A014445 -id:A014445 - cp's OEIS frontend

A103134 a(n) = Fibonacci(6n+4).

3, 55, 987, 17711, 317811, 5702887, 102334155, 1836311903, 32951280099, 591286729879, 10610209857723, 190392490709135, 3416454622906707, 61305790721611591, 1100087778366101931, 19740274219868223167, 354224848179261915075, 6356306993006846248183

Offset: 0

Creighton Dement, Jan 24 2005

Gives those numbers which are Fibonacci numbers in A103135.
Generally, for any sequence where a(0)= Fibonacci(p), a(1) = F(p+q) and Lucas(q)*a(1) +- a(0) = F(p+2q), then a(n) = L(q)*a(n-1) +- a(n-2) generates the following Fibonacci sequence: a(n) = F(q(n)+p). So for this sequence, a(n) = 18*a(n-1) - a(n-2) = F(6n+4): q=6, because 18 is the 6th Lucas number (L(0) = 2, L(1)=1); F(4)=3, F(10)=55 and F(16)=987 (F(0)=0 and F(1)=1). See Lucas sequence A000032. This is a special case where a(0) and a(1) are increasing Fibonacci numbers and Lucas(m)*a(1) +- a(0) is another Fibonacci. - Bob Selcoe, Jul 08 2013
a(n) = x + y where x and y are solutions to x^2 = 5*y^2 - 1. (See related sequences with formula below.) - Richard R. Forberg, Sep 05 2013

Colin Barker, Table of n, a(n) for n = 0..750
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (18,-1).
Index entries for sequences related to Chebyshev polynomials.

Subsequence of A033887.
Cf. A000032, A000045, A001906, A001519, A015448, A014445, A033888, A033889, A033890, A033891, A049310, A049660, A102312, A099100, A134490, A134491, A134492, A134493, A134494, A134495, A103134, A134497, A134498, A134499, A134500, A134501, A134502, A134503, A134504.
Cf. A103135.

[Fibonacci(6*n +4): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[6n+4], {n, 0, 30}]
LinearRecurrence[{18,-1},{3,55},20] (* Harvey P. Dale, Mar 29 2023 *)
Table[ChebyshevU[3*n+1, 3/2], {n, 0, 20}] (* Vaclav Kotesovec, May 27 2023 *)

a(n)=fibonacci(6*n+4) \\ Charles R Greathouse IV, Feb 05 2013

G.f.: (x+3)/(x^2-18*x+1).
a(n) = 18*a(n-1) - a(n-2) for n>1; a(0)=3, a(1)=55. - Philippe Deléham, Nov 17 2008
a(n) = A007805(n) + A075796(n), as follows from comment above. - Richard R. Forberg, Sep 05 2013
a(n) = ((15-7*sqrt(5)+(9+4*sqrt(5))^(2*n)*(15+7*sqrt(5))))/(10*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016
a(n) = S(3*n+1, 3) = 3*S(n,18) + S(n-1,18), with the Chebyshev S polynomials (A049310), S(-1, x) = 0, and S(n, 18) = A049660(n+1). - Wolfdieter Lang, May 08 2023

Edited by N. J. A. Sloane, Aug 10 2010

A274749 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-1,-2) (0,-1) or (-1,0) and new values introduced in order 0..2.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 8, 9, 4, 8, 22, 34, 27, 8, 16, 60, 133, 144, 81, 16, 32, 164, 518, 813, 610, 243, 32, 64, 448, 2017, 4554, 4967, 2584, 729, 64, 128, 1224, 7858, 25585, 40242, 30349, 10946, 2187, 128, 256, 3344, 30605, 143634, 327123, 355504, 185435, 46368

Offset: 1

R. H. Hardin, Jul 04 2016

Table starts
...1.....1......2........4..........8...........16............32
...1.....3......8.......22.........60..........164...........448
...2.....9.....34......133........518.........2017..........7858
...4....27....144......813.......4554........25585........143634
...8....81....610.....4967......40242.......327123.......2661918
..16...243...2584....30349.....355504......4190533......49475642
..32...729..10946...185435....3140840.....53680592.....920432562
..64..2187..46368..1133025...27748676....687685512...17123659885
.128..6561.196418..6922887..245154340...8809672678..318581114142
.256.19683.832040.42299477.2165891856.112857546696.5927090659144

			Some solutions for n=4 k=4
..0..1..0..1. .0..1..0..2. .0..1..2..0. .0..1..0..2. .0..1..0..2
..1..0..1..2. .1..0..2..0. .1..2..1..2. .1..2..1..0. .1..0..1..0
..0..1..2..1. .0..1..0..2. .2..1..0..1. .2..1..0..1. .0..2..0..1
..2..0..1..2. .2..0..2..0. .0..2..1..2. .1..0..1..0. .2..0..2..0

R. H. Hardin, Table of n, a(n) for n = 1..363

Column 1 is A000079(n-2).
Column 2 is A000244(n-1).
Column 3 is A014445.
Row 1 is A000079(n-2).
Row 2 is A028859(n-1).

