A133080 -id:A133080 - cp's OEIS frontend

A094967 Right-hand neighbors of Fibonacci numbers in Stern's diatomic series.

1, 1, 2, 2, 5, 5, 13, 13, 34, 34, 89, 89, 233, 233, 610, 610, 1597, 1597, 4181, 4181, 10946, 10946, 28657, 28657, 75025, 75025, 196418, 196418, 514229, 514229, 1346269, 1346269, 3524578, 3524578, 9227465, 9227465, 24157817, 24157817, 63245986, 63245986, 165580141, 165580141

Offset: 0

Paul Barry, May 26 2004

Fibonacci(2*n+1) repeated. a(n) is the right neighbor of Fibonacci(n+2) in A049456 and A002487 (starts 2,2,5,...). A000045(n+2) = A094966(n) + a(n).
Diagonal sums of A109223. - Paul Barry, Jun 22 2005
The Fi2 sums, see A180662, of triangle A065941 equal the terms of this sequence. - Johannes W. Meijer, Aug 11 2011
a(n) is the last term of (n+1)-th row in Wythoff array A003603. -Reinhard Zumkeller, Jan 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 15.
Ying Wang and Zihao Zhang, Hankel Determinants for a Class of Weighted Lattice Paths, arXiv:2409.18609 [math.CO], 2024. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A001519, A133080.

List([0..50], n -> Fibonacci(n)*(1-(-1)^n)/2 + Fibonacci(n+1)*(1+(-1)^n)/2); # G. C. Greubel, Nov 18 2018

[IsEven(n) select Fibonacci(n+1) else Fibonacci(n): n in [0..70]]; // Vincenzo Librandi, Nov 18 2018

A094967 := proc(n) combinat[fibonacci](2*floor(n/2)+1) ; end proc: seq(A094967(n), n=0..37);

LinearRecurrence[{0,3,0,-1},{1,1,2,2},40] (* Harvey P. Dale, Apr 05 2015 *)
f[n_]:=If[OddQ@n, (Fibonacci[n]), Fibonacci[n+1]]; Array[f, 100, 0] (* Vincenzo Librandi, Nov 18 2018 *)
Table[Fibonacci[n, 0]*Fibonacci[n] + LucasL[n, 0]*Fibonacci[n + 1]/2, {n, 0, 50}] (* G. C. Greubel, Nov 18 2018 *)

vector(50, n, n--; fibonacci(n)*(1-(-1)^n)/2 + fibonacci(n+1)*(1+(-1)^n)/2) \\ G. C. Greubel, Nov 18 2018

[fibonacci(n)*(1-(-1)^n)/2 + fibonacci(n+1)*(1+(-1)^n)/2 for n in range(50)] # G. C. Greubel, Nov 18 2018

G.f.: (1+x-x^2-x^3)/(1-3*x^2+x^4).
a(n) = Fibonacci(n)*(1-(-1)^n)/2 + Fibonacci(n+1)*(1+(-1)^n)/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2)+k, 2*k). - Paul Barry, Jun 22 2005
Starting (1, 2, 2, 5, 5, 13, 13, ...) = A133080 * A000045, where A000045 starts with "1". - Gary W. Adamson, Sep 08 2007
a(n) = Fibonacci(n+1)^(4*k+3) mod Fibonacci(n+2), for any k>-1, n>0. - Gary Detlefs, Nov 29 2010

A114753 First column of A114751.

Original entry on oeis.org

1, 3, 3, 7, 5, 11, 7, 15, 9, 19, 11, 23, 13, 27, 15, 31, 17, 35, 19, 39, 21, 43, 23, 47, 25, 51, 27, 55, 29, 59, 31, 63, 33, 67, 35, 71, 37, 75, 39, 79, 41, 83, 43, 87, 45, 91, 47, 95, 49, 99, 51, 103, 53, 107, 55, 111, 57, 115, 59, 119, 61, 123, 63, 127, 65, 131, 67, 135, 69

Offset: 1

Amarnath Murthy, Nov 15 2005

A114752, (1, 2, 5, 4, 9, 6, ...) + A114753 - (1,1,1,...) = 3n+1: (1, 4, 7, 10, 13, ...). - Gary W. Adamson, Sep 16 2007

First column of table A210530. - Boris Putievskiy, Jan 29 2013

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A114751, A114752.

