21); # Peter Luschny, Dec 17 2015

Table[Fibonacci[n + 1, 2] 2^n, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 08 2016 *)
LinearRecurrence[{4,4},{1,4},30] (* Harvey P. Dale, Aug 17 2017 *)

a(n)=if(n<0, 0, (2*I)^n*subst(I*poltchebi(n+1)+poltchebi(n),'x,-I)/2) /* Michael Somos, Sep 16 2005 */

Vec(1/(1-4*x-4*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015

[lucas_number1(n,4,-4) for n in range(1, 23)] # Zerinvary Lajos, Apr 23 2009

a(n) = 4*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
G.f.: 1/(1 - 4*x - 4*x^2).
a(n) = S(n, 2*i)*(-2*i)^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
a(n) = Sum_{k=0..n} 3^k*A063967(n,k). - Philippe Deléham, Nov 03 2006
a(n) = A000129(n+1)*A000079(n). - R. J. Mathar, Jul 08 2009
From Johannes W. Meijer, Aug 01 2010: (Start)
Limit_{k->oo} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2);
Limit_{n->oo} A084128(n)/A057087(n-1) = sqrt(2). (End)
a(n) = 4^n*hypergeom([1/2-n/2, -n/2], [-n], -1) for n>=1. - Peter Luschny, Dec 17 2015

A086347 On a 3 X 3 board, number of n-move routes of chess king ending in a given side square.

Original entry on oeis.org

1, 5, 24, 116, 560, 2704, 13056, 63040, 304384, 1469696, 7096320, 34264064, 165441536, 798822400, 3857055744, 18623512576, 89922273280, 434183143424, 2096421666816, 10122419240960, 48875363631104, 235991131488256, 1139465980477440, 5501828447862784

Offset: 0

Zak Seidov, Jul 17 2003

Number of aa-avoiding words of length n on alphabet {a,b,c,d,e}. - Tanya Khovanova, Jan 11 2007
Binomial transform of A164589 and second binomial transform of A096886. [Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009]
From Johannes W. Meijer, Aug 01 2010: (Start)
The a(n) represent the number of n-move paths of a chess king on a 3 X 3 board that end or start in a given side square m (m = 2, 4, 6, 8).
Inverse binomial transform of A001109 (without the leading 0).
(End)
Number of independent vertex subsets of the graph obtained by attaching two pendant edges to each vertex of the path graph P_n (see A235116). Example: a(1)=5; indeed, P_1 is the one-vertex graph and after attaching two pendant vertices we obtain the path graph ABC; the independent vertex subsets are: empty, {A}, {B}, {C}, and {A,C}.
Number of simple paths from corner to diagonally opposite corner on a 2 X n grid with king moves allowed. - Andrew Howroyd, Nov 06 2019
Number of 4-compositions of n+1 restricted to parts 1 and 2 (and allowed zeros); see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020

			a(3) = 116 = 5^3 - 9 (aaa, aab, aac, aad, aae, baa, caa, daa, eaa). [Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009]

Indranil Ghosh, Table of n, a(n) for n = 0..1459
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
Joerg Arndt, Matters Computational (The Fxtbook)
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 7.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
Mike Oakes, KingMovesForPrimes.
Zak Seidov, KingMovesForPrimes.
Zak Seidov et al., New puzzle? King moves for primes, digest of 28 messages in primenumbers group, Jul 13 - Jul 23, 2003. [Cached copy]
Sleephound, KingMovesForPrimes.
Index entries for linear recurrences with constant coefficients, signature (4,4).

Row 2 of A329118.
Row sums of A235113.
Cf. A086346, A086348.
Cf. A028859.
Cf. A126473. - Johannes W. Meijer, Aug 01 2010

with(LinearAlgebra): nmax:=19; m:=2; A[5]:= [1,1,1,1,0,1,1,1,1]: A:=Matrix([[0,1,0,1,1,0,0,0,0],[1,0,1,1,1,1,0,0,0],[0,1,0,0,1,1,0,0,0],[1,1,0,0,1,0,1,1,0],A[5],[0,1,1,0,1,0,0,1,1],[0,0,0,1,1,0,0,1,0],[0,0,0,1,1,1,1,0,1],[0,0,0,0,1,1,0,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Aug 01 2010
# second Maple program:
a:= n-> (<<0|1>, <4|4>>^n. <<1, 5>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2022

