A054486 -id:A054486 - cp's OEIS frontend

A152187 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=5.

1, 5, 20, 85, 355, 1490, 6245, 26185, 109780, 460265, 1929695, 8090410, 33919705, 142211165, 596232020, 2499751885, 10480415755, 43940006690, 184222098845, 772366329985, 3238209484180, 13576460102465, 56920427728295

Offset: 0

Philippe Deléham, Nov 28 2008

Unsigned version of A152185.
From Johannes W. Meijer, Aug 01 2010: (Start)
The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 and 8) 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.
The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the corner squares to A015523 and for the central square to A179606.
This sequence belongs to a family of sequences with g.f. (1+2*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k=2), A108981 (k=4), A152187 (k=5; this sequence), A154964 (k=6), A179602 (k=7) and A179598 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A036563 (k=-2), A054486 (k=-1), A084244 (k=0), A108300 (k=1) and A000351 (k=10).
Inverse binomial transform of A015449 (without the first leading 1).
(End)

Index entries for linear recurrences with constant coefficients, signature (3, 5).

Cf. A015523, A072263, A072264, A179606, A197189.

LinearRecurrence[{3,5},{1,5},40] (* Harvey P. Dale, May 03 2013 *)

G.f.: (1+2*x)/(1 - 3*x - 5*x^2).
Lim_{k->infinity} a(n+k)/a(k) = (A072263(n) + A015523(n)*sqrt(29))/2. - Johannes W. Meijer, Aug 01 2010
G.f.: G(0)*(1+2*x)/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

A228208 y-values in the solution to x^2 - 20*y^2 = 176.

Original entry on oeis.org

1, 2, 5, 7, 14, 19, 37, 50, 97, 131, 254, 343, 665, 898, 1741, 2351, 4558, 6155, 11933, 16114, 31241, 42187, 81790, 110447, 214129, 289154, 560597, 757015, 1467662, 1981891, 3842389, 5188658, 10059505, 13584083, 26336126, 35563591, 68948873, 93106690

Offset: 1

Colin Barker, Aug 16 2013

Also y-values in the solution of x^2-5*y^2=44.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A228207, A054486 (bisection).

I:=[1,2,5,7,14]; [n le 4 select I[n] else 3*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Aug 17 2013

CoefficientList[Series[(x + 1) (x^2 + x + 1) / ((x^2 - x - 1) (x^2 + x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 17 2013 *)

Vec(x*(x+1)*(x^2+x+1)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

G.f.: x*(x+1)*(x^2+x+1) / ((x^2-x-1)*(x^2+x-1)).
a(n) = 3*a(n-2)-a(n-4).
Let h(n) = hypergeom([(1 - n)/2, (n + 1) mod 2 - n/2], [1 - n], -4) then a(n) = h(n-1) + h(n) for n > 2. - Peter Luschny, Sep 03 2019

A054492 a(n) = 3*a(n-1) - a(n-2), a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 17, 45, 118, 309, 809, 2118, 5545, 14517, 38006, 99501, 260497, 681990, 1785473, 4674429, 12237814, 32039013, 83879225, 219598662, 574916761, 1505151621, 3940538102, 10316462685, 27008849953, 70710087174, 185121411569, 484654147533, 1268841031030

Offset: 0

Barry E. Williams, May 06 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A002878, A054486.

I:=[1,6]; [n le 2 select I[n] else 3*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 20 2015

CoefficientList[Series[(1 + 3 x) / (1 - 3 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)
LinearRecurrence[{3, -1}, {1, 6}, 100] (* G. C. Greubel, Mar 26 2016 *)

Vec((1+3*x)/(1-3*x+x^2) + O(x^30)) \\ Michel Marcus, Mar 20 2015

a(n) = (6*(((3+sqrt(5))/2)^n - ((3-sqrt(5))/2)^n) - (((3+sqrt(5))/2)^(n-1) - ((3-sqrt(5))/2)^(n-1)))/sqrt(5).
a(n) = 2*Lucas(2*n+1) - Fibonacci(2*n+1).
G.f.: (1+3*x)/(1-3*x+x^2). - Philippe Deléham, Nov 03 2008
a(n) = 5*Fibonacci(2*n) + Fibonacci(2*n-1). - Ehren Metcalfe, Mar 26 2016
E.g.f.: (1/10) * exp((3-sqrt(5))*x/2) * ((5-9*sqrt(5)) + (5+9*sqrt(5)) * exp(sqrt(5)*x) ). - G. C. Greubel, Mar 26 2016

More terms from Vincenzo Librandi, Mar 20 2015

Typo in name fixed by Karl V. Keller, Jr., Jun 23 2015

A055267 a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=7.

Original entry on oeis.org

1, 7, 20, 53, 139, 364, 953, 2495, 6532, 17101, 44771, 117212, 306865, 803383, 2103284, 5506469, 14416123, 37741900, 98809577, 258686831, 677250916, 1773065917, 4641946835, 12152774588, 31816376929, 83296356199, 218072691668, 570921718805, 1494692464747

Offset: 0

Barry E. Williams, May 09 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000032, A000045, A054486, A054492.

List([0..30], n-> Fibonacci(2*n+2) +4*Fibonacci(2*n) ); # G. C. Greubel, Jan 17 2020

[5*Fibonacci(2*n) + Fibonacci(2*n+1): n in [0..30]]; // Vincenzo Librandi, Dec 25 2018

with(combinat); seq(fibonacci(2*n+2) +4*fibonacci(2*n), n=0..30); # G. C. Greubel, Jan 17 2020

Table[5*Fibonacci[2n] + Fibonacci[2n+1],  {n, 0, 30}]
Table[4*Fibonacci[2n-1] + 3*LucasL[2n-1], {n, 0, 30}] (* Rigoberto Florez, Dec 24 2018 *)
LinearRecurrence[{3,-1}, {1,7}, 30] (* Vincenzo Librandi, Dec 25 2018 *)
nxt[{a_,b_}]:={b,3b-a}; NestList[nxt,{1,7},30][[;;,1]] (* Harvey P. Dale, Mar 23 2025 *)

Vec((1+4*x)/(1-3*x+x^2) + O(x^40)) \\ Michel Marcus, Sep 06 2017

[fibonacci(2*n+2) +4*fibonacci(2*n) for n in (0..30)] # G. C. Greubel, Jan 17 2020

a(n) = (7*(((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n) - (((3 + sqrt(5))/2)^(n - 1) - ((3 - sqrt(5))/2)^(n - 1)))/sqrt(5).
G.f.: (1 + 4*x)/(1 - 3*x + x^2).
From Rigoberto Florez, Dec 24 2018: (Start)
a(n) = 5*Fibonacci(2*n) + Fibonacci(2*n+1).
a(n) = 4*Fibonacci(2*n - 1) + 3*Lucas(2*n - 1). (End)
E.g.f.: exp(3*t/2)*( cosh(sqrt(5)*t/2) + (11/sqrt(5))*sinh(sqrt(5)*t/2) ). - G. C. Greubel, Jan 17 2020
a(n) = 4*A001906(n) + A001906(n+1). - R. J. Mathar, Mar 06 2022

A129905 Expansion of g.f.: (1-x)(1+2x)/((1+x)(1-3x+x^2)).

