A294116 -id:A294116 - cp's OEIS frontend

A294157 Fibonacci sequence beginning 2, 8.

2, 8, 10, 18, 28, 46, 74, 120, 194, 314, 508, 822, 1330, 2152, 3482, 5634, 9116, 14750, 23866, 38616, 62482, 101098, 163580, 264678, 428258, 692936, 1121194, 1814130, 2935324, 4749454, 7684778, 12434232, 20119010, 32553242, 52672252, 85225494, 137897746, 223123240

Offset: 0

Bruno Berselli, Oct 24 2017

Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications (2008), page 24 (formula 8).

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).

Cf. A000032, A000045, A000285.

Similar sequences listed in A294116.

a0:=2; a1:=8; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]];

f:= gfun:-rectoproc({a(n)=a(n-1)+a(n-2),a(0)=2,a(1)=8},a(n),remember):
map(f, [$0..100]); # Robert Israel, Oct 24 2017

Mathematica
```
LinearRecurrence[{1, 1}, {2, 8}, 40]
```

Vec(2*(1 + 3*x)/(1 - x - x^2) + O(x^40)) \\ Colin Barker, Oct 25 2017

a = BinaryRecurrenceSequence(1, 1, 2, 8)
print([a(n) for n in range(38)]) # Peter Luschny, Oct 25 2017

G.f.: 2*(1 + 3*x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2).
a(n) = 2*A000285(n).
Let g(r,s;n) be the n-th generalized Fibonacci number with initial values r, s. We have:
a(n) =     Lucas(n) + g(0,7;n), see A022090;
a(n) = Fibonacci(n) + g(2,7;n), see A022113;
a(n) =   2*g(1,8;n) - g(0,8;n);
a(n) =     g(1,k;n) + g(1,8-k;n) for all k in Z.
a(h+k) = a(h)*Fibonacci(k-1) + a(h+1)*Fibonacci(k) for all h, k in Z (see S. Vajda in References section). For h=0 and k=n:
a(n) = 2*Fibonacci(n-1) + 8*Fibonacci(n).
Sum_{j=0..n} a(j) = a(n+2) - 8.
a(n) = (2^(-n)*((1-sqrt(5))^n*(-7+sqrt(5)) + (1+sqrt(5))^n*(7+sqrt(5)))) / sqrt(5). - Colin Barker, Oct 25 2017

A347351 Triangle read by rows: T(n,k) is the number of links of length k in a set of all necklaces A000358 of length n, 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 4, 2, 0, 1, 5, 1, 1, 0, 1, 6, 4, 2, 1, 0, 1, 7, 3, 2, 1, 1, 0, 1, 8, 8, 3, 3, 1, 1, 0, 1, 9, 8, 7, 3, 2, 1, 1, 0, 1, 10, 18, 9, 5, 4, 2, 1, 1, 0, 1, 11, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1, 12, 40, 24, 16, 8, 6, 3, 2, 1, 1, 0, 1, 13, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1

Offset: 0

Maxim Karimov and Vladislav Sulima, Aug 28 2021

Definitions:
1. A link is any 0 in any necklace from A000358 and all 1s following this 0 in this necklace to right until another 0 is encountered.
2. Length of the link is the number of elements in the link.
Sum of all elements n-row is Fibonacci(n-1)+n iff n=1 or n=p (follows from the identity for the sum of the Fibonacci numbers and the formula for the triangle T(n,k)).

			For k > 0:
   n\k |  1   2   3   4   5   6   7   8   9  10  ...
  -----+---------------------------------------
   1   |  1
   2   |  2   1
   3   |  3   0   1
   4   |  4   2   0   1
   5   |  5   1   1   0   1
   6   |  6   4   2   1   0   1
   7   |  7   3   2   1   1   0   1
   8   |  8   8   3   3   1   1   0   1
   9   |  9   8   7   3   2   1   1   0   1
  10   | 10  18   9   5   4   2   1   1   0   1
  ...
If we continue the calculation for nonpositive k, we get a table in which each row is a Fibonacci sequence, in which term(0) = A113166, term(1) = A034748.
For k <= 0:
   n\k |  0   -1   -2   -3   -4   -5   -6   -7   -8   -9 ...
  -----+------------------------------------------------
   1   |  0    1    1    2    3    5    8   13   21   34 ... A000045
   2   |  1    2    3    5    8   13   21   34   55   89 ... A000045
   3   |  1    4    5    9   14   23   37   60   97  157 ... A000285
   4   |  3    6    9   15   24   39   63  102  165  267 ... A022086
   5   |  3    9   12   21   33   54   87  141  228  369 ... A022379
   6   |  8   14   22   36   58   94  152  246  398  644 ... A022112
   7   |  8   19   27   46   73  119  192  311  503  814 ... A206420
   8   | 17   30   47   77  124  201  325  526  851 1377 ... A022132
   9   | 23   44   67  111  178  289  467  756 1223 1979 ... A294116
  10   | 41   68  109  177  286  463  749 1212 1961 3173 ... A022103
  ...

Cf. A000010, A000027, A000045, A000358, A113166, A034748.

function [res] = calcLinks(n,k)
if k==1
    res=n;
else
    d=divisors(n);
    res=0;
    for i=1:length(d)
        if d (i) >= k
            res=res+eulerPhi(n/d(i))*fiboExt(d(i)-k-1);
        end
    end
end
function [s] = fiboExt(m) % extended fibonacci function (including negative arguments)
m=sym(m); % for large fibonacci numbers
if m>=0 || mod(m,2)==1
    s=fibonacci(abs(m));
else
    s=fibonacci(abs(m))*(-1);
end

T(n, k) = if (k==1, n, sumdiv(n, d, if (d>=k, eulerphi(n/d)*fibonacci(d-k-1)))); \\ Michel Marcus, Aug 29 2021

If k=1, T(n,k)=n, otherwise T(n,k) = Sum_{d>=k, d|n} Phi(n/d)*Fibonacci(d-k-1), where Phi=A000010.

cp's OEIS Frontend

A294157 Fibonacci sequence beginning 2, 8.

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