A022086 -id:A022086 - cp's OEIS frontend

A194000 Triangular array: the self-fission of (p(n,x)), where sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

1, 2, 3, 3, 5, 9, 5, 8, 15, 24, 8, 13, 24, 39, 64, 13, 21, 39, 63, 104, 168, 21, 34, 63, 102, 168, 272, 441, 34, 55, 102, 165, 272, 440, 714, 1155, 55, 89, 165, 267, 440, 712, 1155, 1869, 3025, 89, 144, 267, 432, 712, 1152, 1869, 3024, 4895, 7920, 144, 233

Offset: 0

Clark Kimberling, Aug 11 2011

See A193917 for the self-fusion of the same sequence of polynomials.  (Fusion is defined at A193822; fission, at A193842; see A202503 and A202453 for infinite-matrix representations of fusion and fission.)
...
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...1
1...1...2
1...1...2...3
1...1...2...3...5
First eight rows of A194000:
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
21...34...63...102..168...272...441
34...55...102..165..272...440...714..1155
...
col 1:  A000045
col 2:  A000045
col 3:  A022086
col 4:  A022086
col 5:  A022091
col 6:  A022091
right edge, d(n,n):  A064831
d(n,n-1):  A059840
d(n,n-2):  A080097
d(n,n-3):  A080143
d(n,n-4):  A080144
...
Suppose n is an odd positive integer and d(n+1,x) is the polynomial matched to row n+1 of A194000 as in the Mathematica program (and definition of fission at A193842), where the first row is counted as row 0.

			First six rows:
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
...
Referring to the matrix product for fission at A193842,
the row (5,8,15,24) is the product of P(4) and QQ, where
P(4)=(p(4,4), p(4,3), p(4,2), p(4,1))=(5,3,2,1); and
QQ is the 4x4 matrix
(1..1..2..3)
(0..1..1..2)
(0..0..1..1)
(0..0..0..1).

Cf. A193842, A194001, A193917, A193918.

z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A194000 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]  (* A194001 *)

A286311 a(n) = 2*a(n-1) - a(n-2) + a(n-4), n>3, a(0)=0, a(1)=a(2)=1, a(3)=3.

Original entry on oeis.org

0, 1, 1, 3, 5, 8, 12, 19, 31, 51, 83, 134, 216, 349, 565, 915, 1481, 2396, 3876, 6271, 10147, 16419, 26567, 42986, 69552, 112537, 182089, 294627, 476717, 771344, 1248060, 2019403, 3267463, 5286867, 8554331, 13841198, 22395528, 36236725, 58632253, 94868979

Offset: 0

Paul Curtz, May 06 2017

Difference table for a(n):
   0,  1, 1, 3, 5, 8, 12, 19, 31, 51, 83, 134, 216, ...
   1,  0, 2, 2, 3, 4,  7, 12, 20, 32, 51,  82, 133, ...
  -1,  2, 0, 1, 1, 3,  5,  8, 12, 19, 31,  51,  83, ...
   3, -2, 1, 0, 2, 2,  3,  4,  7, 12, 20,  32,  51, ...
etc.
The pair a(n) = 0, 1, 1, 3, 5, 8, 12, 19, 31, 51, ...
     and b(n) = 0, 2, 2, 3, 4, 7, 12, 20, 32, 51, ...
is interesting. a(n) and b(n) are autosequences of the first kind (see Link). a(n) and b(n) have the same first trisection: 3*A001076(n).
a(n) + b(n) = A022086(n) = 3*A000045(n) (Fibonacci).
b(n) - a(n) = 0, 1, 1, 0, -1, -1, 0, ... = A128834(n).
a(n+6) - a(n) = b(n+6) - b(n) = 6*Fib(n+3).
a(n) - a(n) mod 9 = 9*A004699(n) = b(n) - b(n) mod 9.

Colin Barker, Table of n, a(n) for n = 0..1000
OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1).

Cf. A000034, A000045, A001076, A004699, A022086, A128834.

I:=[0,1,1,3]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-2) + Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 15 2018

LinearRecurrence[{2, -1, 0, 1}, {0, 1, 1, 3}, 40] (* or *)
CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x + x^2) (1 - x - x^2)), {x, 0, 39}], x] (* Michael De Vlieger, May 07 2017 *)

concat(0, Vec(x*(1 - x + 2*x^2) / ((1 - x + x^2)*(1 - x - x^2)) + O(x^60))) \\ Colin Barker, May 06 2017

a(n) = 2*a(n-1) - a(n-2) + a(n-4). Valid for b(n).