Cf. A133080, A210530.

Table[If[OddQ[n],n,2n-1],{n,1,120}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011*)

a(2n+1) = 2n+1, a(2n) = 4n-1.
a(n) = 2*a(n-2) - a(n-4). - Joerg Arndt, Apr 02 2011
Equals A133080 * [1,2,3,...]. - Gary W. Adamson, Sep 08 2007
G.f. x*(1+3*x+x^2+x^3) / ( (x-1)^2*(1+x)^2 ). - R. J. Mathar, Apr 04 2012
a(n) = (3*n-1-(n-1)*(-1)^(n-1))/2. - Boris Putievskiy, Jan 29 2013

More terms from Joshua Zucker, May 05 2006

A133125 a(n) = (7*3^n - (-3)^n)/6.

Original entry on oeis.org

1, 4, 9, 36, 81, 324, 729, 2916, 6561, 26244, 59049, 236196, 531441, 2125764, 4782969, 19131876, 43046721, 172186884, 387420489, 1549681956, 3486784401, 13947137604, 31381059609, 125524238436, 282429536481, 1129718145924, 2541865828329, 10167463313316

Offset: 0

Gary W. Adamson, Sep 19 2007

A133647 is a companion sequence.

			a(4) = 3^4 = 81.
a(5) = 324 = 4 * 3^4.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,9).

Cf. A000244, A038754, A133080, A133647.

Table[3^(n - 2) ((-1)^n + 7)/2, {n, 1, 60}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{0,9},{1,4},30] (* Harvey P. Dale, Mar 27 2017 *)

a(n) =  (7*3^n - (-3)^n)/6 \\ Andrew Howroyd, Jul 03 2024

A133080 * A000244, where A000244 = (3^0, 3^1, 3^2, ...).
For even n, a(n) = 3^n. For odd n, a(n) = 4 * 3^(n-1).
From R. J. Mathar, Oct 30 2008: (Start)
G.f.: (1+4*x)/((1+3*x)*(1-3*x)).
a(n) = 9*a(n-2). (End)
a(n) = A038754(n)^2. - T. D. Noe, Jun 10 2011
E.g.f.: (3*exp(3*x) + sinh(3*x))/3. - Andrew Howroyd, Jul 03 2024
From Amiram Eldar, Jun 02 2025: (Start)
Sum_{n>=0} 1/a(n) = 45/32.
Sum_{n>=0} (-1)^n/a(n) = 27/32. (End)

Name changed by Andrew Howroyd, Jul 03 2024

A133585 Expansion of x - x^2(2x+1)(x^2-2) / ( (x^2-x-1)(x^2+x-1) ).

Original entry on oeis.org

1, 2, 4, 5, 10, 13, 26, 34, 68, 89, 178, 233, 466, 610, 1220, 1597, 3194, 4181, 8362, 10946, 21892, 28657, 57314, 75025, 150050, 196418, 392836, 514229, 1028458, 1346269, 2692538, 3524578, 7049156, 9227465, 18454930, 24157817

Offset: 1

Gary W. Adamson, Sep 18 2007

A133585 is a companion to A133586.

			a(4) = F(5) = 5.
a(5) = 2*a(4) = 2*5 = 10.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A133080, A000045, A133586.

A133585aux := proc(n,k)
    add(A133566(n,j)*A133080(j,k),j=k..n) ;
end proc:
A000045 := proc(n)
    combinat[fibonacci](n) ;
end proc:
A133585 := proc(n)
    add(A133585aux(n,j)*A000045(j),j=0..n) ;
end proc: # R. J. Mathar, Jun 20 2015

CoefficientList[Series[1 - x (2 x + 1) (x^2 - 2)/((x^2 - x - 1) (x^2 + x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 21 2015 *)
LinearRecurrence[{0,3,0,-1},{1,2,4,5,10},40] (* Harvey P. Dale, Mar 04 2019 *)

a(n)=if(n>1,([0,1,0,0;0,0,1,0;0,0,0,1;-1,0,3,0]^(n-2)*[2;4;5;10])[1,1],1) \\ Charles R Greathouse IV, Jun 20 2015

Equals the matrix-matrix-vector product A133566 * A133080 * A000045 (previous name).

