A190994 -id:A190994 - cp's OEIS frontend

A190995 Fibonacci sequence beginning 9, 7.

9, 7, 16, 23, 39, 62, 101, 163, 264, 427, 691, 1118, 1809, 2927, 4736, 7663, 12399, 20062, 32461, 52523, 84984, 137507, 222491, 359998, 582489, 942487, 1524976, 2467463, 3992439, 6459902, 10452341, 16912243, 27364584, 44276827, 71641411, 115918238, 187559649

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jun 07 2011

From Wajdi Maaloul, Jun 20 2022: (Start)
For n>0, 2*a(n) is the number of ways to tile this figure below with squares and dominoes (a strip of length n+1 that begins with a length 3 vertical strip and length 4 one).
     _
   ||
  |||
  |||_______     _
  |||_|||_|...|_|
(End)

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

Cf. A000032, A000045, A190994.

[n le 2 select 11-2*n else Self(n-1)+Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

a:= n-> (<<0|1>, <1|1>>^n. <<9, 7>>)[1, 1]:
seq(a(n), n=0..36);  # Alois P. Heinz, Oct 26 2022

Mathematica
```
LinearRecurrence[{1, 1}, {9, 7}, 100]
```

a(n)=7*fibonacci(n)+9*fibonacci(n-1) \\ Charles R Greathouse IV, Jun 08 2011

[7*fibonacci(n) + 9*fibonacci(n-1) for n in range(51)] # G. C. Greubel, Oct 26 2022

a(n) = ((9+sqrt(5))/2)*((1+sqrt(5))/2)^n + ((9-sqrt(5))/2)*((1-sqrt(5))/2)^n. - Antonio Alberto Olivares
G.f.: (9-2*x)/(1-x-x^2). - Colin Barker, Jan 11 2012
a(n) = 7*Fibonacci(n) + 9*Fibonacci(n-1) = 7*Fibonacci(n+1) + 2*Fibonacci(n-1) = 7*Lucas(n) - 5*Fibonacci(n-1) for n>0. - Wajdi Maaloul, Jun 20 2022

A294116 Fibonacci sequence beginning 2, 21.

Original entry on oeis.org

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

A190996 Fibonacci sequence beginning 10, 7.

Original entry on oeis.org

10, 7, 17, 24, 41, 65, 106, 171, 277, 448, 725, 1173, 1898, 3071, 4969, 8040, 13009, 21049, 34058, 55107, 89165, 144272, 233437, 377709, 611146, 988855, 1600001, 2588856, 4188857, 6777713, 10966570, 17744283, 28710853, 46455136, 75165989, 121621125

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jun 07 2011

For n >= 5, the number a(n-3) is the dimension of a commutative Hecke algebra of affine type D_n with independent parameters. See Theorem 1.4, Corollary 1.5, and the table on page 524 in the link "Hecke algebras with independent parameters". - Jia Huang, Jan 20 2019
From Greg Dresden and Yiming Wu, Sep 10 2023: (Start)
For n >= 3, a(n) is the number of ways to tile this shape of length n+2 with squares and dominos:
                        
 ||___________________||
|||_|||_|||_|||_|||
  ||                   ||. (End)
For n >= 3, a(n) is the number of edge covers of the kayak paddle graphs KP(3,3,n-3), where we interpret KP(3,3,0) as two C_3's with one common vertex. - Feryal Alayont, Sep 28 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jia Huang, Hecke algebras with independent parameters, arXiv preprint arXiv:1405.1636 [math.RT], 2014; Journal of Algebraic Combinatorics 43 (2016) 521-551.
Eric Weisstein's World of Mathematics, Kayak Paddle Graph.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A190994, A190995.

[n le 2 select 13-3*n else Self(n-1)+Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

seq(coeff(series((10-3*x)/(1-x-x^2),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Jan 22 2019

Mathematica
```
LinearRecurrence[{1, 1}, {10, 7}, 100]
```

a(n)=7*fibonacci(n)+10*fibonacci(n-1) \\ Charles R Greathouse IV, Jun 08 2011

[7*fibonacci(n+1) +3*fibonacci(n-1) for n in range(51)] # G. C. Greubel, Oct 26 2022

a(n) = (5 + 2*sqrt(5)/5)*((1 + sqrt(5))/2)^n + (5 - 2*sqrt(5)/5)*((1 - sqrt(5))/2)^n. - Antonio Alberto Olivares, Jun 07 2011
a(n) = 7*Fibonacci(n) + 10*Fibonacci(n-1). - Charles R Greathouse IV, Jun 08 2011
G.f.: (10-3*x)/(1-x-x^2). - Colin Barker, Jan 11 2012
a(n) = 4*Fibonacci(n+1) + 3*LucasL(n). - G. C. Greubel, Oct 26 2022
a(n) = A000285(n)+3*A000285(n-1). - Feryal Alayont, Sep 28 2024
a(2*n) = A000285(n)^2 + A000285(n-1)^2. - Greg Dresden, Feb 28 2025

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A190995 Fibonacci sequence beginning 9, 7.

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A294116 Fibonacci sequence beginning 2, 21.

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A190996 Fibonacci sequence beginning 10, 7.

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Magma

Maple

Mathematica

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