G.f.: x*(1 - x + 2*x^2) / ((1 - x + x^2)*(1 - x - x^2)). - Colin Barker, May 06 2017

More terms from Colin Barker, May 06 2017

A036605 a(n) = a(n-2) + 2*a(n-3) + a(n-4).

Original entry on oeis.org

1, 4, 4, 7, 13, 19, 31, 52, 82, 133, 217, 349, 565, 916, 1480, 2395, 3877, 6271, 10147, 16420, 26566, 42985, 69553, 112537, 182089, 294628, 476716, 771343, 1248061, 2019403, 3267463, 5286868, 8554330, 13841197, 22395529, 36236725

Offset: 0

N. J. A. Sloane

D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.4.2, Eq. (25).

Index entries for linear recurrences with constant coefficients, signature (0,1,2,1).

Cf. A004695.

A036605 := proc(n) option remember; if n <= 0 then 1 else A036605(n-2)+2*A036605(n-3)+A036605(n-4); fi; end;

LinearRecurrence[{0, 1, 2, 1}, {1, 4, 4, 7}, 36] (* Jean-François Alcover, Apr 01 2020 *)

3 * [Fibonacci(n+2)/2] + 1. - Ralf Stephan, Dec 02 2004

a(n) = (A099837(n+2)+A022086(n+2))/2. G.f. ( -1-4*x-3*x^2-x^3 ) / ( (1+x+x^2)*(x^2+x-1) ). - R. J. Mathar, Mar 21 2011

A036999 Restricted permutations.

Original entry on oeis.org

6, 9, 12, 18, 27, 42, 66, 105, 168, 270, 435, 702, 1134, 1833, 2964, 4794, 7755, 12546, 20298, 32841, 53136, 85974, 139107, 225078, 364182, 589257, 953436, 1542690, 2496123, 4038810, 6534930, 10573737, 17108664, 27682398, 44791059

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. S. Mendelsohn, Permutations with confined displacement, Canad. Math. Bull., 4 (1961), 29-38.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1).

Equals A022086(n+3) + 3.

[6] cat [3*Fibonacci(n+3)+3: n in [0..40]]; // Vincenzo Librandi, Jul 01 2017

Join[{6}, Table[3 Fibonacci[n+3] + 3, {n, 0, 40}]] (* or *) CoefficientList[Series[3 (2 - x - 2 x^2) / ((x^2 + x - 1) (x - 1)), {x, 0, 33}], x] (* Vincenzo Librandi, Jul 01 2017 *)

3*A000381.

G.f.: 3*(2-x-2*x^2)/((x^2+x-1)*(x-1)). - Vincenzo Librandi, Jul 01 2017

A099255 Expansion of g.f. (7+6x-6x^2-3x^3)/((x^2+x-1)(x^2-x-1)).

Original entry on oeis.org

7, 6, 15, 15, 38, 39, 99, 102, 259, 267, 678, 699, 1775, 1830, 4647, 4791, 12166, 12543, 31851, 32838, 83387, 85971, 218310, 225075, 571543, 589254, 1496319, 1542687, 3917414, 4038807, 10255923, 10573734, 26850355, 27682395, 70295142, 72473451

Offset: 0

Creighton Dement, Oct 09 2004

One of two sequences involving the Lucas/Fibonacci numbers.

This sequence consists of pairs of numbers more or less close to each other with "jumps" in between pairs. "pos((Ex)^n)" sums up over all floretion basis vectors with positive coefficients for each n. The following relations appear to hold: a(2n) - (a(2n-1) + a(2n-2)) = 2*Luc(2n) a(2n+1) - a(2n) = Fib(2n), apart from initial term a(2n+1)/a(2n-1) -> 2 + golden ratio phi a(2n)/a(2n-2) -> 2 + golden ratio phi An identity: (1/2)a(n) - (1/2)A099256(n) = ((-1)^n)A000032(n)

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A099256, A000032.

LinearRecurrence[{0,3,0,-1},{7,6,15,15},40] (* Harvey P. Dale, Dec 29 2012 *)

a(n) = 2*pos((Ex)^n)
a(0) = 7, a(1) = 6, a(2) = a(3) = 15, a(n+4) = 3a(n+2) - a(n).
a(2n) = A022097(2n+1), a(2n+1) = A022086(2n+3).
a(n) = A061084(n+1)+A013655(n+2). [R. J. Mathar, Nov 30 2008]

More terms from Creighton Dement, Apr 19 2005

A099256 Expansion of g.f. (3-x)(1+3x+x^2)/((1-x-x^2)*(1+x-x^2)).

