A033887 -id:A033887 - cp's OEIS frontend

A117647 a(2n) = A014445(n), a(2n+1) = A015448(n+1).

0, 1, 2, 5, 8, 21, 34, 89, 144, 377, 610, 1597, 2584, 6765, 10946, 28657, 46368, 121393, 196418, 514229, 832040, 2178309, 3524578, 9227465, 14930352, 39088169, 63245986, 165580141, 267914296, 701408733, 1134903170, 2971215073, 4807526976, 12586269025, 20365011074, 53316291173

Offset: 0

Creighton Dement, Apr 10 2006

Because of g.f. in Formula section, numbers k + 1 such that A088207(k)/k is an integer. - Ctibor O. Zizka, Apr 07 2025

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

Cf. A000045, A001950, A007494, A014445, A015448, A033887, A059973, A088207.

I:=[0,1,2,5]; [n le 4 select I[n] else 4*Self(n-2) +Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 12 2021

Table[Fibonacci[(6*n+1 -(-1)^n)/4], {n, 0, 40}] (* G. C. Greubel, Jul 12 2021 *)

[fibonacci((6*n+1-(-1)^n)/4) for n in [0..40]] # G. C. Greubel, Jul 12 2021

a(n) = A059973(n+2) - A059973(n+1).
G.f.: x*(x+1)^2/(1 -4*x^2 -x^4).
a(n) = Fibonacci((6*n + 1 - (-1)^n)/4) = Fibonacci(A007494(n)). - G. C. Greubel, Jul 12 2021

A209084 a(n) = 2a(n-1) + 4a(n-2) with n>1, a(0)=0, a(1)=4.

Original entry on oeis.org

0, 4, 8, 32, 96, 320, 1024, 3328, 10752, 34816, 112640, 364544, 1179648, 3817472, 12353536, 39976960, 129368064, 418643968, 1354760192, 4384096256, 14187233280, 45910851584, 148570636288, 480784678912, 1555851902976, 5034842521600, 16293092655104

Offset: 0

Seiichi Kirikami, Mar 06 2012

a(n)/A063727(n) are convergents for A134972.

Abs(Sum_{i=0..n} C(n,n-i)*a(i)-(sqrt(5)-1)* A033887(n))->0. - Seiichi Kirikami, Jan 20 2016

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

Bruno Berselli, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (2,4).

Cf. A063727, A033887, A085449, A103435, A134972, A269991.

Cf. A086344 (this sequence with signs).

I:=[0,4]; [n le 2 select I[n] else 2*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 16 2016

RecurrenceTable[{a[n]==2*a[n-1]+4*a[n-2], a[0]==0, a[1]==4}, a, {n, 30}]
LinearRecurrence[{2, 4}, {0, 4}, 40] (* Vincenzo Librandi, Jan 16 2016 *)

concat(0, Vec(4*x/(1-2*x-4*x^2) + O(x^40))) \\ Michel Marcus, Jan 16 2016

a(n) = (2/sqrt(5))*((1+sqrt(5))^n-(1-sqrt(5))^n).
G.f.: 4*x/(1-2*x-4*x^2). - Bruno Berselli, Mar 08 2012
a(n) = 4*A085449(n) = 2*A103435(n). - Bruno Berselli, Mar 08 2012
Sum_{n>=1} 1/a(n) = (1/4) * A269991. - Amiram Eldar, Feb 01 2021

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

A099015 a(n) = Fib(n+1)(2Fib(n)^2 + Fib(n)*Fib(n-1) + Fib(n-1)^2).

Original entry on oeis.org

1, 2, 8, 33, 140, 592, 2509, 10626, 45016, 190685, 807764, 3421728, 14494697, 61400482, 260096680, 1101787113, 4667245276, 19770767984, 83750317589, 354772037730, 1502838469496, 6366125914117, 26967342128548

Offset: 0

Paul Barry, Sep 22 2004

Form the matrix A=[1,1,1,1;3,2,1,0;3,1,0,0;1,0,0,0]=(binomial(3-j,i)). Then a(n)=(2,2)-element of A^n.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..172 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Cf. A000045, A056570, A066258, A066259, A099014, A033887.

[Fibonacci(n+1)*(2*Fibonacci(n)^2 + Fibonacci(n)*Fibonacci(n-1) + Fibonacci(n-1)^2): n in [0..30]]; // Vincenzo Librandi, Jun 05 2011

LinearRecurrence[{3,6,-3,-1},{1,2,8,33},30] (* Harvey P. Dale, Nov 28 2015 *)
CoefficientList[Series[(1-x-4*x^2)/((1+x-x^2)*(1-4*x-x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 31 2017 *)

a(n)=my(e=fibonacci(n-1),f=fibonacci(n));(e+f)*(2*f^2+f*e+e^2) \\ Charles R Greathouse IV, Jun 05 2011

first(n) = Vec((1 - x - 4*x^2)/(1 - 3*x - 6*x^2 + 3*x^3 + x^4) + O(x^n)) \\ Iain Fox, Dec 31 2017

