A033888 -id:A033888 - cp's OEIS frontend

A221693 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some horizontal or antidiagonal neighbor, without consecutive moves in the same direction.

1, 3, 1, 6, 21, 1, 13, 152, 144, 1, 28, 1336, 3112, 987, 1, 60, 11036, 95076, 63676, 6765, 1, 129, 92660, 2627760, 7033429, 1302720, 46368, 1, 277, 774380, 74531185, 659020617, 520770016, 26650988, 317811, 1, 595, 6479664, 2098932856

Offset: 1

R. H. Hardin Jan 22 2013

Table starts
.1.........3............6..............13................28................60
.1........21..........152............1336.............11036.............92660
.1.......144.........3112...........95076...........2627760..........74531185
.1.......987........63676.........7033429.........659020617.......64509677154
.1......6765......1302720.......520770016......164441398718....55433846144035
.1.....46368.....26650988.....38558774282....41023076707062.47681199145323119
.1....317811....545221260...2854945130275.10233875156794211
.1...2178309..11154026300.211383822013331
.1..14930352.228186749552
.1.102334155
.1

			Some solutions for n=3 k=4
..0..3..0..1....0..4..0..1....2..0..2..0....0..3..2..0....2..1..0..1
..0..1..0..1....1..1..1..1....1..0..0..2....0..1..1..1....0..2..3..0
..1..2..3..0....1..1..0..1....1..3..0..1....2..0..1..1....1..1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..70

Column 2 is A033888

Row 1 is A002478

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

A337928 Numbers w such that (F(2n+1)^2, -F(2n)^2, -w) are primitive solutions of the Diophantine equation 2x^3 + 2y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).

Original entry on oeis.org

1, 5, 31, 209, 1429, 9791, 67105, 459941, 3152479, 21607409, 148099381, 1015088255, 6957518401, 47687540549, 326855265439, 2240299317521, 15355239957205, 105246380382911, 721369422723169, 4944339578679269, 33889007628031711, 232278713817542705

Offset: 0

XU Pingya, Sep 30 2020

			2*(F(5)^2)^3 + 2*(-F(4)^2)^3 + (-31)^3 = 2*(25)^3 + 2*(-9)^3 + (-31)^3 = 1, a(2) = 31.

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045, A000578, A056573, A081018, A337929.

Table[(2*Fibonacci[2n+1]^6 - 2*Fibonacci[2n]^6 - 1)^(1/3), {n, 0, 21}]
Table[(Fibonacci[2n+1]*Fibonacci[2n+2]- Fibonacci[2n]^2), {n, 0, 21}] (* Wolfgang Berndt, May 26 2023 *)
LinearRecurrence[{8,-8,1},{1,5,31},30] (* Harvey P. Dale, Dec 17 2023 *)

Vec((1 - 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)) + O(x^20)) \\ Colin Barker, Oct 01 2020

a(n) = (2*F(2*n+1)^6 - 2*F(2*n)^6 - 1)^(1/3).
From Colin Barker, Oct 01 2020: (Start)
G.f.: (1 - 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)).
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>2.
(End)
a(n) = 2*A081018(n) + 1. - Hugo Pfoertner, Oct 01 2020
a(n) = A064170(n+2) + A033888(n). - Flávio V. Fernandes, Jan 10 2021
a(n) = F(2*n+1)*F(2*n+2) - F(2*n)^2. - Wolfgang Berndt, May 26 2023
a(2*n-1) = 5 + 6*Sum_{k=1..n-1} F(8*k+1), a(2*n) = 1 + 6*Sum_{k=1..n} F(8*k-3). - XU Pingya, Jun 09 2024

A337929 Numbers w such that (F(2n-1)^2, -F(2n)^2, w) are primitive solutions of the Diophantine equation 2x^3 + 2y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).

Original entry on oeis.org

1, 11, 79, 545, 3739, 25631, 175681, 1204139, 8253295, 56568929, 387729211, 2657535551, 18215019649, 124847601995, 855718194319, 5865179758241, 40200540113371, 275538601035359, 1888569667134145, 12944449068903659, 88722573815191471, 608113567637436641

Offset: 1

XU Pingya, Sep 30 2020

			2*(F(3)^2)^3 + 2*(-F(4)^2)^3 + 11^3 = 2*4^3 + 2*(-9)^3 + 11^3 = 1, 11 is a term.