Original entry on oeis.org

3, 8, 9, 23, 24, 61, 63, 160, 165, 419, 432, 1097, 1131, 2872, 2961, 7519, 7752, 19685, 20295, 51536, 53133, 134923, 139104, 353233, 364179, 924776, 953433, 2421095, 2496120, 6338509, 6534927, 16594432, 17108661, 43444787, 44791056, 113739929, 117264507, 297775000, 307002465, 779585071

Offset: 0

Creighton Dement, Oct 18 2004

One of two sequences involving the Lucas/Fibonacci numbers. This sequence consists of pairs of numbers more or less close to each other with "jumps" in between pairs.
a(n+3) + a(n) - a(n+2) appears to be mysteriously connected with a(n+1).
Both this sequence and A099255 were created using "Floretion dynamical symmetries" (see link for further details).

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A000045, A099255, A000032, A055273 (bisection), A097134 (bisection).

LinearRecurrence[{0,3,0,-1},{3,8,9,23},40] (* Harvey P. Dale, Apr 22 2012 *)

a(2n+2) - a(2n+1) = Fibonacci(2n-1).
A099255(n)/2 - a(n)/2 = (-1)^n*A000032(n)
a(0) = 3, a(1) = 8, a(2) = 9, a(3) = 23, a(n+4) = 3a(n+2) - a(n).
a(2n) = A022086(2n+2), a(2n+1) = A022097(2n+2).
a(n) = A013655(n+2)-A061084(n+1).

Definition corrected, extended. - R. J. Mathar, Nov 13 2008

A199535 Clark Kimberling's even first column Stolarsky array read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 5, 11, 9, 10, 8, 18, 15, 17, 12, 13, 29, 24, 27, 19, 14, 21, 47, 39, 44, 31, 23, 16, 34, 76, 63, 71, 50, 37, 25, 20, 55, 123, 102, 115, 81, 60, 41, 33, 22, 89, 199, 165, 186, 131, 97, 66, 53, 35, 26, 144, 322, 267, 301, 212, 157, 107, 86, 57, 43, 28

Offset: 1

Casey Mongoven, Nov 07 2011

The rows of the array can be seen to have the form A(n, k) = p(n)*Fibonacci(k) + q(n)*Fibonacci(k+1) where p(n) is the sequence {0, 1, 3, 3, 3, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, ...}{n >= 1} and q(n) is the sequence {1, 3, 3, 7, 2, 9, 9, 13, 13, 17, 17, 19, 19, 23, 23, 25, ...}{n >= 1}. - G. C. Greubel, Jun 23 2022

			The even first column stolarsky array (EFC array), northwest corner:
  1......2.....3.....5.....8....13....21....34....55....89...144 ... A000045;
  4......7....11....18....29....47....76...123...199...322...521 ... A000032;
  6......9....15....24....39....63...102...165...267...432...699 ... A022086;
  10....17....27....44....71...115...186...301...487...788..1275 ... A022120;
  12....19....31....50....81...131...212...343...555...898..1453 ... A013655;
  14....23....37....60....97...157...254...411...665..1076..1741 ... A000285;
  16....25....41....66...107...173...280...453...733..1186..1919 ... A022113;
  20....33....53....86...139...225...364...589...953..1542..2495 ... A022096;
  22....35....57....92...149...241...390...631..1021..1652..2673 ... A022130;
Antidiagonal rows (T(n, k)):
   1;
   2,   4;
   3,   7,   6;
   5,  11,   9,  10;
   8,  18,  15,  17, 12;
  13,  29,  24,  27, 19, 14;
  21,  47,  39,  44, 31, 23, 16;
  34,  76,  63,  71, 50, 37, 25, 20;
  55, 123, 102, 115, 81, 60, 41, 33, 22;

Clark Kimberling, The first column of an interspersion, Fibonacci Quarterly 32 (1994), pp. 301-314.

Cf. A000032, A000045, A000285, A013655, A022086, A022088, A022095.
Cf. A022096, A022097, A022113, A022120, A022121, A022122, A022130.
Cf. A022388, A022390, A035506, A035513, A097135, A199536, A199537.

From G. C. Greubel, Jun 23 2022: (Start)
T(n, 1) = A000045(n+1).
T(n, 2) = A000032(n+1), n >= 2.
T(n, 3) = A022086(n) = A097135(n), n >= 3.
T(n, 4) = A022120(n-2), n >= 4.
T(n, 5) = A013655(n-1), n >= 5.
T(n, 6) = A000285(n-2), n >= 6.
T(n, 7) = A022113(n-4), n >= 7.
T(n, 8) = A022096(n-4), n >= 8.
T(n, 9) = A022130(n-6), n >= 9.
T(n, 10) = A022098(n-5), n >= 10.
T(n, 11) = A022095(n-7), n >= 11.
T(n, 12) = A022121(n-8), n >= 12.
T(n, 13) = A022388(n-10), n >= 13.
T(n, 14) = A022122(n-10), n >= 14.
T(n, 15) = A022097(n-10), n >= 15.
T(n, 16) = A022088(n-10), n >= 16.
T(n, 17) = A022390(n-14), n >= 17.
T(n, n) = A199536(n).
T(n, n-1) = A199537(n-1), n >= 2. (End)

More terms added by G. C. Greubel, Jun 23 2022

A286350 a(n) = 2*a(n-1) - a(n-2) + a(n-4) for n>3, a(0)=0, a(1)=a(2)=2, a(3)=3.