Table[(Sqrt[2]/32)((2+Sqrt[8])^(n+2)-(2-Sqrt[8])^(n+2)), {n, 0, 19}]

a(n) = (sqrt(2)/32)*((2+sqrt(8))^(n+2)-(2-sqrt(8))^(n+2)).
From Ralf Stephan, Feb 01 2004: (Start)
G.f.: (1+x)/(1-4*x-4*x^2).
a(n) = A057087(n) + A057087(n-1). (End)
a(n) = 4*a(n-1) + 4*a(n-2). - Tanya Khovanova, Jan 11 2007
Limit_{k->oo} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2). - Johannes W. Meijer, Aug 01 2010
E.g.f.: exp(2*x)*(4*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x))/4. - Stefano Spezia, Mar 17 2025

Offset changed and edited by Johannes W. Meijer, Jul 15 2010

A090390 Repeatedly multiply (1,0,0) by ([1,2,2],[2,1,2],[2,2,3]); sequence gives leading entry.

Original entry on oeis.org

1, 1, 9, 49, 289, 1681, 9801, 57121, 332929, 1940449, 11309769, 65918161, 384199201, 2239277041, 13051463049, 76069501249, 443365544449, 2584123765441, 15061377048201, 87784138523761, 511643454094369, 2982076586042449, 17380816062160329, 101302819786919521, 590436102659356801

Offset: 0

Vim Wenders, Jan 30 2004

The values of a and b in (a,b,c)*A give all (positive integer) solutions to Pell equation a^2 - 2*b^2 = -1; the values of c are A000129(2n)
Binomial transform of A086348. - Johannes W. Meijer, Aug 01 2010
All values of a(n) are squares.  sqrt(a(n+1)) = A001333(n). The ratio a(n+1)/a(n) converges to 3 + 2*sqrt(2). - Richard R. Forberg, Aug 14 2013

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 60 at p. 123.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Cf. A000129, A001333, A001541, A079291, A086348, A095344, A123270, A302946.

a090390 n = a090390_list !! n
a090390_list = 1 : 1 : 9 : zipWith (-) (map (* 5) $
   tail $ zipWith (+) (tail a090390_list) a090390_list) a090390_list
-- Reinhard Zumkeller, Aug 17 2013

[Evaluate(DicksonFirst(n,-1),2)^2/4: n in [0..40]]; // G. C. Greubel, Aug 21 2022

a:= n-> (<<1|0|0>>. <<1|2|2>, <2|1|2>, <2|2|3>>^n)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 17 2013

CoefficientList[Series[(1-4x-x^2)/((1+x)(1-6x+x^2)),{x, 0, 30}], x] (* Harvey P. Dale, May 20 2012 *)
LinearRecurrence[{5,5,-1}, {1,1,9}, 30] (* Harvey P. Dale, May 20 2012 *)
Table[(ChebyshevT[n,3]+(-1)^n)/2, {n,0,30}] (* Eric W. Weisstein, Apr 17 2018 *)
(LucasL[Range[0, 40], 2]/2)^2 (* G. C. Greubel, Aug 21 2022 *)

a(n)=polcoeff((1-4*x-x^2)/((1+x)*(1-6*x+x^2))+x*O(x^n),n)

a(n)=if(n<0,0,([1,2,2;2,1,2;2,2,3]^n)[1,1])

Vec( (1-4*x-x^2)/((1+x)*(1-6*x+x^2)) + O(x^66) ) \\ Joerg Arndt, Aug 16 2013

use Math::Matrix; use Math::BigInt; $a = new Math::Matrix ([ 1, 2, 2], [ 2, 1, 2], [ 2, 2, 3]); $p = new Math::Matrix ([1, 0, 0]); $p->print(); for ($i=1; $i<20;$i++) { $p = $p->multiply($a); $p->print(); }

