Yeshwant Shivrai Valaulikar - cp's OEIS frontend

A224508 a(n+2) = a(n+1) + a(n) + A*t^n, with A = 1 and t = -2.

0, 1, 2, 1, 7, 0, 23, -9, 78, -59, 275, -296, 1003, -1341, 3758, -5775, 14367, -24176, 55727, -99521, 218350, -405459, 861467, -1641144, 3414627, -6615125, 13576718, -26592839, 54092743, -106717824, 215810375, -427778361, 861773838, -1713488171, 3443252963

Offset: 0

Chandrakant N Phadte and Yeshwant Shivrai Valaulikar, Sep 18 2013

With these particular values, a(n) turns out to be Fibonacci(n) + (-2)^n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. N. Phadte and S. P. Pethe, On second order non homogeneous recurrence relation, Annales Mathematicae et Infomaticae, 41 (2013), 205-210.
Index entries for linear recurrences with constant coefficients, signature (-1,3,2).

Cf. A227200.

nn = 40; A = 1; t = -2; CoefficientList[Series[(x + x^2 (A - t))/((1 - x*t) (1 - x - x^2)), {x, 0, nn}], x] (* T. D. Noe, Sep 21 2013 *) (* or *)
LinearRecurrence[{-1,3,2}, {0,1,2}, 35] (* Georg Fischer, Jan 26 2022 *)

G.f.: (x+x^2*(A-t))/((1-x*t)*(1-x-x^2)).

A228789 a(n) = 2^L(n) + 1, where L(n) is A000032(n).

Original entry on oeis.org

5, 3, 9, 17, 129, 2049, 262145, 536870913, 140737488355329, 75557863725914323419137, 10633823966279326983230456482242756609, 803469022129495137770981046170581301261101496891396417650689

Offset: 0

M. Tamba and Yeshwant Shivrai Valaulikar, Sep 04 2013

Cf. A000032.

Table[1 + 2^LucasL[n], {n, 0, 11}] (* T. D. Noe, Sep 17 2013 *)

a(n+2) = a(n)*a(n+1) - a(n) - a(n+1) + 2.

Binet type formula: log_2(a(n)-1) = r^n - s^n where r and s are the roots of x^2-x-1 = 0. (This is true the definition).

A228778 a(n) = 2^Fibonacci(n) + 1.

Original entry on oeis.org

2, 3, 3, 5, 9, 33, 257, 8193, 2097153, 17179869185, 36028797018963969, 618970019642690137449562113, 22300745198530623141535718272648361505980417, 13803492693581127574869511724554050904902217944340773110325048447598593

Offset: 0

Yeshwant Shivrai Valaulikar and M. Tamba, Sep 04 2013

nonn

Cf. A000045, A000301, A063896.

a:= n-> 1 + 2^(<<0|1>, <1|1>>^n)[1,2]:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 12 2017

Table[2^Fibonacci[n] + 1, {n, 0, 13}] (* T. D. Noe, Sep 07 2013 *)

a(n+2) = a(n+1)*a(n) - a(n) - a(n+1) + 2, a(0)=2, a(1)=3.
Binet type formula: log_2(a(n)-1) = (1/sqrt(5)) * (r^n - s^n), where r and s are the roots of x^2-x-1. (this is true by definition).
a(n) = A000301(n) + 1 = A063896(n) + 2. - Alois P. Heinz, Aug 12 2017

A228763 a(n) = 2^L(n) - 1, where L(n) is the n-th Lucas number (A000032).

Original entry on oeis.org

3, 1, 7, 15, 127, 2047, 262143, 536870911, 140737488355327, 75557863725914323419135, 10633823966279326983230456482242756607, 803469022129495137770981046170581301261101496891396417650687

Offset: 0

Yeshwant Shivrai Valaulikar, Sep 03 2013

nonn

Cf. A000032, A063896.

[2^Lucas(n)-1: n in [0..15]]; // Bruno Berselli, Sep 03 2013

Table[2^LucasL[n] - 1, {n, 0, 15}] (* Bruno Berselli, Sep 03 2013 *)

a(n)=2^(fibonacci(n-1)+fibonacci(n+1))-1 \\ Charles R Greathouse IV, Sep 03 2013

a(n) = a(n-1) + a(n-2) + a(n-1) * a(n-2) for n>1, a(0)=3, a(1)=1.

More terms from Bruno Berselli, Sep 03 2013

cp's OEIS Frontend

User: Yeshwant Shivrai Valaulikar

A224508 a(n+2) = a(n+1) + a(n) + A*t^n, with A = 1 and t = -2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A228789 a(n) = 2^L(n) + 1, where L(n) is A000032(n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A228778 a(n) = 2^Fibonacci(n) + 1.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A228763 a(n) = 2^L(n) - 1, where L(n) is the n-th Lucas number (A000032).

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions