A022378 -id:A022378 - 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

A030139 a(n+1) = sum of digits of (a(n) + a(n-1)).

Original entry on oeis.org

1, 4, 5, 9, 5, 5, 1, 6, 7, 4, 2, 6, 8, 5, 4, 9, 4, 4, 8, 3, 2, 5, 7, 3, 1, 4, 5, 9, 5, 5, 1, 6, 7, 4, 2, 6, 8, 5, 4, 9, 4, 4, 8, 3, 2, 5, 7, 3, 1, 4, 5, 9, 5, 5, 1, 6, 7, 4, 2, 6, 8, 5, 4, 9, 4, 4, 8, 3, 2, 5, 7, 3, 1, 4, 5, 9, 5, 5, 1, 6, 7, 4, 2, 6, 8, 5, 4, 9, 4, 4, 8, 3, 2, 5, 7, 3, 1, 4, 5

Offset: 0

N. J. A. Sloane

This is also the digital root of A022378, Fibonacci starting with 2 and 32, beginning from the 20th term 2: [2, 5, 7, 3, 1, 4, 5, 9, 5, 5, 1, 6, 7, 4, 2, 6, 8, 5, 4, 9, 4, 4, 8, 3.] Like the digital root of A000045, sequence is period 24, and likewise, its period also adds up to 117.Peter M. Chema, Apr 28 2016

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

A[0]:= 1: A[1]:= 4:
for i from 2 to 100 do
  t:= A[i-2]+A[i-1];
  A[i]:=(t + 9*(t mod 10))/10;
od:
seq(A[i],i=0..100); # Robert Israel, Apr 28 2016

a[0] = 1; a[1] = 4; a[n_] := a[n] = Total@ IntegerDigits[a[n - 1] + a[n - 2]]; Table[a@ n, {n, 0, 120}] (* Michael De Vlieger, Apr 28 2016 *)
nxt[{a_,b_}]:={b,Total[IntegerDigits[a+b]]}; NestList[nxt,{1,4},100][[All,1]] (* or *) PadRight[{},100,{1,4,5,9,5,5,1,6,7,4,2,6,8,5,4,9,4,4,8,3,2,5,7,3}] (* Harvey P. Dale, Apr 27 2018 *)

a(n)=n=n%24;my(a=3,b=1);while(n,[a,b]=[b,sumdigits(a+b)]; n--);b \\ Charles R Greathouse IV, Apr 28 2016

G.f.: (1+4*x+5*x^2+9*x^3+5*x^4+5*x^5+x^6+6*x^7+7*x^8+4*x^9+2*x^10+6*x^11+8*x^12+5*x^13+4*x^14+9*x^15+4*x^16+4*x^17+8*x^18+3*x^19+2*x^20+5*x^21+7*x^22+3*x^23)/(1-x^24). - Robert Israel, Apr 28 2016

cp's OEIS Frontend

A294116 Fibonacci sequence beginning 2, 21.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula