A015448 -id:A015448 - cp's OEIS frontend

A107857 a(n) = floor[(phi + n mod 2)*a(n-1)], a(1)=1.

1, 1, 2, 3, 7, 11, 28, 45, 117, 189, 494, 799, 2091, 3383, 8856, 14329, 37513, 60697, 158906, 257115, 673135, 1089155, 2851444, 4613733, 12078909, 19544085, 51167078, 82790071, 216747219, 350704367, 918155952, 1485607537, 3889371025

Offset: 1

Roger L. Bagula, Jun 12 2005

A switched sequence with alternating limits of the golden mean and its square. The sequence uses only one initial term. Note that lim_{n->oo} a(n)/a(n-1) does not exist.

The consecutive pairs (2,3), (7,11), (28,45) occur as pairs in columns 2 and 3 of the Wythoff array, A035513. Suppose (l(n)) and (u(n)) are the lower and upper Beatty sequences of positive irrational numbers rClark Kimberling, Nov 24 2010

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

Cf. A000045, A000201, A001950, A015448, A033887.

[ n eq 1 select 1 else Floor(((Sqrt(5)+1)/2+(n mod 2))*Self(n-1)): n in [1..35] ];

Phi = N[(Sqrt[5] + 1)/2] F[1] = 1; F[n__] := F[n] = If[Mod[n, 2] == 0, Floor[Phi*F[n - 1]], Floor[(Phi + 1)*F[n -1]]] a = Table[F[n], {n, 1, 50}]
LinearRecurrence[{1,4,-4,1,-1},{1,1,2,3,7},40] (* Harvey P. Dale, Mar 31 2023 *)

PARI
```
a(n)=if(n<2,1,floor((phi+n%2)*a(n-1)))
```

G.f.: -x*(-1+3*x^2-x^3+x^4) / ( (x-1)*(x^4+4*x^2-1) ). - R. J. Mathar, Sep 11 2011
a(2n+2) = (1/2)*(Fib(3n+2) + 1), a(2n+1) = (1/2)*(Fib(3n+1) + 1).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) + a(n-4) - a(n-5). - Wesley Ivan Hurt, May 04 2025

Edited and better name by Ralf Stephan, Nov 24 2010

A048878 Generalized Pellian with second term of 9.

Original entry on oeis.org

1, 9, 37, 157, 665, 2817, 11933, 50549, 214129, 907065, 3842389, 16276621, 68948873, 292072113, 1237237325, 5241021413, 22201322977, 94046313321, 398386576261, 1687592618365, 7148757049721, 30282620817249, 128279240318717, 543399582092117, 2301877568687185

Offset: 0

Barry E. Williams

			a(n) = 4a(n-1) + a(n-2); a(0)=1, a(1)=9.

Harry J. Smith, Table of n, a(n) for n = 0..1584
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000045, A015448, A001077, A001076, A033887.

with(combinat): a:=n->5*fibonacci(n-1,4)+fibonacci(n,4): seq(a(n), n=1..16); # Zerinvary Lajos, Apr 04 2008

LinearRecurrence[{4,1},{1,9},31] (* or *) CoefficientList[ Series[ (1+5x)/(1-4x-x^2),{x,0,30}],x] (* Harvey P. Dale, Jul 12 2011 *)

{ default(realprecision, 2000); for (n=0, 2000, a=round(((7+sqrt(5))*(2+sqrt(5))^n - (7-sqrt(5))*(2-sqrt(5))^n )/10*sqrt(5)); if (a > 10^(10^3 - 6), break); write("b048878.txt", n, " ", a); ); } \\ Harry J. Smith, May 31 2009

a(n) = ( (7+sqrt(5))(2+sqrt(5))^n - (7-sqrt(5))(2-sqrt(5))^n )/2*sqrt(5).
G.f.: (1+5*x)/(1-4*x-x^2). - Philippe Deléham, Nov 03 2008
a(n) = F(3*n+3) + F(3*n-2); F = A000045. - Yomna Bakr and Greg Dresden, May 25 2024

A110526 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 3.