[lucas_number2(n,2,-1)^2/4 for n in (0..40)] # G. C. Greubel, Aug 21 2022

G.f.: (1-4*x-x^2)/((1+x)*(1-6*x+x^2)).
a(n) = A001333(n)^2
(a, b, c) = (1, 0, 0). Recursively multiply (a, b, c)*( [1, 2, 2], [2, 1, 2], [2, 2, 3] ).
M^n * [ 1 1 1] = [a(n+1) q a(n)], where M = the 3 X 3 matrix [4 4 1 / 2 1 0 / 1 0 0]. E.g. M^5 * [1 1 1] = [9801 4059 1681] where 9801 = a(6), 1681 = a(5). Similarly, M^n * [1 0 0] generates A079291 (Pell number squares). - Gary W. Adamson, Oct 31 2004
a(n) = (((1+sqrt(2))^(2*n) + (1-sqrt(2))^(2*n)) + 2*(-1)^n)/4 - Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 09 2005
a(n) = (A001541(n) + (-1)^n)/2. - R. J. Mathar, Nov 20 2009
a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3), with a(0)=1, a(1)=1, a(2)=9. - Harvey P. Dale, May 20 2012
(a(n)) = tesseq(- .5'j + .5'k - .5j' + .5k' - 2'ii' + 'jj' - 'kk' + .5'ij' + .5'ik' + .5'ji' + 'jk' + .5'ki' + 'kj' + e), apart from initial term. - Creighton Dement, Nov 16 2004
a(n) = A302946(n)/4. - Eric W. Weisstein, Apr 17 2018
E.g.f.: exp(-x)*(1 + exp(4*x)*cosh(2*sqrt(2)*x))/2. - Stefano Spezia, Aug 03 2024

A084128 a(n) = 4a(n-1) + 4a(n-2), a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 12, 56, 272, 1312, 6336, 30592, 147712, 713216, 3443712, 16627712, 80285696, 387653632, 1871757312, 9037643776, 43637604352, 210700992512, 1017354387456, 4912221519872, 23718303629312, 114522100596736, 552961616904192, 2669934870003712

Offset: 0

Paul Barry, May 16 2003

Original name was: Generalized Fibonacci sequence.

Binomial transform of A084058.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,4).

Cf. A001333 A001541, A002203, A057087, A084058, A084128, A201730.

Appears in A086346, A086347 and A086348. - Johannes W. Meijer, Aug 01 2010

[2^(n-1)*Evaluate(DicksonFirst(n,-1), 2): n in [0..40]]; // G. C. Greubel, Oct 13 2022

a:=proc(n) option remember; if n=0 then 1 elif n=1 then 2 else
4*a(n-1)+4*a(n-2); fi; end: seq(a(n), n=0..40); # Wesley Ivan Hurt, Jan 31 2017
a := n -> (2*I)^n*ChebyshevT(n, -I):
seq(simplify(a(n)), n = 0..23); # Peter Luschny, Dec 03 2023

CoefficientList[Series[(2 z - 1)/(4 z^2 + 4 z - 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
Table[2^(n-1) LucasL[n, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 07 2016 *)
LinearRecurrence[{4,4},{1,2},30] (* Harvey P. Dale, Mar 01 2018 *)

a(n)=if(n<0,0,polsym(4+4*x-x^2,n)[n+1]/2)

[lucas_number2(n,4,-4)/2 for n in range(0, 23)] # Zerinvary Lajos, May 14 2009

a(n) = 2^n * A001333(n).
G.f.: (1-2*x)/(1-4*x-4*x^2).
a(n) = 4*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
a(n) = (2 + 2*sqrt(2))^n/2 + (2 - 2*sqrt(2))^n/2.
E.g.f.: exp(2*x)*cosh(2*x*sqrt(2)).
From Johannes W. Meijer, Aug 01 2010: (Start)
Lim_{k->infinity} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2).
Lim_{n->infinity} A084128(n)/A057087(n-1) = sqrt(2). (End)
a(n) = Sum_{k=0..n} A201730(n,k)*7^k. - Philippe Deléham, Dec 06 2011
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(4*k-2)/(x*(4*k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
a(n) = 2^(n-1)*A002203(n). - Vladimir Reshetnikov, Oct 07 2016

A179597 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 5x + 2x^2)/(1 - 2x - 11x^2 - 6*x^3).