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>2
k=2: a(n) = 3*a(n-1)
k=3: a(n) = 4*a(n-1) +a(n-2)
k=4: a(n) = 6*a(n-1) +a(n-2) -2*a(n-3) for n>4
k=5: a(n) = 9*a(n-1) -14*a(n-3) +10*a(n-4) -2*a(n-5) for n>6
k=6: [order 9] for n>11
k=7: [order 13] for n>15
Empirical for row n:
n=1: a(n) = 2*a(n-1) for n>2
n=2: a(n) = 2*a(n-1) +2*a(n-2)
n=3: a(n) = 2*a(n-1) +7*a(n-2) +2*a(n-3) -2*a(n-4)
n=4: [order 8]
n=5: [order 16] for n>17
n=6: [order 36] for n>38
n=7: [order 80] for n>83

A134504 a(n) = Fibonacci(7n + 6).

Original entry on oeis.org

8, 233, 6765, 196418, 5702887, 165580141, 4807526976, 139583862445, 4052739537881, 117669030460994, 3416454622906707, 99194853094755497, 2880067194370816120, 83621143489848422977, 2427893228399975082453

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (29,1).

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888, A033889, A033890, A033891, A102312, A099100, A134490, A134491, A134492, A134493, A134494, A134495, A103134, A134497, A134498, A134499, A134500, A134501, A134502, A134503, A134504.

[Fibonacci(7*n +6): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[7n+6], {n, 0, 30}]
LinearRecurrence[{29,1},{8,233},20] (* Harvey P. Dale, Jul 21 2021 *)

a(n)=fibonacci(7*n+6) \\ Charles R Greathouse IV, Jun 11 2015

G.f.: (-8-x) / (-1 + 29*x + x^2). - R. J. Mathar, Jul 04 2011
a(n) = A000045(A017053(n)). - Michel Marcus, Nov 08 2013
a(n) = 29*a(n-1) + a(n-2). - Wesley Ivan Hurt, Mar 15 2023

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

A134494 a(n) = Fibonacci(6n+2).

Original entry on oeis.org

1, 21, 377, 6765, 121393, 2178309, 39088169, 701408733, 12586269025, 225851433717, 4052739537881, 72723460248141, 1304969544928657, 23416728348467685, 420196140727489673, 7540113804746346429, 135301852344706746049, 2427893228399975082453

Offset: 0

Artur Jasinski, Oct 28 2007

Colin Barker, Table of n, a(n) for n = 0..750
Index entries for linear recurrences with constant coefficients, signature (18,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000045, A001906, A001519, A015448, A014445, A033887-A033891, A049310, A049660, A099100, A102312, A103134, A134490 - A134504.

[Fibonacci(6*n +2): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

seq( combinat[fibonacci](6*n+2),n=0..10) ; # R. J. Mathar, Apr 17 2011

Table[Fibonacci[6n+2], {n, 0, 30}]
Table[ChebyshevU[3*n, 3/2], {n, 0, 20}] (* Vaclav Kotesovec, May 27 2023 *)

a(n)=fibonacci(6*n+2) \\ Charles R Greathouse IV, Jun 11 2015

Vec((1+3*x)/(1-18*x+x^2) + O(x^100)) \\ Altug Alkan, Jan 24 2016

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: ( 1+3*x ) / ( 1-18*x+x^2 ).
a(n) = 3*A049660(n)+A049660(n+1). (End)
a(n) = A000045(A016933(n)). - Michel Marcus, Nov 07 2013
a(n) = ((5-3*sqrt(5)+(5+3*sqrt(5))*(9+4*sqrt(5))^(2*n)))/(10*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016
a(n) = S(3*n, 3) = S(n,18) + 3*S(n-1,18), with the Chebyshev S polynomials (A049310), S(-1, x) = 0, and S(n, 18) = A049660(n+1). - Wolfdieter Lang, May 08 2023

Index in definition corrected by T. D. Noe, Joerg Arndt, Apr 17 2011

A102310 Square array read by antidiagonals: Fibonacci(k*n).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 8, 8, 3, 5, 21, 34, 21, 5, 8, 55, 144, 144, 55, 8, 13, 144, 610, 987, 610, 144, 13, 21, 377, 2584, 6765, 6765, 2584, 377, 21, 34, 987, 10946, 46368, 75025, 46368, 10946, 987, 34, 55, 2584, 46368, 317811, 832040, 832040, 317811, 46368, 2584, 55

Offset: 1

Ralf Stephan, Jan 06 2005

			1,  1,   2,    3,     5, ...
1,  3,   8,   21,    55, ...
2,  8,  34,  144,   610, ...
3, 21, 144,  987,  6765, ...
5, 55, 610, 6765, 75025, ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. 2nd Edition. Addison-Wesley, Reading, MA, 1994, p. 294.