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045, A000578, A003482, A056573, A337928.

Table[(2*Fibonacci[2n]^6 - 2*Fibonacci[2n-1]^6 + 1)^(1/3), {n, 22}]
LinearRecurrence[{8,-8,1},{1,11,79},30] (* Harvey P. Dale, Aug 23 2021 *)

a(n) = (2*F(2*n)^6 - 2*F(2*n-1)^6 + 1)^(1/3).
From Colin Barker, Oct 01 2020: (Start)
G.f.: x*(1 + 3*x - x^2) / ((1 - x)*(1 - 7*x + x^2)).
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) for n>3.
(End)
a(n) = 2*A003482(n) + 1. - Hugo Pfoertner, Oct 01 2020
a(n) = A033888(n) - A064170(n+2). - Flávio V. Fernandes, Jan 10 2021

A003481 a(n) = 7*a(n-1) - a(n-2) + 5.

Original entry on oeis.org

2, 20, 143, 986, 6764, 46367, 317810, 2178308, 14930351, 102334154, 701408732, 4807526975, 32951280098, 225851433716, 1548008755919, 10610209857722, 72723460248140, 498454011879263, 3416454622906706, 23416728348467684, 160500643816367087, 1100087778366101930

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
John Riordan and N. J. A. Sloane, Correspondence, 1974.
S. M. Tanny and M. Zuker, On a unimodal sequence of binomial coefficients, Discrete Math. 9 (1974), 79-89.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A033888.

t = {2, 20}; Do[AppendTo[t, 7*t[[-1]] - t[[-2]] + 5], {n, 2, 30}] (* T. D. Noe, Oct 07 2013 *)
nxt[{a_,b_}]:={b,7b-a+5}; NestList[nxt,{2,20},30][[All,1]] (* Harvey P. Dale, Aug 11 2019 *)

G.f.: ( -2-4*x+x^2 ) / ( (x-1)*(x^2-7*x+1) ). - Simon Plouffe in his 1992 dissertation

a(n) = Fibonacci(4(n+1)) - 1 = A033888(n+1) - 1. - Ralf Stephan, Feb 24 2004, index corrected R. J. Mathar, Sep 18 2008

More terms from Ralf Stephan, Feb 24 2004

A254884 a(n) = Fibonacci(2n) + ((-1)^n-1)Fibonacci(n).

Original entry on oeis.org

0, -1, 3, 4, 21, 45, 144, 351, 987, 2516, 6765, 17533, 46368, 120927, 317811, 830820, 2178309, 5699693, 14930352, 39079807, 102334155, 267892404, 701408733, 1836254589, 4807526976, 12586118975, 32951280099, 86267178436, 225851433717, 591285701421, 1548008755920

Offset: 0

Peter Luschny, Mar 09 2015

Index entries for linear recurrences with constant coefficients, signature (3, 2, -9, 2, 3, -1).

Cf. A000032, A000045, A005522, A006172, A022112, A033888.

gf := x -> x/(x^2-3*x+1) + x/(x^2-x-1) + x/(x^2+x-1):
seq(coeff(series(gf(x),x,n+1),x,n), n=0..30);

LinearRecurrence[{4,-1,-11,11,1,-4,1}, {0,-1,3,4,21,45,144}, 31]
LinearRecurrence[{3, 2, -9, 2, 3, -1},{0, -1, 3, 4, 21, 45},31] (* Ray Chandler, Aug 03 2015 *)

A254884 = lambda n: fibonacci(2*n) + ((-1)^n-1)*fibonacci(n)
[A254884(n) for n in range(31)]

Let phi = (1+sqrt(5))/2, p(n) = phi^n - (-phi)^(-n) and FL(n) = 1 + (p(n-1) + p(n+1) + p(2*n-1)) / sqrt(5).
a(n) = FL(-n) - FL(n). By this definition a(n) is a doubly infinite sequence.
a(n) = -a(-n) for all n in Z.
a(n) = A006172(n) - A005522(n).
a(2*n) = A033888(n).
G.f.: x/(x^2-3*x+1) + x/(x^2-x-1) + x/(x^2+x-1).
a(n) = 4*a(n-1) - a(n-2) - 11*a(n-3) + 11*a(n-4) + a(n-5) - 4*a(n-6) + a(n-7).