Original entry on oeis.org

1, 7, 27, 137, 613, 2895, 13355, 62233, 288741, 1342175, 6233899, 28964169, 134554277, 625117807, 2904117675, 13491856889, 62679715045, 291194561919, 1352817130667, 6284852732713, 29197861274277, 135646005392399

Offset: 0

Johannes W. Meijer, Jul 28 2010, Aug 10 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
For the central square the 512 red kings lead to 47 different red king sequences, see the cross-references for some examples.
The sequence above corresponds to four A[5] vectors with decimal [binary] values 367 [1,0,1,1,0,1,1,1,1], 463 [1,1,1,0,0,1,1,1,1], 487 [1,1,1,1,0,0,1,1,1] and 493 [1,1,1,1,0,1,1,0,1]. These vectors lead for the corner squares to A179596 and for the side squares to A126473.
This sequence belongs to a family of sequences with g.f. (1 + (k+2)*x + (2*k-4)*x^2)/(1 - 2*x - (k+8)*x^2 - (2*k)*x^3). Red king sequences that are members of this family are A179607 (k=0), A179605 (k=1), A179601 (k=2), A179597 (k=3; this sequence) and A086348 (k=4). Another member of this family is A179609 (k = -4).

Index entries for linear recurrences with constant coefficients, signature (2,11,6).

Red king sequences central square [numerical value A[5]]: A086348 [495], A179599 [239], A179597 [367], A179601 [335], A179603 [95], A154964 [31], A179605 [327], A179606 [27], A179611 [15], A179607 [325], A015521 [11], A007483 [2], A000012 [16], A000007 [0].

with(LinearAlgebra): nmax:=21; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:=[1,1,0,0,1,0,1,1,0]: A[5]:= [1,0,1,1,0,1,1,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5], A[6],A[7],A[8],A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{2,11,6},{1,7,27},30] (* Harvey P. Dale, Mar 01 2015 *)

G.f.: (1 + 5*x + 2*x^2)/(1 - 2*x - 11*x^2 - 6*x^3).
a(n) = 2*a(n-1) + 11*a(n-2) + 6*a(n-3) with a(0) = 1, a(1) = 7 and a(2) = 27.
a(n) = 8*(-1/2)^(-n+1)/9 + ((7+11*sqrt(7))*A^(-n-1) + (7-11*sqrt(7))*B^(-n-1))/126 with A = (-2+sqrt(7))/3 and B = (-2-sqrt(7))/3.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*(A000244(n)/(A015530(n)*sqrt(7) - A108851(n))).

A086346 On a 3 X 3 board, the number of n-move paths for a chess king ending in a given corner square.

Original entry on oeis.org

1, 3, 18, 80, 400, 1904, 9248, 44544, 215296, 1039104, 5018112, 24227840, 116985856, 564850688, 2727354368, 13168803840, 63584665600, 307013812224, 1482394042368, 7157631156224, 34560101318656, 166870928850944, 805724122775552, 3890380202311680, 18784417308737536, 90699190027419648

Offset: 0

Zak Seidov, Jul 17 2003

From Johannes W. Meijer, Aug 01 2010: (Start)
The a(n) represent the number of n-move paths of a chess king on a 3 X 3 board that end or start in a given corner square m (m = 1, 3, 7, 9). To determine the a(n) we can either sum the components of the column vector A^n[k,m], with A the adjacency matrix of the king's graph, or we can sum the components of the row vector A^n[m,k], see the Maple program.
Inverse binomial transform of A079291 (without the leading 0).
(End)
From R. J. Mathar, Oct 12 2010: (Start)
The row n=3 of an array counting king walks on an n X n board with k steps, starting from a corner:
  1, 3,  9,  27,  81,  243,   729,   2187,    6561,    19683,    59049, ...;
  1, 3, 18,  80, 400, 1904,  9248,  44544,  215296,  1039104,  5018112, ...;
  1, 3, 18, 105, 615, 3600, 21075, 123375,  722250,  4228125, 24751875, ...;
  1, 3, 18, 105, 684, 4359, 28278, 182349, 1179792,  7622667, 49283802, ...;
  1, 3, 18, 105, 684, 4550, 30807, 209867, 1434279,  9815190, 67209723, ...;
  1, 3, 18, 105, 684, 4550, 31340, 218056, 1533712, 10829360, 76720288, ...;
  1, 3, 18, 105, 684, 4550, 31340, 219555, 1559835, 11177190, 80573373, ...;
  1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11259785, 81765550, ...;
  1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82025163, ...;
  1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82059768, ...;
  1, 3, 18, 105, 684, 4550, 31340, 219555, 1564080, 11271876, 82059768, ...;
The partial sums along the rows are documented in A123109 (king walks with between 1 and k steps). (End)