Freddy Barrera, Antidiagonals n = 1..50, flattened

Equals A000045(A003991(k, n)).
Columns include A000045, A001906, A014445, A033888, A102312.
Main diagonal is in A054783. Antidiagonal sums are in A102311.

/* As triangle */ [[Fibonacci(k*(n-k+1)): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jul 04 2019

Table[Fibonacci[k*(n-k+1)], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 10 2017 *)

F = fibonacci # A000045
def A(n, k):
    return F((n-1)*k)*F(k+1) + F((n-1)*k - 1)*F(k)
[A(n, k) for d in (1..10) for n, k in zip((d..1, step=-1), (1..d))] # Freddy Barrera, Jun 24 2019

For prime p, the formula holds: Fibonacci(k*p) = Fibonacci(p) * Sum_{i=0..floor((k-1)/2)} C(k-i-1, i)*(-1)^(i*p+i)*Lucas(p)^(k-2i-1).

A(n, k) = F((n-1)*k)*F(k+1) + F((n-1)*k-1)*F(k), where F(n) = A000045(n). - Freddy Barrera, Jun 24 2019

A134501 a(n) = Fibonacci(7n + 3).

Original entry on oeis.org

2, 55, 1597, 46368, 1346269, 39088169, 1134903170, 32951280099, 956722026041, 27777890035288, 806515533049393, 23416728348467685, 679891637638612258, 19740274219868223167, 573147844013817084101, 16641027750620563662096

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (29,1).

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888, A033889, A033890, A033891, A102312, A099100, A134490-A134495, A103134, A134497 - A134504.

[Fibonacci(7*n+3): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011

Mathematica
```
Table[Fibonacci[7n+3], {n, 0, 30}]
```

a(n)=fibonacci(7*n+3) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-2+3*x) / (-1 + 29*x + x^2).
a(n) = 2*A049667(n+1) - 3*A049667(n). (End)
a(n) = A000045(A017017(n)). - Michel Marcus, Nov 07 2013

Offset changed to 0 by Vincenzo Librandi, Apr 16 2011

A134502 a(n) = Fibonacci(7n + 4).

Original entry on oeis.org

3, 89, 2584, 75025, 2178309, 63245986, 1836311903, 53316291173, 1548008755920, 44945570212853, 1304969544928657, 37889062373143906, 1100087778366101931, 31940434634990099905, 927372692193078999176, 26925748508234281076009

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (29,1)

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888, A033889, A033890, A033891, A102312, A099100, A134490-A134495, A103134, A134497-A134504.

[Fibonacci(7*n +4): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Mathematica
```
Table[Fibonacci[7n+4], {n, 0, 30}]
```

a(n)=fibonacci(7*n+4) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-3-2*x) / (-1 + 29*x + x^2).
a(n) = 2*A049667(n) + 3*A049667(n+1). (End)
a(n) = A000045(A017029(n)). - Michel Marcus, Nov 07 2013

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

A163063 Lucas(3n+2) = Fibonacci(3n+1) + Fibonacci(3n+3).

Original entry on oeis.org

3, 11, 47, 199, 843, 3571, 15127, 64079, 271443, 1149851, 4870847, 20633239, 87403803, 370248451, 1568397607, 6643838879, 28143753123, 119218851371, 505019158607, 2139295485799, 9062201101803, 38388099893011

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

Binomial transform of A163062. Second binomial transform of A163114. Inverse binomial transform of A098648 without initial 1.

