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

0,1

References

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

Links

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).

Formula

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

Mathematica

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

Prog

(Magma) a0:=2; a1:=21; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]];
(PARI) Vec((2 + 19*x)/(1 - x - x^2) + O(x^40)) \\ Colin Barker, Oct 25 2017
(Sage)
a = BinaryRecurrenceSequence(1, 1, 2, 21)
print([a(n) for n in range(38)]) # Peter Luschny, Oct 25 2017

Crossrefs

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).

Sequence in context: A110301 A233135 A322925 * A041513 A135053 A042565

Adjacent sequences: A294113 A294114 A294115 * A294117 A294118 A294119

Keyword

nonn,easy

Author

Bruno Berselli, Oct 23 2017

Status

approved