Original entry on oeis.org

1, 3, 6, 17, 43, 114, 297, 779, 2038, 5337, 13971, 36578, 95761, 250707, 656358, 1718369, 4498747, 11777874, 30834873, 80726747, 211345366, 553309353, 1448582691, 3792438722, 9928733473, 25993761699, 68052551622, 178163893169

Offset: 0

Creighton Dement, Jun 04 2007

Floretion Algebra Multiplication Program, FAMP Code: tesseq[A*B] with A = + .5'i + .5'j + .5'k + 'ji' + .5e ; B = + .5i' + .5j' + .5k' + 'ij' + .5e (apart from initial term)
From Andrew Rupinski, Jan 31 2011: (Start)
Form the infinite recursive array R(i,j) as follows: R(1,j) = F(j), R(2,j) = L(j) and for i > 2, R(i,j) = R(i-1,j) + R(i-2,j) where F(j) is the j-th Fibonacci number and L(j) is the j-th Lucas number. Then for i > 0, R(i,i) = a(i-1):
   1   1   2   3    5    8   13  ...
   1   3   4   7   11   18   29  ...
   2   4   6  10   16   26   42  ...
   3   7  10  17   27   44   71  ...
   5  11  16  27   43   70  113  ...
   8  18  26  44   70  114  184  ...
  13  29  42  71  113  184  297  ...
  ...
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Cf. A001906, A054486.

List([0..30], n -> (Fibonacci(n-2)^2 + Fibonacci(n+2)^2 + Fibonacci(2*n))/2); # G. C. Greubel, Jan 07 2019

[(Fibonacci(n-2)^2 + Fibonacci(n+2)^2 + Fibonacci(2*n))/2: n in [0..30]]; // G. C. Greubel, Jan 07 2019

CoefficientList[ Series[(1+x-2x^2)/(1-2x-2x^2+x^3), {x, 0, 27}], x] (* Or *)
t[1, k_] := Fibonacci@ k; t[2, k_] := LucasL@ k; t[n_, k_] := t[n, k] = t[n - 1, k] + t[n - 2, k]; Table[ t[n, n], {n, 28}] (* Robert G. Wilson v *)

vector(30, n, n--; (fibonacci(n-2)^2 + fibonacci(n+2)^2 + fibonacci(2*n))/2) \\ G. C. Greubel, Jan 07 2019

[(fibonacci(n-2)^2 + fibonacci(n+2)^2 + fibonacci(2*n))/2 for n in (0..30)] # G. C. Greubel, Jan 07 2019

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n+2) - a(n) = A054486(n+1).
a(n) = ( (4-sqrt(5))*((1+sqrt(5))/2)^(2*n) + (4 + sqrt(5))*((1-sqrt(5))/2 )^(2*n) + 2*(-1)^n)/5.
a(n) = -2*(-1)^n/5-8*A001906(n)/5+7*A001906(n+1)/5. - R. J. Mathar, Nov 10 2009
a(n) = (Fibonacci(n-2)^2 + Fibonacci(n+2)^2 + Fibonacci(2*n))/2. - Gary Detlefs Dec 20 2010

A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)T(n-1,k) + 1]/T(n-2,k-1) for 0=0 and T(n,0) = Fibonacci(2n-1) for n>=1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 13, 13, 8, 4, 1, 34, 34, 21, 11, 5, 1, 89, 89, 55, 29, 14, 6, 1, 233, 233, 144, 76, 37, 17, 7, 1, 610, 610, 377, 199, 97, 45, 20, 8, 1, 1597, 1597, 987, 521, 254, 118, 53, 23, 9, 1, 4181, 4181, 2584, 1364, 665, 309, 139, 61, 26, 10, 1

Offset: 0

Paul D. Hanna, Jul 04 2008

			Generating rule.
Given nonzero elements W, X, Y, Z, relatively arranged like so:
.. W .....
.. X Y ...
.... Z ...
then Z = (X*Y + 1)/W.
Triangle begins:
1;
1, 1;
2, 2, 1;
5, 5, 3, 1;
13, 13, 8, 4, 1;
34, 34, 21, 11, 5, 1;
89, 89, 55, 29, 14, 6, 1;
233, 233, 144, 76, 37, 17, 7, 1;
610, 610, 377, 199, 97, 45, 20, 8, 1;
1597, 1597, 987, 521, 254, 118, 53, 23, 9, 1;
4181, 4181, 2584, 1364, 665, 309, 139, 61, 26, 10, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080, as a flattened table of rows 0..45

Cf. row sums: A141752; columns: A001519, A001906, A002878, A054486, A054492, A055267, A055273, A055849.

