A111569 -id:A111569 - cp's OEIS frontend

A111573 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 0,1,3,3.

0, 1, 3, 3, 4, 8, 14, 21, 33, 55, 90, 144, 232, 377, 611, 987, 1596, 2584, 4182, 6765, 10945, 17711, 28658, 46368, 75024, 121393, 196419, 317811, 514228, 832040, 1346270, 2178309, 3524577, 5702887, 9227466, 14930352, 24157816, 39088169

Offset: 0

Creighton Dement, Aug 10 2005

See comment and FAMP code for A111569.

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

Cf. A001638, A111569, A111570, A111571, A111572, A111574, A111575, A111576.

Table[Fibonacci[n + 1] - Cos[n*Pi/2], {n, 0, 40}] (* Greg Dresden, Oct 16 2021 *)

G.f.: -x*(1+2*x)/((x^2+x-1)*(x^2+1)).
a(n) = A056594(n+3) + A000045(n+1). - R. J. Mathar, Nov 10 2009
From Greg Dresden, Jan 15 2024: (Start)
a(2*n) = Fibonacci(n)*Lucas(n+1);
a(2*n+1) = Fibonacci(2*n+1). (End)

Name clarified by Robert C. Lyons, Feb 06 2025

A111570 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 2,5,4,7.

Original entry on oeis.org

2, 5, 4, 7, 14, 23, 34, 55, 92, 149, 238, 385, 626, 1013, 1636, 2647, 4286, 6935, 11218, 18151, 29372, 47525, 76894, 124417, 201314, 325733, 527044, 852775, 1379822, 2232599, 3612418, 5845015, 9457436, 15302453, 24759886

Offset: 0

Creighton Dement, Aug 10 2005

See comment and FAMP code for A111569.

Floretion Algebra Multiplication Program, FAMP Code: 1vesseq[B+H] with B = - .25'i + .25'j - .25i' + .25j' + k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and H = + .75'ii' + .75'jj' + .75'kk' + .75e

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

Cf. A001638, A111569, A111571, A111572, A111573, A007598, A000045.

G.f.: (-2-3*x+x^2-x^3)/((1+x^2)*(x^2+x-1)).

Name clarified by Robert C. Lyons, Feb 06 2025

A111571 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 1,1,-2,-1.

Original entry on oeis.org

1, 1, -2, -1, 1, 0, -3, -3, -2, -5, -11, -16, -23, -39, -66, -105, -167, -272, -443, -715, -1154, -1869, -3027, -4896, -7919, -12815, -20738, -33553, -54287, -87840, -142131, -229971, -372098, -602069, -974171, -1576240, -2550407, -4126647, -6677058, -10803705

Offset: 0

Creighton Dement, Aug 10 2005

See comment and FAMP code for A111569.

Floretion Algebra Multiplication Program, FAMP Code: 1jesseq[B+H] with B = - .25'i + .25'j - .25i' + .25j' + k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and H = + .75'ii' + .75'jj' + .75'kk' + .75e

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

Cf. A001638, A111569, A111570, A111572, A111573, A007598, A000045.

LinearRecurrence[{1,0,1,1},{1,1,-2,-1},40] (* or *) CoefficientList[ Series[ (-1+3*x^2)/((1+x^2)*(x^2+x-1)),{x,0,40}],x]  (* Harvey P. Dale, Apr 23 2011 *)

G.f.: (-1+3*x^2)/((1+x^2)*(x^2+x-1)).

Name clarified by Robert C. Lyons, Feb 06 2025

A111572 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms -1,3,2,1.

Original entry on oeis.org

-1, 3, 2, 1, 3, 8, 11, 15, 26, 45, 71, 112, 183, 299, 482, 777, 1259, 2040, 3299, 5335, 8634, 13973, 22607, 36576, 59183, 95763, 154946, 250705, 405651, 656360, 1062011, 1718367, 2780378, 4498749, 7279127, 11777872, 19056999, 30834875, 49891874, 80726745

Offset: 0

Creighton Dement, Aug 10 2005

See comment and FAMP code for A111569.
Floretion Algebra Multiplication Program, FAMP Code: 4ibaseseq[B+H] with B = - .25'i + .25'j - .25i' + .25j' + k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and H = + .75'ii' + .75'jj' + .75'kk' + .75e
From Greg Dresden and Jiaqi Wang, Jun 24 2023: (Start)
For n >= 5, a(n) is also the number of ways to tile this "central staircase" figure of length n-2 with squares and dominoes. This is the picture for length 9; there are a(11)=112 ways to tile it:
             _
   _______|_|___
  |||_|||_|||_|
        |_|            (End)

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

Cf. A001638, A111569, A111570, A111571, A111573, A111574, A111575, A111576.

Cf. A000032, A000045.