For even-indexed terms, a(n) = F(n+1). For odd-indexed terms (n>1), a(n) = 2*a(n-1), A126358.

Previous name corrected and new name from R. J. Mathar, Jun 20 2015

A133086 Row sums of triangle A133085.

Original entry on oeis.org

1, 4, 10, 26, 64, 152, 352, 800, 1792, 3968, 8704, 18944, 40960, 88064, 188416, 401408, 851968, 1802240, 3801088, 7995392, 16777216, 35127296, 73400320, 153092096, 318767104, 662700032, 1375731712, 2852126720, 5905580032, 12213813248, 25232932864

Offset: 0

Gary W. Adamson, Sep 08 2007

nonn

			a(3) = 26 = sum of row 3 of triangle A133085: (12 + 8, + 5 + 1).
a(3) = 26 = (1, 3, 3, 1) dot (1, 3, 3, 7) = (1 + 9 + 9 + 7).

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

Cf. A133085, A114753, A133080.

[1,4] cat [2^n+3*n*2^(n-2): n in [2..30]]; // Vincenzo Librandi, Oct 21 2017

Join[{1, 4}, Table[2^n + 3*n*2^(n - 2), {n, 2, 50}]] (* G. C. Greubel, Oct 21 2017 *)
LinearRecurrence[{4,-4},{1,4,10,26},40] (* Harvey P. Dale, Jul 19 2020 *)

concat([1,4], for(n=2,50, print1(2^n + 3*n*2^(n-2), ", "))) \\ G. C. Greubel, Oct 21 2017

Binomial transform of A114753: (1, 3, 3, 7, 5, 11, 7, 15, ...).

For n>1, a(n) = 2^n + 3*n*2^(n-2). - R. J. Mathar, Apr 04 2012

A133586 Expansion of x(1+2x)/( (x^2-x-1)*(x^2+x-1) ).

Original entry on oeis.org

1, 2, 3, 6, 8, 16, 21, 42, 55, 110, 144, 288, 377, 754, 987, 1974, 2584, 5168, 6765, 13530, 17711, 35422, 46368, 92736, 121393, 242786, 317811, 635622, 832040, 1664080, 2178309, 4356618, 5702887, 11405774, 14930352, 29860704, 39088169, 78176338, 102334155

Offset: 1

Gary W. Adamson, Sep 18 2007

For n>1 A133585(n) + a(n) = A000032(n+1).

			a(5) = F(6) = 8.
a(6) = 2*a(5) = 2*8 = 16.

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

Cf. A001906 (bisection), A025169 (bisection), A000032, A133586.

A133586aux := proc(n,k)
    add(A133080(n,j)*A133566(j,k),j=k..n) ;
end proc:
A000045 := proc(n)
    combinat[fibonacci](n) ;
end proc:
A133586 := proc(n)
    add(A133586aux(n,j)*A000045(j),j=0..n) ;
end proc: # R. J. Mathar, Jun 20 2015

CoefficientList[Series[(1 + 2 x)/((x^2 - x - 1) (x^2 + x - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 21 2015 *)
LinearRecurrence[{0,3,0,-1},{1,2,3,6},40] (* Harvey P. Dale, Dec 10 2017 *)

{a(n) = if( n%2, fibonacci(n+1), 2*fibonacci(n))}; /* Michael Somos, Jun 20 2015 */

Vec(x*(1+2*x)/((x^2-x-1)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, Mar 28 2016