PARI
```
T(n,k)=if(n
				
```

T(n,k)=fibonacci(2*(n-k))*k+fibonacci(2*(n-k)-1)
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

T(n,k) = Fibonacci(2*(n-k)-1) + k*Fibonacci(2*(n-k)) for 0<=k<=n.

A109165 a(n) = 5a(n-2) - 2a(n-4), n >= 4.

Original entry on oeis.org

1, 2, 5, 10, 23, 46, 105, 210, 479, 958, 2185, 4370, 9967, 19934, 45465, 90930, 207391, 414782, 946025, 1892050, 4315343, 8630686, 19684665, 39369330, 89792639, 179585278, 409593865, 819187730, 1868384047, 3736768094, 8522732505

Offset: 0

Creighton Dement, Aug 18 2005

Floretion Algebra Multiplication Program, FAMP Code: 4kbaseksigcycsumseq[ - .25'i - .25i' + .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' + .25e], sumtype: (Y[15], *, vesy)

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-2).

Cf. A107839, A106709, A005824, A054486.

a(2n) = A107839(n), a(2n+1) = A106709(n+1), a(n) - a(n-1) = A005824(n+2).

G.f.: (2*x+1)/(1-5*x^2+2*x^4).

A109164 a(n) = 4a(n-1) - 4a(n-2) + a(n-3), n >= 3; a(0)=1, a(1)=6, a(2)=20.

Original entry on oeis.org

1, 6, 20, 57, 154, 408, 1073, 2814, 7372, 19305, 50546, 132336, 346465, 907062, 2374724, 6217113, 16276618, 42612744, 111561617, 292072110, 764654716, 2001892041, 5241021410, 13721172192, 35922495169, 94046313318, 246216444788

Offset: 0

Creighton Dement, Aug 18 2005

Floretion Algebra Multiplication Program, FAMP Code: 4tessumseq[ + .25'i + .25i' + .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' + .25e], sumtype: (Y[15], *, vesy)

Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A054486.

Join[{a=1,b=6},Table[c=3*b-1*a+3;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2011 *)
LinearRecurrence[{4,-4,1},{1,6,20},30] (* Harvey P. Dale, Apr 14 2016 *)

a(n) - a(n-1) = A054486(n+1).
G.f.: (2*x+1)/((1-x)*(x^2-3*x+1)).
a(n) = A027941(n+1) +2*A027941(n). - R. J. Mathar, Sep 11 2019

cp's OEIS Frontend

A152187 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=5.

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A228208 y-values in the solution to x^2 - 20*y^2 = 176.

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A054492 a(n) = 3*a(n-1) - a(n-2), a(0)=1, a(1)=6.

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A055267 a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=7.

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A129905 Expansion of g.f.: (1-x)*(1+2*x)/((1+x)*(1-3*x+x^2)).

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A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)*T(n-1,k) + 1]/T(n-2,k-1) for 0=0 and T(n,0) = Fibonacci(2*n-1) for n>=1.

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A109165 a(n) = 5*a(n-2) - 2*a(n-4), n >= 4.

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A152187 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=5.

A129905 Expansion of g.f.: (1-x)(1+2x)/((1+x)(1-3x+x^2)).

A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)T(n-1,k) + 1]/T(n-2,k-1) for 0=0 and T(n,0) = Fibonacci(2n-1) for n>=1.

A109165 a(n) = 5a(n-2) - 2a(n-4), n >= 4.

A109164 a(n) = 4a(n-1) - 4a(n-2) + a(n-3), n >= 3; a(0)=1, a(1)=6, a(2)=20.