G.f.: (1-4*x+x^2)/((1+x^2)*(x^2+x-1))
From Greg Dresden and Jiaqi Wang, Jun 24 2023: (Start)
a(2*n) = F(n+1)*L(n-1) + F(n)*F(n-1),
a(2*n+1) = F(n+1)*(F(n+1) + 2*F(n-1)), for F(n) and L(n) the Fibonacci and Lucas numbers.
(End)

Name clarified by Robert C. Lyons, Feb 06 2025

A260259 a(n) = F(n)*F(n+1) - (-1)^n, where F = A000045.

Original entry on oeis.org

-1, 2, 1, 7, 14, 41, 103, 274, 713, 1871, 4894, 12817, 33551, 87842, 229969, 602071, 1576238, 4126649, 10803703, 28284466, 74049689, 193864607, 507544126, 1328767777, 3478759199, 9107509826, 23843770273, 62423800999, 163427632718, 427859097161, 1120149658759

Offset: 0

Bruno Berselli, Oct 31 2015

Primes in sequence for n = 1, 3, 5, 6, 9, 24, 42, 48, 53, 71, 86, 102, 138, 182, 302, 438, 506, 926, ...

Bruno Berselli, Table of n, a(n) for n = 0..500
A. Bremner, R. Høibakk, D. Lukkassen, Crossed ladders and Euler’s quartic, Annales Mathematicae et Informaticae, 36 (2009) pp. 29-41. See p. 33.
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

First bisection of A111569.
Cf. A226205: numbers of the form F(n)*F(n+1)+(-1)^n.
Cf. A000045, A001654, A003482, A059929, A089508 (first bisection, without -1), A206351.

[Fibonacci(n)*Fibonacci(n+1)-(-1)^n: n in [0..30]];

with(combinat): A260259:=n->fibonacci(n)*fibonacci(n+1)-(-1)^n: seq(A260259(n), n=0..50); # Wesley Ivan Hurt, Feb 04 2017

Table[Fibonacci[n] Fibonacci[n + 1] - (-1)^n, {n, 0, 30}]

makelist(fib(n)*fib(n+1)-(-1)^n,n,0,30);

for(n=0, 30, print1(fibonacci(n)*fibonacci(n+1)-(-1)^n", "));

a(n) = round((2^(-1-n)*(-3*(-1)^n*2^(2+n)-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n))/5) \\ Colin Barker, Sep 29 2016

Vec(-(1-4*x+x^2)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 29 2016

[fibonacci(n)*fibonacci(n+1)-(-1)^n for n in (0..30)]

G.f.: (-1 + 4*x - x^2)/((1 + x)*(1 - 3*x + x^2)).
a(n) = -a(-n-1) = 2*a(n-1) + 2*a(n-2) - a(n-3) for all n in Z.
a(n) = F(n+2)^2 - 2*F(n+1)^2.
a(n) = A059929(n) - A059929(n-1) with A059929(-1)=1.
a(n) = -A001654(n+1) + 4*A001654(n) - A001654(n-1).
a(n) = A206351((n+2)/2)-2 for even n; a(n) = A003482((n-1)/2)+2 for odd n.
Sum_{i>=0} 1/a(i) = .754301907697893871765121109686...
a(n) = (2^(-1-n)*(-3*(-1)^n*2^(2+n)-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n))/5. - Colin Barker, Sep 29 2016

A111574 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 1,-1,2,3.

Original entry on oeis.org

1, -1, 2, 3, 3, 4, 9, 15, 22, 35, 59, 96, 153, 247, 402, 651, 1051, 1700, 2753, 4455, 7206, 11659, 18867, 30528, 49393, 79919, 129314, 209235, 338547, 547780, 886329, 1434111, 2320438, 3754547, 6074987, 9829536, 15904521, 25734055, 41638578, 67372635

Offset: 0

Creighton Dement, Aug 10 2005

See comment and FAMP code for A111569.

Floretion Algebra Multiplication Program, FAMP Code: -4baseiseq[B+H] with B = - .25'i + .25'j - .25i' + .25j' + k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and H = + .75'ii' + .75'jj' + .75'kk' + .75e

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

Cf. A001638, A111569, A111570, A111571, A111572, A111573, A111575, A111576.

LinearRecurrence[{1,0,1,1},{1,-1,2,3},40] (* Harvey P. Dale, Jan 24 2017 *)

G.f.: (-1+2*x-3*x^2)/((x^2+x-1)*(x^2+1)).

Name clarified by Robert C. Lyons, Feb 06 2025

cp's OEIS Frontend

A111573 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 0,1,3,3.

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A111570 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 2,5,4,7.

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A111571 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 1,1,-2,-1.

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A111572 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms -1,3,2,1.

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A260259 a(n) = F(n)*F(n+1) - (-1)^n, where F = A000045.

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A111574 a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4, with initial terms 1,-1,2,3.

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