Equals A133080 * A133566 * A000045, where A133080 and A133566 are infinite lower triangular matrices and the Fibonacci sequence as a vector (previous definition).
For odd-indexed terms, a(n) = F(n+1). For even-indexed terms, a(n) = 2*a(n-1).
For n>1 A133585(n) + a(n) = A000032(n+1).
a(n) = A147600(n) + 2*A147600(n-1). - R. J. Mathar, Jun 20 2015
a(n) = (2^(-2-n)*((1-sqrt(5))^n*(-5+sqrt(5)) - (-1-sqrt(5))^n*(-3+sqrt(5)) - (-1+sqrt(5))^n*(3+sqrt(5)) + (1+sqrt(5))^n*(5+sqrt(5))))/sqrt(5). - Colin Barker, Mar 28 2016

New definition and A-number in previous definition corrected by R. J. Mathar, Jun 20 2015

A092809 Expansion of (1+x-x^2) / ((1-x^2)(1-4x^2)).

Original entry on oeis.org

1, 1, 4, 5, 16, 21, 64, 85, 256, 341, 1024, 1365, 4096, 5461, 16384, 21845, 65536, 87381, 262144, 349525, 1048576, 1398101, 4194304, 5592405, 16777216, 22369621, 67108864, 89478485, 268435456, 357913941, 1073741824, 1431655765, 4294967296, 5726623061

Offset: 0

Paul Barry, Mar 10 2004

Partial sums of A092808.

Apply matrix A133080 to A001045(n+1). - Paul Barry, Oct 08 2009

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4)

Cf. A001045.

Vec((1+x-x^2)/((1-x)*(1+x)*(1-2*x)*(1+2*x)) + O(x^40)) \\ Colin Barker, Sep 09 2016

a(n) = if(n%2,2,3)<Charles R Greathouse IV, Sep 09 2016

a(n) = 5*2^n/6+(-2)^n/6+(-1)^n/6-1/6.
a(2*n) = 4^n = A000302(n).
a(2*n+1) = (4*4^n-1)/3 = A002450(n+1).
From Colin Barker, Sep 09 2016: (Start)
a(n) = 5*a(n-2)-4*a(n-4) for n>3.
G.f.: (1+x-x^2) / ((1-x)*(1+x)*(1-2*x)*(1+2*x))
(End)

A133085 A133084 * A000012.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 12, 8, 5, 1, 24, 19, 15, 5, 1, 48, 42, 37, 17, 7, 1, 96, 89, 83, 48, 28, 7, 1, 192, 184, 177, 121, 86, 30, 9, 1, 384, 375, 367, 283, 227, 101, 45, 9, 1, 768, 758, 749, 629, 545, 293, 167, 47, 11, 1

Offset: 1

Gary W. Adamson, Sep 08 2007

Left column = A003945: (1, 3, 6, 12, 24, 48, ...). Row sums = A133086: (1, 4, 10, 26, 64, 152, ...).

			First few rows of the triangle are:
1;
3, 1;
6, 3, 1;
12, 8, 5, 1;
24, 19, 15, 5, 1;
48, 42, 37, 17, 7, 1;
96, 89, 83, 48, 28, 7, 1;
...

Cf. A133084, A133086, A133080, A003945.

A133084 * A000012 as infinite lower triangular matrices.

a(28)=1 inserted by Georg Fischer, Oct 10 2021

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A094967 Right-hand neighbors of Fibonacci numbers in Stern's diatomic series.

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A114753 First column of A114751.

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A133125 a(n) = (7*3^n - (-3)^n)/6.

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A133585 Expansion of x - x^2*(2*x+1)*(x^2-2) / ( (x^2-x-1)*(x^2+x-1) ).

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A133086 Row sums of triangle A133085.

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A133586 Expansion of x*(1+2*x)/( (x^2-x-1)*(x^2+x-1) ).

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A133585 Expansion of x - x^2(2x+1)(x^2-2) / ( (x^2-x-1)(x^2+x-1) ).

A133586 Expansion of x(1+2x)/( (x^2-x-1)*(x^2+x-1) ).

A092809 Expansion of (1+x-x^2) / ((1-x^2)(1-4x^2)).