Original entry on oeis.org

0, 1, 3, 14, 58, 247, 1045, 4428, 18756, 79453, 336567, 1425722, 6039454, 25583539, 108373609, 459077976, 1944685512, 8237820025, 34895965611, 147821682470, 626182695490, 2652552464431, 11236392553213, 47598122677284

Offset: 0

Creighton Dement, Jul 24 2005

A001076(n) = a(n) + a(n+1). Program "Superseeker" finds: A033887(n+1) = a(n+2) - a(n); Elements of even index in the sequence: A049661(n) = (F(6n+1)-1)/4; A015448(n+2) = a(n+2) + 2*a(n+1) + a(n)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, 5, 1).

Cf. A110526, A110527, A033887, A001076, A049661, A033887, A000045.

seriestolist(series(-x/((1+x)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 1jbaseseq[(- 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj')(+ .5'i + .5i' + .5'jj' + .5'kk')]

Table[(Fibonacci[3n+1]-(-1)^n)/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)

concat(0, Vec(x/((1+x)*(1-x^2-4*x)) + O(x^100))) \\ Altug Alkan, Oct 28 2015

G.f.: -x/((1+x)*(x^2+4*x-1)).
a(n) = (-1)^n/2 * Sum_{k=0..n} (-1)^k*Fibonacci(3*k). - Gary Detlefs, Jan 03 2013
a(n) = (Fibonacci(3*n+1)-(-1)^n)/4, where Fibonacci(n) = A000045(n). - Vladimir Reshetnikov, Oct 28 2015

A110527 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.

Original entry on oeis.org

0, 1, 8, 29, 128, 537, 2280, 9653, 40896, 173233, 733832, 3108557, 13168064, 55780809, 236291304, 1000946021, 4240075392, 17961247585, 76085065736, 322301510525, 1365291107840, 5783465941881, 24499154875368

Offset: 0

Creighton Dement, Jul 24 2005

A048878(n) = a(n) + a(n+1). Compare with A110526.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5,1).

Cf. A110526, A110528, A033887, A001076, A049661, A033887.

seriestolist(series(-x*(1+5*x)/((1+x)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 1lesseq[(- 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj')(+ .5'i + .5i' + .5'jj' + .5'kk')], apart from initial term.

LinearRecurrence[{3,5,1},{0,1,8},30] (* Harvey P. Dale, Feb 12 2015 *)

x='x+O('x^50); concat(0, Vec(-x*(1+5*x)/((1+x)*(x^2+4*x-1)))) \\ G. C. Greubel, Aug 30 2017

G.f.: -x*(1+5*x)/((1+x)*(x^2+4*x-1)).
a(n) = (-1)^n + 3*A001076(n) - A015448(n). - Ehren Metcalfe, Nov 18 2017
a(n) = (-1)^n + 2*A110526(n) + A110679(n-2) for n >= 2. - Yomna Bakr and Greg Dresden, May 25 2024

A213893 Fixed points of a sequence h(n) defined by the minimum number of 4's in the relation n*[n,4,4,...,4,n] = [x,...,x] between simple continued fractions.

Original entry on oeis.org

3, 7, 43, 67, 103, 127, 163, 223, 283, 367, 463, 487, 523, 547, 607, 643, 727, 787, 823, 883, 907, 1063, 1123, 1303, 1327, 1423, 1447, 1543, 1567, 1627, 1663, 1723, 1747, 1783, 1867, 1987, 2083, 2143, 2203, 2287, 2347, 2383, 2467, 2683, 2707, 2767, 2803, 2887