G.f.: (1-x-4*x^2)/((1+x-x^2)*(1-4*x-x^2)).
G.f.: (1-x-4*x^2)/(1-3*x-6*x^2+3*x^3+x^4).
a(n) = (3*Fib(3*n+1) + (-1)^n*Fib(n-3))/5.
a(n) = (2+sqrt(5))^n*(3/10 + 3*sqrt(5)/50) + (2-sqrt(5))^n*(3/10 - 3*sqrt(5)/50) + (-1)^n*((1/2 - sqrt(5)/2)^n*(1/5 + 2*sqrt(5)/25) + (1/5 - 2*sqrt(5)/25)*(1/2 + sqrt(5)/2)^n).

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

A020712 Pisot sequences E(5,8), P(5,8).

Original entry on oeis.org

5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141

Offset: 0

David W. Wilson

Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60,.. - R. J. Mathar, Aug 10 2012

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

Subsequence of A020701 and hence A020695, A000045. See A008776 for definitions of Pisot sequences.

Trisections: A015448, A014445, A033887.

CoefficientList[Series[(-3 z - 5)/(z^2 + z - 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{1,1},{5,8},40] (* Harvey P. Dale, Dec 28 2013 *)

a(n)=fibonacci(n+5) \\ Charles R Greathouse IV, Jan 17 2012

a(n) = Fib(n+5). a(n) = a(n-1) + a(n-2).
O.g.f.: (5+3x)/(1-x-x^2). a(n)=A020701(n+1). - R. J. Mathar, May 28 2008
a(n) = (2^(-1-n)*((1-sqrt(5))^n*(-11+5*sqrt(5))+(1+sqrt(5))^n*(11+5*sqrt(5))))/sqrt(5). - Colin Barker, Jun 05 2016

A290750 Inverse Euler transform of [3, 13, 55, 233, 987, 4181, 17711, 75025, 317811, ...], Fibonacci(3*k+1).

Original entry on oeis.org

3, 7, 24, 76, 272, 948, 3496, 12920, 48792, 185912, 716472, 2781600, 10878640, 42789292, 169181280, 671865840, 2678679360, 10716650484, 43007271768, 173072547360, 698235684336, 2823329204964, 11439823954664, 46440709197120, 188856966713360, 769241291697640, 3137871076653336, 12817512478814400

Offset: 1

N. J. A. Sloane, Aug 12 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Latham Boyle, Paul J. Steinhardt, Self-Similar One-Dimensional Quasilattices, arXiv preprint arXiv:1608.08220 [math-ph], 2016. See Table 2, column 8.
N. J. A. Sloane, Transforms

Cf. A033887.

read(transforms): with(combinat); F:=fibonacci;
s1:=[seq(F(3*n+1),n=1..40)];
EULERi(s1);

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i b[[i]] - Sum[c[[d]] b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d] c[[d]], {d, 1, i}]]]; Return[a]];
EULERi[Table[Fibonacci[3k + 1], {k, 1, 30}]] (* Jean-François Alcover, Aug 06 2018 *)

a(n) ~ (2 + sqrt(5))^n / n. - Vaclav Kotesovec, Oct 09 2019

A271388 a(n) = 4*a(n-1) + a(n-2) - n for n > 1, with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, 6, 22, 89, 372, 1570, 6644, 28137, 119182, 504854, 2138586, 9059185, 38375312, 162560418, 688616968, 2917028273, 12356730042, 52343948422, 221732523710, 939274043241, 3978828696652, 16854588829826, 71397184015932, 302443324893529, 1281170483590022

Offset: 0

Ilya Gutkovskiy, Apr 06 2016

Index entries for linear recurrences with constant coefficients, signature (6,-8,2,1).

Cf. A030119, A033887, A048776, A098317.

RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == 4 a[n - 1] + a[n - 2] - n}, a, {n, 28}]
LinearRecurrence[{6, -8, 2, 1}, {0, 1, 2, 6}, 29]
nxt[{n_,a_,b_}]:={n+1,b,4b+a-n-1}; NestList[nxt,{1,0,1},30][[;;,2]] (* Harvey P. Dale, Feb 07 2025 *)

x='x+O('x^99); concat(0, Vec(x*(1-4*x+2*x^2)/((1-x)^2*(1-4*x-x^2)))) \\ Altug Alkan, Apr 06 2016

a(n) = (3*fibonacci(3*n-2) + 2*n+3) >> 3; \\ Kevin Ryde, May 16 2021

G.f.: x*(1 - 4*x + 2*x^2)/((1 - x)^2*(1 - 4*x - x^2)).
E.g.f.: (1/80)*(10*exp(x)*(2*x + 3) - 3*(5 + 3*sqrt(5))*exp((2 - sqrt(5))*x) + 3*(3*sqrt(5) - 5)*exp((2 + sqrt(5))*x)).
a(n) = 6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4).
a(n) = (1/80)*(20*n - 3*(5 + 3*sqrt(5))*(2 - sqrt(5))^n + 3*(3*sqrt(5) - 5)*(2 + sqrt(5))^n + 30).
Lim_{n->infinity} a(n + 1)/a(n) = 2 + sqrt(5) = phi^3 = A098317, where phi is the golden ratio (A001622).
a(n) = (2*n + 3 + 3*A033887(n-1))/8. - R. J. Mathar, Mar 12 2017