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984. [From Johannes W. Meijer, Aug 01 2010]

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mike Oakes, KingMovesForPrimes.
Zak Seidov et al., New puzzle? King moves for primes, digest of 28 messages in primenumbers group, Jul 13 - Jul 23, 2003. [Cached copy]
Zak Seidov, KingMovesForPrimes.
Sleephound, KingMovesForPrimes.
Index entries for linear recurrences with constant coefficients, signature (2,12,8).

Cf. A086347, A086348, A086349.
Cf. A179596. - Johannes W. Meijer, Aug 01 2010
Cf. A002203, A057087, A079291, A084128, A110048, A123109.

[2^(n-3)*(Evaluate(DicksonFirst(n+2,-1), 2) +2*(-1)^n): n in [0..30]]; // G. C. Greubel, Aug 18 2022

with(LinearAlgebra):
nmax:=19; m:=1;
A[5]:= [1, 1, 1, 1, 0, 1, 1, 1, 1]:
A:=Matrix([[0, 1, 0, 1, 1, 0, 0, 0, 0], [1, 0, 1, 1, 1, 1, 0, 0, 0], [0, 1, 0, 0, 1, 1, 0, 0, 0], [1, 1, 0, 0, 1, 0, 1, 1, 0], A[5], [0, 1, 1, 0, 1, 0, 0, 1, 1], [0, 0, 0, 1, 1, 0, 0, 1, 0], [0, 0, 0, 1, 1, 1, 1, 0, 1], [0, 0, 0, 0, 1, 1, 0, 1, 0]]):
for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m, k], k=1..9): od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Aug 01 2010

Table[(1/32)(2(-2)^(n+2)+(2+Sqrt[8])^(n+2)+(2-Sqrt[8])^(n+2)), {n, 0, 19}] // FullSimplify
LinearRecurrence[{2,12,8}, {1,3,18}, 31] (* G. C. Greubel, Aug 18 2022 *)

Vec((1+x)/((1+2*x)*(1-4*x-4*x^2))+O(x^30)) \\ Joerg Arndt, Jan 29 2024

[2^(n-3)*(lucas_number2(n+2,2,-1) +2*(-1)^n) for n in (0..30)] # G. C. Greubel, Aug 18 2022

a(n) = (1/32)*(2*(-2)^(n+2) + (2+sqrt(8))^(n+2) + (2-sqrt(8))^(n+2)).
From R. J. Mathar, Jul 22 2010: (Start)
a(n) = 2*a(n-1) + 12*a(n-2) + 8*a(n-3).
G.f.: (1+x) / ( (1+2*x)*(1-4*x-4*x^2) ).
a(n) = (2*A057087(n-1) + 3*A057087(n) + (-2)^n)/4. (End)
Limit_{k->oo} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2). - Johannes W. Meijer, Aug 01 2010
a(n) = A110048(n) + A110048(n-1). - R. J. Mathar, Mar 08 2021
a(n) = 2^(n-3)*(A002203(n+2) + 2*(-1)^n). - G. C. Greubel, Aug 18 2022

Offset changed and edited by Johannes W. Meijer, Jul 15 2010

A110048 Expansion of 1/((1+2x)(1-4x-4x^2)).

Original entry on oeis.org

1, 2, 16, 64, 336, 1568, 7680, 36864, 178432, 860672, 4157440, 20070400, 96915456, 467935232, 2259419136, 10909384704, 52675280896, 254338531328, 1228055511040, 5929575645184, 28630525673472, 138240403177472

Offset: 0

Creighton Dement, Jul 10 2005

Floretion Algebra Multiplication Program, FAMP Code:
-kbasejseq[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'
See also comment for A110047.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (2,12,8).