Offset: 1

Art DuPre, Jun 23 2012

nonn

In a variant of A213891, multiply n by a number with simple continued fraction [n,4,4,...,4,n] and increase the number of 4's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are
2*[2,4,2] = [4,2,4],
3*[3,4,4,4,3] = [9,1,2,2,2,1,9],
4*[4,4,4] = [16,1,16],
5*[5,4,4,4,4,5] = [26,5,1,1,5,26].
The number of 4's needed defines the sequence h(n) = 1, 3, 1, 4, 3, 7, 3, 3, 9, ... (n>=2).
The current sequence contains the fixed points of h, i.e., those n where h(n)=n.
We conjecture that this sequence contains prime numbers analogous to the sequence of prime numbers A000057, in the sense that, instead of referring to the Fibonacci sequences(sequences satisfying f(n) = f(n-1) + f(n-2) with arbitrary positive integer values for f(1) and f(2)) it refers to the sequences satisfying f(n) = 4*f(n-1) + f(n-2), A001076, A001077, A015448, etc. This would mean that a prime is in the sequence if and only if it divides some term in each of the sequences satisfying f(n) = 4*f(n-1) + f(n-2).
The above sequence h() is recorded as A262214. - M. F. Hasler, Sep 15 2015

Cf. A000057, A213891, A213892, A213894 - A213899, A261311; A213358.

Cf. A213648, A262212 - A262220, A213900, A262211.

f[m_, n_] := Block[{c, k = 1}, c[x_, y_] := ContinuedFraction[x FromContinuedFraction[Join[{x}, Table[m, {y}], {x}]]]; While[First@ c[n, k] != Last@ c[n, k], k++]; k]; Select[Range[2, 1000], f[4, #] == # &] (* Michael De Vlieger, Sep 16 2015 *)

{a(n) = local(t, m=1); if( n<2, 0, while( 1,
   t = contfracpnqn( concat([n, vector(m,i,4), n]));
   t = contfrac(n*t[1,1]/t[2,1]);
   if(t[1]

A226328 a(0)=1, a(1)=-2; a(n+2) = a(n+1) + a(n) + (period 3: repeat 3, 1, -1).

Original entry on oeis.org

1, -2, 2, 1, 2, 6, 9, 14, 26, 41, 66, 110, 177, 286, 466, 753, 1218, 1974, 3193, 5166, 8362, 13529, 21890, 35422, 57313, 92734, 150050, 242785, 392834, 635622, 1028457, 1664078, 2692538, 4356617, 7049154, 11405774, 18454929, 29860702, 48315634, 78176337

Offset: 0

Paul Curtz, Jun 04 2013

a(n+1)/a(n) -> the golden ratio, A001622.

a(3*n)+a(3*n+1)+a(3*n+2) = 1,9,49,217,929,... = b(n), and b(n+1)-b(n) = 8*A015448(n+1).

			a(2)=-2+1+3=2, a(3)=2-2+1=1, a(4)=1+2-1=2, a(5)=2+1+3=6.
a(0)=F(-3)+F(n)-1=2+0-1=1,  a(1)=-1+1-2=-2, a(2)=1+1-0=2.
a(3)=1+4*0=1, a(4)=-2+4*1=2, a(5)=2+4*1=6, a(6)=1+4*2=9.

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

Cf. A156279, A022120, A022353.

I:=[1,-2,2,1,2]; [n le 5 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)-Self(n-4)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, Jun 05 2013

CoefficientList[Series[(2 x^4 + 3 x^2 - 3 x + 1) / (x^5 + x^4 - x^3 - x^2 - x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 05 2013 *)
LinearRecurrence[{1, 1, 1, -1, -1}, {1, -2, 2, 1, 2}, 40] (* Hugo Pfoertner, Feb 12 2024 *)

Vec((2*x^4+3*x^2-3*x+1)/(x^5+x^4-x^3-x^2-x+1)+O(x^99)) \\ Charles R Greathouse IV, Jun 04 2013

a(n) = F(n-3) + F(n) - A010872(n+1).
a(n+3) = a(n) + 4*F(n).
G.f.: (2*x^4+3*x^2-3*x+1)/( (x-1)*(x^2+x-1)*(1+x+x^2) ). [Charles R Greathouse IV, Jun 04 2013]
a(n) = A057078(n+1) +2*A212804(n) -1. - R. J. Mathar, Jun 26 2013

a(23) corrected by Charles R Greathouse IV, Jun 04 2013

More terms from Bruno Berselli, Jun 04 2013

A048877 a(n) = 4*a(n-1) + a(n-2); a(0)=1, a(1)=8.

