A097136 -id:A097136 - cp's OEIS frontend

A097135 a(0) = 1; for n>0, a(n) = 3*Fibonacci(n).

1, 3, 3, 6, 9, 15, 24, 39, 63, 102, 165, 267, 432, 699, 1131, 1830, 2961, 4791, 7752, 12543, 20295, 32838, 53133, 85971, 139104, 225075, 364179, 589254, 953433, 1542687, 2496120, 4038807, 6534927, 10573734, 17108661, 27682395, 44791056, 72473451, 117264507

Offset: 0

Paul Barry, Jul 26 2004

Binomial transform is A097136.

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

Cf. A000032, A000045.

Essentially the same as A022086.

Join[{1}, Table[3*Fibonacci[n], {n, 70}]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)

a(n)=if(n,3*finonacci(n),1) \\ Charles R Greathouse IV, Oct 16 2015

G.f. : (1+2*x-x^2)/(1-x-x^2).
a(n) = a(n-1)+a(n-2) for n>2.
a(2n) = A097134(n); a(2n+1) = 3*F(2n+1).

Definition rewritten by N. J. A. Sloane, Jan 24 2010

A097133 a(n) = 3*Fibonacci(n) + (-1)^n.

Original entry on oeis.org

1, 2, 4, 5, 10, 14, 25, 38, 64, 101, 166, 266, 433, 698, 1132, 1829, 2962, 4790, 7753, 12542, 20296, 32837, 53134, 85970, 139105, 225074, 364180, 589253, 953434, 1542686, 2496121, 4038806, 6534928, 10573733, 17108662, 27682394, 44791057, 72473450, 117264508

Offset: 0

Paul Barry, Jul 26 2004

Binomial transform is A097134.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A000045, A097134, A097136.

a097133 n = a097133_list !! n
a097133_list = 1 : 2 : 4 : zipWith (+)
               (map (* 2) $ tail a097133_list) a097133_list
-- Reinhard Zumkeller, Feb 24 2015

CoefficientList[Series[(1+2x+2x^2)/((1+x)(1-x-x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{0,2,1},{1,2,4},40] (* Harvey P. Dale, May 07 2011 *)

G.f.: (1+2*x+2*x^2)/((1+x)*(1-x-x^2));
a(n) = 2*a(n-2)+a(n-3);
a(2*n) = 3*F(2*n)+1 = A097136(n).

A192908 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 1, 3, 7, 17, 43, 111, 289, 755, 1975, 5169, 13531, 35423, 92737, 242787, 635623, 1664081, 4356619, 11405775, 29860705, 78176339, 204668311, 535828593, 1402817467, 3672623807, 9615053953, 25172538051, 65902560199

Offset: 0

Clark Kimberling, Jul 12 2011

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x + 1.

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

Cf. A192232, A192744, A192872, A192904.
Cf. A000045; A052995: 2*Fibonacci(2*n-1) for n>0.
CF. A055588, A097136.

Concatenation([1], List([1..30], n -> 1+2*Fibonacci(2*(n-1)))); # G. C. Greubel, Jan 11 2019

[1] cat [1+2*Fibonacci(2*(n-1)): n in [1..30]]; // G. C. Greubel, Jan 11 2019

u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 0; f = 1;
q = x^2; s = u*x + v; z = 26;
p[0, x_] := a;  p[1, x_] := b*x + c
p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
Table[Expand[p[n, x]], {n, 0, 8}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]    (* A192908 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]    (* A069403 *)
Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
LinearRecurrence[{4,-4,1}, {1,1,3,7}, 30] (* G. C. Greubel, Jan 11 2019 *)

vector(30, n, n--; if(n==0,1,1+2*fibonacci(2*n-2))) \\ G. C. Greubel, Jan 11 2019

[1]+[1+2*fibonacci(2*(n-1)) for n in (1..30)] # G. C. Greubel, Jan 11 2019

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n>3.
G.f.: 1 + x*(1 - x - x^2)/((1 - x)*(1 - 3*x + x^2)). - R. J. Mathar, Jul 13 2011
a(n) = 2*Fibonacci(2*n-2) + 1 for n>0, a(0)=1. - Bruno Berselli, Dec 27 2016
a(n) = -1 + 3*a(n-1) - a(n-2) with a(1) = 1 and a(2) = 3. Cf. A055588 and A097136. - Peter Bala, Nov 12 2017

A097132 a(n) = Sum_{k=0..n} Fibonacci(k) + (-1)^k*Fibonacci(k-1).

Original entry on oeis.org

1, 2, 4, 5, 10, 12, 25, 30, 64, 77, 166, 200, 433, 522, 1132, 1365, 2962, 3572, 7753, 9350, 20296, 24477, 53134, 64080, 139105, 167762, 364180, 439205, 953434, 1149852, 2496121, 3010350, 6534928, 7881197, 17108662, 20633240, 44791057

Offset: 0

Paul Barry, Jul 26 2004

Partial sums of A097131.

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

Cf. A000032, A000045, A097131, A097136.

LinearRecurrence[{1,3,-3,-1,1},{1,2,4,5,10},40] (* Harvey P. Dale, Nov 12 2022 *)

G.f.: (1 + x - x^2 - 2*x^3)/((1 - 3*x^2 + x^4)*(1-x));
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - a(n-4) + a(n-5);
a(n) = 1 + (1/2 - sqrt(5)/2)^n*(1/2 - 3*sqrt(5)/10) - (sqrt(5)/2 - 1/2)^n*(3*sqrt(5)/10 + 1/2) + (-sqrt(5)/2 - 1/2)^n*(3*sqrt(5)/10 - 1/2) + (sqrt(5)/2 + 1/2)^n*(3*sqrt(5)/10 + 1/2);
a(2n) = 1 + 3*Fibonacci(2n) = A097136(n);
a(2n+1) = 1 + Fibonacci(2n) + Fibonacci(2n+2) = 1 + Lucas(2n).

cp's OEIS Frontend

A097135 a(0) = 1; for n>0, a(n) = 3*Fibonacci(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A097133 a(n) = 3*Fibonacci(n) + (-1)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A192908 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A097132 a(n) = Sum_{k=0..n} Fibonacci(k) + (-1)^k*Fibonacci(k-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula