A111573 -id:A111573 - cp's OEIS frontend

A111569 a(n) = a(n-1) + a(n-3) + a(n-4) for n > 3, a(0) = -1, a(1) = 1, a(2) = 2, a(3) = 1.

-1, 1, 2, 1, 1, 4, 7, 9, 14, 25, 41, 64, 103, 169, 274, 441, 713, 1156, 1871, 3025, 4894, 7921, 12817, 20736, 33551, 54289, 87842, 142129, 229969, 372100, 602071, 974169, 1576238, 2550409, 4126649, 6677056, 10803703, 17480761, 28284466, 45765225

Offset: 0

Creighton Dement, Aug 07 2005

In reference to the program code given, 4*tesseq[A*H] = A001638 (a Fielder sequence) where A001638(2n) = L(n)^2. Here we have: a(2n+1) = A007598(n+1) = Fibonacci(n+1)^2.
Floretion Algebra Multiplication Program, FAMP Code: 4kbaseiseq[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
First bisection is A260259 (see previous comment for the second bisection). [Bruno Berselli, Nov 02 2015]

Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Robert Munafo, Sequences Related to Floretions.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 1).

Cf. A000045, A001638, A007598, A111570, A111571, A111572, A111573, A260259.

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

a(n) = 2*A056594(n+3)/5 - 6*A056594(n)/5 + A000032(n+1)/5. [R. J. Mathar, Nov 12 2009]

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

A255935 Triangle read by rows: a(n) = Pascal's triangle A007318(n) + A197870(n+1).

Original entry on oeis.org

0, 1, 2, 1, 2, 0, 1, 3, 3, 2, 1, 4, 6, 4, 0, 1, 5, 10, 10, 5, 2, 1, 6, 15, 20, 15, 6, 0, 1, 7, 21, 35, 35, 21, 7, 2, 1, 8, 28, 56, 70, 56, 28, 8, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 2, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0

Offset: 0

Paul Curtz, Mar 11 2015

Consider the difference table of a sequence with A000004(n)=0's as main diagonal. (Example: A000045(n).) We call this sequence an autosequence of the first kind.
Based on Pascal's triangle, a(n) =
0,                   T1
1, 2,
1, 2, 0,
1, 3, 3, 2,
etc.
transforms every sequence s(n) in an autosequence of the first kind via the multiplication by the triangle
s0,                  T2
s0, s1,
s0, s1, s2,
s0, s1, s2, s3,
etc.
Examples.
1) s(n) = A198631(n)/A006519(n+1), the second fractional Euler numbers (see A209308). This yields 0*1, 1*1+2*1/2=2, 1*1+2*1/2+0*0=2, 1*1+3*1/2++3*0+2*(-1/4)=2, ... .
The autosequence is 0 followed by 2's or 2*(0,1,1,1,1,1,1,1,... = b(n)).
b(n), the basic autosequence of the first kind, is not in the OEIS (see A140575 and A054977).
2) s(n) = A164555(n)/A027642(n), the second Bernoulli numbers, yields 0,2,2,3,4,5,6,7,... = A254667(n).
Row sums of T1: A062510(n) = 3*A001045(n).
Antidiagonal sums of T1: A111573(n).
With 0's instead of the spaces, every column, i.e.,
0, 1, 1, 1, 1,  1,  1,  1,  1,  1,   1, ...
0, 2, 2, 3, 4,  5,  6,  7,  8,  9,  10, ... = A001477(n) with 0 instead of 1 = A254667(n)
0, 0, 0, 3, 6, 10, 15, 21, 28, 36,  45, ... = A161680(n) with 0 instead of 1
0, 0, 0, 2, 4, 10, 20, 35, 56, 84, 120, ...
etc., is an autosequence of the first kind.
With T(0,0) = 1, it is (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 24 2015

			Triangle starts:
0;
1, 2;
1, 2, 0;
1, 3, 3, 2;
1, 4, 6, 4, 0;
1, 5, 10, 10, 5, 2;
1, 6, 15, 20, 15, 6, 0;
...

Cf. A001045, A001477, A010673, A011973, A054977, A062510, A007318, A197870, A006519, A198631, A140575, A209308, A164555, A027642, A111573, A161680, A254667.

a[n_, k_] := If[k == n, 2*Mod[n, 2], Binomial[n, k]]; Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 23 2015 *)

a(n) = Pascal's triangle A007318(n) with main diagonal A010673(n) (= period 2: repeat 0, 2) instead of 1's=A000012(n).
a(n) = reversal abs(A140575(n)).
a(n) = A007318(n) + A197870(n+1).
T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k>n or if k<0 . - Philippe Deléham, May 24 2015
G.f.: (-1-2*x*y+x^2*y+x^2*y^2)/((x*y+1)*(x*y+x-1)) - 1. - R. J. Mathar, Aug 12 2015

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

A111569 a(n) = a(n-1) + a(n-3) + a(n-4) for n > 3, a(0) = -1, a(1) = 1, a(2) = 2, a(3) = 1.

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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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A255935 Triangle read by rows: a(n) = Pascal's triangle A007318(n) + A197870(n+1).

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Formula

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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