Nathaniel Johnston, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000032, A000045, A163062, A163114, A098648, A001077 (L(3*n)/L(2)), A048876 (L(3*n+1)).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((3+r)*(2+r)^n+(3-r)*(2-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

[Lucas(3*n+2): n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

with(combinat):A163063:=proc(n)return fibonacci(3*n+1) + fibonacci(3*n+3): end:seq(A163063(n), n=0..21); # Nathaniel Johnston, Apr 18 2011

Table[Fibonacci[3n + 1] + Fibonacci[3n + 3], {n, 0, 21}] (* Alonso del Arte, Nov 29 2010 *)
LinearRecurrence[{4,1},{3,11},30] (* Harvey P. Dale, Apr 14 2021 *)

Vec((3-x)/(1-4*x-x^2) + O(x^100)) \\ Altug Alkan, Dec 10 2015

a(n) = 4*a(n-1)+a(n-2) for n > 1; a(0) = 3, a(1) = 11.
G.f.: (3-x)/(1-4*x-x^2).
a(n) = A033887(n) + A014445(n+1).
a(n) = ((3+sqrt(5))*(2+sqrt(5))^n+(3-sqrt(5))*(2-sqrt(5))^n)/2.
a(n) = A000032(3*n+2), n>=0, (Lucas trisection). - Wolfdieter Lang, Mar 09 2011.
a(n) = 5*F(n)*F(n+1)*L(n+1) + L(n+2)*(-1)^n with F(n)=A000045(n) and L(n)=A000032(n). - J. M. Bergot, Dec 10 2015

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

A140413 a(2n) = A000045(6n) + 1, a(2n+1) = A000045(6n+3) - 1.

Original entry on oeis.org

1, 1, 9, 33, 145, 609, 2585, 10945, 46369, 196417, 832041, 3524577, 14930353, 63245985, 267914297, 1134903169, 4807526977, 20365011073, 86267571273, 365435296161, 1548008755921, 6557470319841, 27777890035289, 117669030460993, 498454011879265, 2111485077978049

Offset: 0

Paul Curtz, Jun 17 2008

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5,1).

Cf. A000045, A014445, A141325.

a:=[1,1,9];; for n in [4..30] do a[n]:=3*a[n-1]+5*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Jun 08 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)^2/((1+x)*(1-4*x-x^2)) )); // G. C. Greubel, Jun 08 2019

LinearRecurrence[{3,5,1},{1,1,9},30] (* or *) CoefficientList[Series[ (1-x)^2/((1+x)(1-4*x-x^2)),{x,0,30}],x] (* Harvey P. Dale, Jun 20 2011 *)

Vec((1-x)^2/((1+x)*(1-4*x-x^2)) + O(x^30)) \\ Colin Barker, Jun 06 2017

((1-x)^2/((1+x)*(1-4*x-x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 08 2019

a(n) = A141325(3*n) = (-1)^n + A014445(n).
a(n) = +3*a(n-1) +5*a(n-2) +a(n-3). - R. J. Mathar, Dec 17 2010
G.f.: (1-x)^2 / ( (1+x)*(1-4*x-x^2) ). - R. J. Mathar, Dec 17 2010
a(n) = ((-1)^n + (-(2-sqrt(5))^n + (2+sqrt(5))^n) / sqrt(5)). - Colin Barker, Jun 06 2017
a(n) = -A033887(n) + 2*Sum_{k=0..n} A033887(k)*(-1)^(n-k). - Yomna Bakr and Greg Dresden, Jun 03 2024

A353135 Primes having Fibonacci prime gaps to both neighbor primes.

Original entry on oeis.org

3, 5, 10007, 11777, 12163, 17291, 20443, 20477, 37781, 41333, 47743, 47777, 49991, 59887, 59921, 61091, 61331, 64271, 77417, 88177, 88609, 88643, 89363, 91639, 93337, 97073, 105863, 106453, 107507, 108463, 108497, 112363, 113383, 113717, 125149, 133631, 134293

Offset: 1

Alois P. Heinz, Apr 25 2022

nonn

			Prime 10007 is a term, the gap to the previous prime 9973 is 34 and the gap to the next prime 10009 is 2 and both gaps are Fibonacci numbers.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A000045, A014445, A353088, A353136, A353137.

f:= proc(n) option remember; (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:
q:= n-> isprime(n) and andmap(f, [n-prevprime(n), nextprime(n)-n]):
select(q, [$3..150000])[];

f[n_] := f[n] = With[{t = 5n^2}, IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4]];
q[n_] := PrimeQ[n] && f[n-NextPrime[n, -1]] && f[NextPrime[n]-n];
Select[Range[3, 150000], q] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A103134 a(n) = Fibonacci(6n+4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A274749 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-1,-2) (0,-1) or (-1,0) and new values introduced in order 0..2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A134504 a(n) = Fibonacci(7n + 6).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A134494 a(n) = Fibonacci(6n+2).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A102310 Square array read by antidiagonals: Fibonacci(k*n).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A134501 a(n) = Fibonacci(7n + 3).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A134502 a(n) = Fibonacci(7n + 4).

Views

Author

Keywords

Links

Crossrefs