Original entry on oeis.org

1, 8, 33, 140, 593, 2512, 10641, 45076, 190945, 808856, 3426369, 14514332, 61483697, 260449120, 1103280177, 4673569828, 19797559489, 83863807784, 355252790625, 1504874970284, 6374752671761

Offset: 0

Barry E. Williams

Generalized Pellian with second term of 8.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A015448, A001076, A001077, A033887.

a048877 n = a048877_list !! n
a048877_list = 1 : 8 : zipWith (+) a048877_list (map (* 4) $ tail a048877_list)
-- Reinhard Zumkeller, May 01 2013

with(combinat): a:=n->4*fibonacci(n-1,4)+fibonacci(n,4): seq(a(n), n=1..16); # Zerinvary Lajos, Apr 04 2008

CoefficientList[Series[(1+4x)/(1-4x-x^2),{x,0,20}],x]  (* Harvey P. Dale, Mar 30 2011 *)
LinearRecurrence[{4,1},{1,8},30] (* Harvey P. Dale, Nov 03 2013 *)

a(n) = ((6+sqrt(5))*(2+sqrt(5))^n - (6-sqrt(5))*(2-sqrt(5))^n )/(2*sqrt(5)).
G.f.: (1+4*x)/(1-4*x-x^2). - Philippe Deléham, Nov 03 2008
a(n)=4*a(n-1) + a(n-2); a(0)=1, a(1)=8.

A048879 Generalized Pellian with second term of 10.

Original entry on oeis.org

1, 10, 41, 174, 737, 3122, 13225, 56022, 237313, 1005274, 4258409, 18038910, 76414049, 323695106, 1371194473, 5808472998, 24605086465, 104228818858, 441520361897, 1870310266446, 7922761427681, 33561355977170, 142168185336361, 602234097322614

Offset: 0

Barry E. Williams

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,1)

Cf. A015448, A001077, A001076, A033887.

a048879 n = a048879_list !! n
a048879_list = 1 : 10 : zipWith (+)
                        a048879_list (map (* 4) $ tail a048879_list)
-- Reinhard Zumkeller, Mar 03 2014

with(combinat): a:=n->6*fibonacci(n-1,4)+fibonacci(n,4): seq(a(n), n=1..16); # Zerinvary Lajos, Apr 04 2008

LinearRecurrence[{4,1},{1,10},30] (* Harvey P. Dale, Jul 18 2011 *)

a(n) = ((8+sqrt(5))*(2+sqrt(5))^n - (8-sqrt(5))*(2-sqrt(5))^n)2*sqrt(5).
From Philippe Deléham, Nov 03 2008: (Start)
a(n) = 4*a(n-1) + a(n-2); a(0)=1, a(1)=10.
G.f.: (1+6*x)/(1-4*x-x^2). (End)
For n >= 1, a(n) equals the denominator of the continued fraction [4, 4, ..., 4, 10] (with n copies of 4). The numerator of that continued fraction is a(n+1). - ZhenShu Luan, Aug 05 2019

More terms from Harvey P. Dale, Jul 18 2011

A099843 A transform of the Fibonacci numbers.

Original entry on oeis.org

1, -5, 21, -89, 377, -1597, 6765, -28657, 121393, -514229, 2178309, -9227465, 39088169, -165580141, 701408733, -2971215073, 12586269025, -53316291173, 225851433717, -956722026041, 4052739537881, -17167680177565, 72723460248141, -308061521170129, 1304969544928657

Offset: 0

Paul Barry, Oct 27 2004

The g.f. is the transform of the g.f. of A000045 under the mapping G(x) -> (-1/(1+x))*G((x-1)/(x+1)). In general this mapping transforms x/(1-k*x-k*x^2) into (1-x)/(1 + 2(k+1)*x - (2*k-1)*x^2).