A323013 Form of Zorach additive triangle T(n,k) (see A035312) where each number is sum of west and northwest numbers, with the additional condition that the first element T(n,1) is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 8, 13, 20, 30, 21, 29, 42, 62, 92, 34, 55, 84, 126, 188, 280, 89, 123, 178, 262, 388, 576, 856, 144, 233, 356, 534, 796, 1184, 1760, 2616, 377, 521, 754, 1110, 1644, 2440, 3624, 5384, 8000, 610, 987, 1508, 2262, 3372, 5016, 7456, 11080, 16464, 24464

Offset: 1

Michel Lagneau, Jan 02 2019

Conjecture: Let F(i) be the i-th Fibonacci number. Each number of T(n, k), k = 1, 2, 3 is the difference between two Fibonacci numbers F(i) - F(j) for some i, j, where F(i) is the smallest Fibonacci number greater than T(n, k). The case T(n, 1) is trivial. Examples: 10 = 13 - 3, 29 = 34 - 5, 20 = 21 - 1, 42 = 55 - 13, 84 = 89 - 5, ...
We observe interesting properties:
T(n,1) = A117647(n) = 1, 2, 5, 8, 21, ... where n = 1, 2, ...
T(2n,2) = A033887(n) = 3, 13, 55, ... (Fibonacci(3n+1)), and T(2n+1,2) = A048876(n) = 7, 29, 123, ... (Generalized Pell equation with second term of 7) where n = 1, 2, ...
T(3n,3) = 10, 84, 754, 6388,... If n = 2m - 1, T(6m - 3, 3) = F(9m - 2) - F(9m - 5) and if n = 2m, T(6m, 3) =  F(9m + 2) - F(9m - 4).
T(3n+1,3) = 20, 178, 1508, 13530, ... If n = 2m - 1, T(6m - 2, 3) = F(9m - 1) - F(9m - 7) and if n = 2m, T(6m+1, 3) =  F(9m + 4) - F(9m + 1).
T(3n+2,3) = 42, 356, 3194, 27060, ... If n = 2m - 1, T(6m - 1, 3) = F(9m + 1) - F(9m - 2) and if n = 2m, T(6m + 2, 3) =  F(9m + 5) - F(9m - 1).
Other property:
T(2m, 1) + T(2m, 2) = T(2m +1, 1) with T(2m, 1)= F(3m), T(2m, 2) = F(3m + 1) and T(2m + 1, 1) = F(3m + 2).
T(2m + 1, 1) + T(2m + 1, 2) = F(3m + 4) - F(3m - 1).

			The start of the sequence as a triangular array T(n, k) read by rows:
   1;
   2,   3;
   5,   7,  10;
   8,  13,  20,   30;
  21,  29,  42,   62,   92;
  34,  55,  84,  126,  188,  280;
  ...

Cf. A000045, A002878, A033887, A035312, A036561, A117647.

with(combinat,fibonacci):
lst:={1}:lst2:=lst:
for n from 2 to 15 do :
lst1:={}:ii:=0:
  for j from 1 to 1000 while(ii=0) do:
     i:=fibonacci(j):
     if {i} intersect lst2 = {} and {i+lst[1]} intersect lst2 = {}
      then
      lst1:=lst1 union {i}:ii:=1:
      else
     fi:
   od:
    for k from 1 to n-1 do:
      lst1:=lst1 union {lst1[k]+lst[k]}:
    od:
    lst:=lst1:lst2:=lst2 union lst:
    print(lst1):
   od:

cp's OEIS Frontend

A117647 a(2n) = A014445(n), a(2n+1) = A015448(n+1).

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A209084 a(n) = 2*a(n-1) + 4*a(n-2) with n>1, a(0)=0, a(1)=4.

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

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A048879 Generalized Pellian with second term of 10.

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A099015 a(n) = Fib(n+1)*(2*Fib(n)^2 + Fib(n)*Fib(n-1) + Fib(n-1)^2).

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

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A020712 Pisot sequences E(5,8), P(5,8).

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A209084 a(n) = 2a(n-1) + 4a(n-2) with n>1, a(0)=0, a(1)=4.

A099015 a(n) = Fib(n+1)(2Fib(n)^2 + Fib(n)*Fib(n-1) + Fib(n-1)^2).