A081072 Fibonacci(4n) + 3, or Fibonacci(2n+2)Lucas(2n-2).

Original entry on oeis.org

3, 6, 24, 147, 990, 6768, 46371, 317814, 2178312, 14930355, 102334158, 701408736, 4807526979, 32951280102, 225851433720, 1548008755923, 10610209857726, 72723460248144, 498454011879267, 3416454622906710, 23416728348467688, 160500643816367091, 1100087778366101934

Offset: 0

R. K. Guy, Mar 04 2003

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045 (Fibonacci numbers), A004187, A033888.

[Fibonacci(4*n)+3: n in [0..50]]; // Vincenzo Librandi, Apr 20 2011

with(combinat): for n from 0 to 40 do printf(`%d,`,fibonacci(4*n)+3) od: # James Sellers, Mar 05 2003

Fibonacci[4*Range[0, 25]] + 3 (* Paolo Xausa, Jul 03 2025 *)

makelist(fib(4*n)+3, n, 0, 30); /* Martin Ettl, Nov 11 2012 */

Vec((-3+18*x)/((x-1)*(x^2-7*x+1)) + O(x^30)) \\ Michel Marcus, Dec 23 2014

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
From R. J. Mathar, Sep 03 2010: (Start)
G.f.: ( -3+18*x ) / ( (x-1)*(x^2-7*x+1) ).
a(n) = 3+A033888(n). (End)
a(n) = (A004187(n)+1)*3. - Martin Ettl, Nov 11 2012

A134489 a(n) = Fibonacci(5*n + 2).

Original entry on oeis.org

1, 13, 144, 1597, 17711, 196418, 2178309, 24157817, 267914296, 2971215073, 32951280099, 365435296162, 4052739537881, 44945570212853, 498454011879264, 5527939700884757, 61305790721611591, 679891637638612258

Offset: 0

Artur Jasinski, Oct 28 2007

The o.g.f. of {F(m*n + 2)}_{n>=0}, for m = 1, 2, ..., is

G(m,x) = (1 + F(m - 2)*x) / (1 - L(m)*x + (-1)^m*x^2), with F = A000045 and F(-1) = 1, and L = A000032. - Wolfdieter Lang, Feb 06 2023

Index entries for linear recurrences with constant coefficients, signature (11,1).

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

[Fibonacci(5*n+2): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011

Table[Fibonacci[5n + 2], {n, 0, 30}]
LinearRecurrence[{11,1},{1,13},20] (* Harvey P. Dale, May 05 2022 *)

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-1-2*x) / (-1 + 11*x + x^2).
a(n) = 2*A049666(n) + A049666(n+1). (End)
a(n) = A000045(A016873(n)). - Michel Marcus, Nov 05 2013

cp's OEIS Frontend

A221693 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some horizontal or antidiagonal neighbor, without consecutive moves in the same direction.

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

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Magma

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

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Magma

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PARI

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

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Magma

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A337928 Numbers w such that (F(2n+1)^2, -F(2n)^2, -w) are primitive solutions of the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).

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Examples

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Programs

Mathematica

PARI

Formula

A337929 Numbers w such that (F(2*n-1)^2, -F(2*n)^2, w) are primitive solutions of the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).

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Keywords

Examples

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Crossrefs

Programs

Mathematica

Formula

A003481 a(n) = 7*a(n-1) - a(n-2) + 5.

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Mathematica

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Extensions

A254884 a(n) = Fibonacci(2*n) + ((-1)^n-1)*Fibonacci(n).

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A337928 Numbers w such that (F(2n+1)^2, -F(2n)^2, -w) are primitive solutions of the Diophantine equation 2x^3 + 2y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).

A337929 Numbers w such that (F(2n-1)^2, -F(2n)^2, w) are primitive solutions of the Diophantine equation 2x^3 + 2y^3 + z^3 = 1, where F(n) is the n-th Fibonacci number (A000045).

A254884 a(n) = Fibonacci(2n) + ((-1)^n-1)Fibonacci(n).

A081072 Fibonacci(4n) + 3, or Fibonacci(2n+2)Lucas(2n-2).