Cf. A079291, A084159, A086346, A086348, A110046, A110047, A110049, A110050.

[2^(n-2)*(Evaluate(DicksonFirst(n+1,-1), 2) +2*(-1)^n): n in [0..40]]; // G. C. Greubel, Aug 18 2022

seriestolist(series(1/((1+2*x)*(1-4*x-4*x^2)), x=0,40));

CoefficientList[Series[1/((1+2x)(1-4x-4x^2)), {x,0,40}], x] (* or *) LinearRecurrence[{2,12,8}, {1,2,16}, 41] (* Harvey P. Dale, Nov 02 2011 *)

[2^(n-2)*(lucas_number2(n+1,2,-1) +2*(-1)^n) for n in (0..40)] # G. C. Greubel, Aug 18 2022

Superseeker finds: a(n+1) = 2*A086348(n+1) (A086348's offset is 1: On a 3 X 3 board, number of n-move routes of chess king ending at central cell); binomial transform matches A084159 (Pell oblongs); j-th coefficient of g.f.*(1+x)^j matches A079291 (Squares of Pell numbers); a(n) + a(n+1) = A086346(n+2) (A086346's offset is 1: On a 3 X 3 board, the number of n-move paths for a chess king ending in a given corner cell.)
From Maksym Voznyy (voznyy(AT)mail.ru), Jul 24 2008: (Start)
a(n) = 2*a(n-1) + 12*a(n-2) + 8*a(n-3), where a(1)=1, a(2)=2, a(3)=16.
a(n) = 2^(n-3)*( 4*(-1)^(1-n) + (sqrt(2)-1)^(-n) + (-sqrt(2)-1)^(-n)) . (End)
a(n) = 2^n*A097076(n+1). - R. J. Mathar, Mar 08 2021

A086349 On a 3 X 3 board, the number of n-move paths for a chess king.

Original entry on oeis.org

1, 9, 40, 200, 952, 4624, 22272, 107648, 519552, 2509056, 12113920, 58492928, 282425344, 1363677184, 6584401920, 31792332800, 153506906112, 741197021184, 3578815578112, 17280050659328, 83435464425472

Offset: 0

Zak Seidov, Jul 17 2003

			a(1)=9 because there are 9 cells in the 3 X 3 board;
a(2)=40 because from each of 4 corner cells, king can move to 3 cells, this gives 4*3=12 moves, from each of 4 side cells, king can move to 5 cells, this gives 4*5=20 moves and from the central cell, king can move to 8 cells, this gives 8 moves and the total is 12+20+8=40.

Mike Oakes, KingMovesForPrimes.
Zak Seidov, KingMovesForPrimes.
Zak Seidov et al., New puzzle? King moves for primes, digest of 28 messages in primenumbers group, Jul 13 - Jul 23, 2003. [Cached copy]
Sleephound, KingMovesForPrimes.

Cf. A086346, A086347, A086348.

a(n) = 4*A086346(n) + 4*A086347(n) + A086349(n).

a(n) = 2*a(n-1)+12*a(n-2)+8*a(n-3) for n>3. G.f.: (1+x)*(1+6*x+4*x^2)/((1+2*x)*(1-4*x-4*x^2)). - Colin Barker, Apr 12 2012

Edited by Johannes W. Meijer, Jul 15 2010

cp's OEIS Frontend

A057087 Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence.

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A086347 On a 3 X 3 board, number of n-move routes of chess king ending in a given side square.

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A090390 Repeatedly multiply (1,0,0) by ([1,2,2],[2,1,2],[2,2,3]); sequence gives leading entry.

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A084128 a(n) = 4*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.

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A179597 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 5*x + 2*x^2)/(1 - 2*x - 11*x^2 - 6*x^3).

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A086346 On a 3 X 3 board, the number of n-move paths for a chess king ending in a given corner square.

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A084128 a(n) = 4a(n-1) + 4a(n-2), a(0)=1, a(1)=2.

A179597 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 5x + 2x^2)/(1 - 2x - 11x^2 - 6*x^3).

A110048 Expansion of 1/((1+2x)(1-4x-4x^2)).