A055267
a(n) = 3*a(n-1) - a(n-2) with a(0)=1, a(1)=7.
Original entry on oeis.org
1, 7, 20, 53, 139, 364, 953, 2495, 6532, 17101, 44771, 117212, 306865, 803383, 2103284, 5506469, 14416123, 37741900, 98809577, 258686831, 677250916, 1773065917, 4641946835, 12152774588, 31816376929, 83296356199, 218072691668, 570921718805, 1494692464747
Offset: 0
- A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
- E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
- Tanya Khovanova, Recursive Sequences
- Index entries for linear recurrences with constant coefficients, signature (3,-1).
-
List([0..30], n-> Fibonacci(2*n+2) +4*Fibonacci(2*n) ); # G. C. Greubel, Jan 17 2020
-
[5*Fibonacci(2*n) + Fibonacci(2*n+1): n in [0..30]]; // Vincenzo Librandi, Dec 25 2018
-
with(combinat); seq(fibonacci(2*n+2) +4*fibonacci(2*n), n=0..30); # G. C. Greubel, Jan 17 2020
-
Table[5*Fibonacci[2n] + Fibonacci[2n+1], {n, 0, 30}]
Table[4*Fibonacci[2n-1] + 3*LucasL[2n-1], {n, 0, 30}] (* Rigoberto Florez, Dec 24 2018 *)
LinearRecurrence[{3,-1}, {1,7}, 30] (* Vincenzo Librandi, Dec 25 2018 *)
nxt[{a_,b_}]:={b,3b-a}; NestList[nxt,{1,7},30][[;;,1]] (* Harvey P. Dale, Mar 23 2025 *)
-
Vec((1+4*x)/(1-3*x+x^2) + O(x^40)) \\ Michel Marcus, Sep 06 2017
-
[fibonacci(2*n+2) +4*fibonacci(2*n) for n in (0..30)] # G. C. Greubel, Jan 17 2020
A141751
Triangle, read by rows, where T(n,k) = [T(n-1,k-1)*T(n-1,k) + 1]/T(n-2,k-1) for 0=0 and T(n,0) = Fibonacci(2*n-1) for n>=1.
Original entry on oeis.org
1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 13, 13, 8, 4, 1, 34, 34, 21, 11, 5, 1, 89, 89, 55, 29, 14, 6, 1, 233, 233, 144, 76, 37, 17, 7, 1, 610, 610, 377, 199, 97, 45, 20, 8, 1, 1597, 1597, 987, 521, 254, 118, 53, 23, 9, 1, 4181, 4181, 2584, 1364, 665, 309, 139, 61, 26, 10, 1
Offset: 0
Generating rule.
Given nonzero elements W, X, Y, Z, relatively arranged like so:
.. W .....
.. X Y ...
.... Z ...
then Z = (X*Y + 1)/W.
Triangle begins:
1;
1, 1;
2, 2, 1;
5, 5, 3, 1;
13, 13, 8, 4, 1;
34, 34, 21, 11, 5, 1;
89, 89, 55, 29, 14, 6, 1;
233, 233, 144, 76, 37, 17, 7, 1;
610, 610, 377, 199, 97, 45, 20, 8, 1;
1597, 1597, 987, 521, 254, 118, 53, 23, 9, 1;
4181, 4181, 2584, 1364, 665, 309, 139, 61, 26, 10, 1; ...
-
T(n,k)=if(n
-
T(n,k)=fibonacci(2*(n-k))*k+fibonacci(2*(n-k)-1)
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
A192916
Constant term in the reduction by (x^2 -> x+1) of the polynomial C(n)*x^n, where C=A022095.
Original entry on oeis.org
1, 0, 6, 11, 34, 84, 225, 584, 1534, 4011, 10506, 27500, 72001, 188496, 493494, 1291979, 3382450, 8855364, 23183649, 60695576, 158903086, 416013675, 1089137946, 2851400156, 7465062529, 19543787424, 51166299750, 133955111819, 350699035714, 918141995316
Offset: 0
-
F:=Fibonacci;; List([0..30], n-> F(2*n) +2*F(n)*F(n-1) +(-1)^n); # G. C. Greubel, Jul 28 2019
-
F:=Fibonacci; [F(2*n) +2*F(n)*F(n-1) +(-1)^n: n in [0..30]]; // G. C. Greubel, Jul 28 2019
-
(* First program *)
q = x^2; s = x + 1; z = 28;
p[0, x_]:= 1; p[1, x_]:= 5 x;
p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^2;
Table[Expand[p[n, x]], {n, 0, 7}]
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}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192914 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* see A192878 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[2*n] +2*F[n]*F[n-1] +(-1)^n, {n,0,30}]] (* G. C. Greubel, Jul 28 2019 *)
-
a(n) = round((2^(-n)*(7*(-2)^n+(3+sqrt(5))^n*(-1+2*sqrt(5))-(3-sqrt(5))^n*(1+2*sqrt(5))))/5) \\ Colin Barker, Oct 01 2016
-
Vec((1+4*x^2-2*x)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Oct 01 2016
-
vector(30, n, n--; f=fibonacci; f(2*n) +2*f(n)*f(n-1) +(-1)^n) \\ G. C. Greubel, Jul 28 2019
-
f=fibonacci; [f(2*n) +2*f(n)*f(n-1) +(-1)^n for n in (0..30)] # G. C. Greubel, Jul 28 2019
Showing 1-3 of 3 results.
Comments