A022117 -id:A022117 - cp's OEIS frontend

A294116 Fibonacci sequence beginning 2, 21.

2, 21, 23, 44, 67, 111, 178, 289, 467, 756, 1223, 1979, 3202, 5181, 8383, 13564, 21947, 35511, 57458, 92969, 150427, 243396, 393823, 637219, 1031042, 1668261, 2699303, 4367564, 7066867, 11434431, 18501298, 29935729, 48437027, 78372756, 126809783, 205182539, 331992322, 537174861

Offset: 0

Bruno Berselli, Oct 23 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.
Subsequence of A047201, A047592, A113763.
Sequences of the type g(2,k;n): A118658 (k=0), A000032 (k=1), 2*A000045 (k=2,4), A020695 (k=3), A001060 (k=5), A022112 (k=6), A022113 (k=7), A294157 (k=8), A022114 (k=9), A022367 (k=10), A022115 (k=11), A022368 (k=12), A022116 (k=13), A022369 (k=14), A022117 (k=15), A022370 (k=16), A022118 (k=17), A022371 (k=18), A022119 (k=19), A022372 (k=20), this sequence (k=21), A022373 (k=22); A022374 (k=24); A022375 (k=26); A022376 (k=28), A190994 (k=29), A022377 (k=30); A022378 (k=32).

a0:=2; a1:=21; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]];

Mathematica
```
LinearRecurrence[{1, 1}, {2, 21}, 40]
```

Vec((2 + 19*x)/(1 - x - x^2) + O(x^40)) \\ Colin Barker, Oct 25 2017

a = BinaryRecurrenceSequence(1, 1, 2, 21)
print([a(n) for n in range(38)]) # Peter Luschny, Oct 25 2017

G.f.: (2 + 19*x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2).
Let g(r,s;n) be the n-th generalized Fibonacci number with initial values r, s. We have:
a(n) =     Lucas(n) + g(0,20;n), see A022354;
a(n) = Fibonacci(n) + g(2,20;n), see A022372;
a(n) =  2*g(1,21;n) - g(0,21;n);
a(n) =     g(1,k;n) + g(1,21-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) + 21*Fibonacci(n).
Sum_{j=0..n} a(j) = a(n+2) - 21.
a(n) = (2^(-n)*((1-sqrt(5))^n*(-20+sqrt(5)) + (1+sqrt(5))^n*(20+sqrt(5)))) / sqrt(5). - Colin Barker, Oct 25 2017

A022387 Fibonacci sequence beginning 4, 30.

Original entry on oeis.org

4, 30, 34, 64, 98, 162, 260, 422, 682, 1104, 1786, 2890, 4676, 7566, 12242, 19808, 32050, 51858, 83908, 135766, 219674, 355440, 575114, 930554, 1505668, 2436222, 3941890, 6378112, 10320002, 16698114, 27018116, 43716230, 70734346, 114450576, 185184922, 299635498, 484820420

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1, 1).

Equals 2 * A022117.

List([0..40],n->4*Fibonacci(n+2)+22*Fibonacci(n)); # Muniru A Asiru, Mar 03 2018

I:=[4,30]; [n le 2 select I[n] else Self(n-1) + Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 17 2012

[4*Fibonacci(n+2) + 22*Fibonacci(n): n in [0..40]]; // G. C. Greubel, Mar 01 2018

with(combinat,fibonacci):  seq(4*fibonacci(n+2)+22*fibonacci(n),n=0..35); # Muniru A Asiru, Mar 03 2018

LinearRecurrence[{1, 1}, {4, 30}, 30] (* Harvey P. Dale, Oct 16 2012 *)
CoefficientList[Series[(4 + 26 * x)/(1 - x - x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 17 2012 *)
Table[4 * Fibonacci[n + 2] + 22 * Fibonacci[n], {n, 0, 50}] (* G. C. Greubel, Mar 02 2018 *)

for(n=0, 40, print1(4*fibonacci(n+2) + 22*fibonacci(n), ", ")) \\ G. C. Greubel, Mar 01 2018

G.f.: (4+26*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n) = 4*Fibonacci(n+2) + 22*fibonacci(n) = 4*Fibonacci(n-1) + 30*Fibonacci(n). - G. C. Greubel, Mar 02 2018

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A294116 Fibonacci sequence beginning 2, 21.

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A022387 Fibonacci sequence beginning 4, 30.

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