Pisano period lengths: 1, 1, 8, 2, 20, 8, 16, 4, 8, 20, 10, 8, 28, 16, 40, 8, 12, 8, 6, 20, ... - R. J. Mathar, Aug 10 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (-4,1).

Cf. A000045, A001076, A015448, A099842.

Cf. A084326 (shifted unsigned inverse binomial transform), A152174 (binomial transform).

[(-1)^n*Fibonacci(3*n+2): n in [0..40]]; // G. C. Greubel, Apr 20 2023

a:= n-> (<<0|1>, <1|-4>>^n.<<1, -5>>)[1,1]:
seq(a(n), n=0..24);  # Alois P. Heinz, Apr 21 2023

CoefficientList[Series[(1-x)/(1+4*x-x^2), {x,0,30}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{-4,1},{1,-5},30] (* Harvey P. Dale, Aug 13 2015 *)

[(-1)^n*fibonacci(3*n+2) for n in range(41)] # G. C. Greubel, Apr 20 2023

G.f.: (1-x)/(1+4*x-x^2).
a(n) = (sqrt(5)-2)^n * (1/2 - 3*sqrt(5)/10) + (-sqrt(5)-2)^n * (1/2 + 3*sqrt(5)/10).
a(n) = (-1)^n*Fibonacci(3*n+2).
a(n) = -4*a(n-1) + a(n-2), a(0)=1, a(1)=-5. - Philippe Deléham, Nov 03 2008
a(n) = (-1)^n*(A001076(n) + A001076(n+1)). - R. J. Mathar, Aug 10 2012
a(n) = (-1)^n*A015448(n+1). - R. J. Mathar, May 07 2019

A134489 a(n) = Fibonacci(5*n + 2).

Original entry on oeis.org

1, 13, 144, 1597, 17711, 196418, 2178309, 24157817, 267914296, 2971215073, 32951280099, 365435296162, 4052739537881, 44945570212853, 498454011879264, 5527939700884757, 61305790721611591, 679891637638612258

Offset: 0

Artur Jasinski, Oct 28 2007

The o.g.f. of {F(m*n + 2)}_{n>=0}, for m = 1, 2, ..., is

G(m,x) = (1 + F(m - 2)*x) / (1 - L(m)*x + (-1)^m*x^2), with F = A000045 and F(-1) = 1, and L = A000032. - Wolfdieter Lang, Feb 06 2023

Index entries for linear recurrences with constant coefficients, signature (11,1).

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888-A033891, A102312, A099100, A134490-A134495, A103134, A134497-A134504.

[Fibonacci(5*n+2): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011

Table[Fibonacci[5n + 2], {n, 0, 30}]
LinearRecurrence[{11,1},{1,13},20] (* Harvey P. Dale, May 05 2022 *)

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-1-2*x) / (-1 + 11*x + x^2).
a(n) = 2*A049666(n) + A049666(n+1). (End)
a(n) = A000045(A016873(n)). - Michel Marcus, Nov 05 2013

cp's OEIS Frontend

A107857 a(n) = floor[(phi + n mod 2)*a(n-1)], a(1)=1.

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A048878 Generalized Pellian with second term of 9.

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A110526 a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 3.

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A110527 a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.

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A213893 Fixed points of a sequence h(n) defined by the minimum number of 4's in the relation n*[n,4,4,...,4,n] = [x,...,x] between simple continued fractions.

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A226328 a(0)=1, a(1)=-2; a(n+2) = a(n+1) + a(n) + (period 3: repeat 3, 1, -1).

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A048877 a(n) = 4*a(n-1) + a(n-2); a(0)=1, a(1)=8.

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A110526 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 3.

A110527 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.