Original entry on oeis.org

0, 2, 2, 3, 4, 7, 12, 20, 32, 51, 82, 133, 216, 350, 566, 915, 1480, 2395, 3876, 6272, 10148, 16419, 26566, 42985, 69552, 112538, 182090, 294627, 476716, 771343, 1248060, 2019404, 3267464, 5286867, 8554330, 13841197, 22395528, 36236726, 58632254, 94868979

Offset: 0

Paul Curtz, May 08 2017

This is b(n) in A286311(n). As mentioned in A286311, the pair A286311(n) and, here a(n), are autosequences of the first kind.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1).

Cf. A022086, A128834, A226956 (same recurrence), A286311.

I:=[0,2,2,3]; [n le 4 select I[n] else 2*Self(n-1) - Self(n-2) + Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 15 2018

LinearRecurrence[{2, -1, 0, 1}, {0, 2, 2, 3}, 40] (* or *)
CoefficientList[Series[x (2 - 2 x + x^2)/((1 - x + x^2) (1 - x - x^2)), {x, 0, 39}], x] (* Michael De Vlieger, May 09 2017 *)

concat(0, Vec(x*(2 - 2*x + x^2) / ((1 - x + x^2)*(1 - x - x^2)) + O(x^60))) \\ Colin Barker, May 09 2017

a(n) = A286311(n) + A128834(n).
a(n) = A022086(n) - A286311(n).
a(n) = (A022086(n) + A128834(n))/2.
G.f.: x*(2 - 2*x + x^2) / ((1 - x + x^2)*(1 - x - x^2)). - Colin Barker, May 09 2017

More terms from Colin Barker, May 09 2017

A026390 Expansion of (2 + x + x^2)/((1 - x)*(1 - x - x^2)).

Original entry on oeis.org

2, 5, 11, 20, 35, 59, 98, 161, 263, 428, 695, 1127, 1826, 2957, 4787, 7748, 12539, 20291, 32834, 53129, 85967, 139100, 225071, 364175, 589250, 953429, 1542683, 2496116, 4038803, 6534923, 10573730

Offset: 0

N. J. A. Sloane

Cf. A022086.

Fibonacci[Range[3,5!]]*3-4 (* Vladimir Joseph Stephan Orlovsky, Nov 23 2010 *)

Vec((2+x+x^2)/((1-x)*(1-x-x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n) = A022086(n+3) - 4.

A258121 Number of vertices of degree n in all Lucas cubes.

Original entry on oeis.org

2, 5, 15, 39, 102, 267, 699, 1830, 4791, 12543, 32838, 85971, 225075, 589254, 1542687, 4038807, 10573734, 27682395, 72473451, 189737958, 496740423, 1300483311, 3404709510, 8913645219, 23336226147, 61095033222, 159948873519, 418751587335, 1096305888486, 2870166078123

Offset: 0

Emeric Deutsch, Jun 23 2015

Column sums of A245960.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sandi Klavzar, Michel Mollard and Marko Petkovsek, The degree sequence of Fibonacci and Lucas cubes, Discrete Math., Vol. 311, No. 14 (2011), pp. 1310-1322.
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000045, A001519, A022086, A245960.

I:=[2,5,15,39]; [n le 4 select I[n] else 3*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 19 2017

g := (2-x)*(1+x^2)/(1-3*x+x^2): gser := series(g, x = 0, 35): seq(coeff(gser, x, n), n = 0 .. 32);
with(combinat): 2, 5, seq(3*fibonacci(2*n+1), n = 2 .. 32);

CoefficientList[Series[(2 - x)*(1 + x^2)/(1 - 3 x + x^2), {x, 0, 50}], x] (* G. C. Greubel, Oct 19 2017 *)
Join[{2, 5}, LinearRecurrence[{3, -1}, {15, 39}, 30]] (* Vincenzo Librandi, Oct 19 2017 *)

my(x='x+O('x^50)); Vec((2-x)*(1+x^2)/(1-3*x+x^2)) \\ G. C. Greubel, Oct 19 2017

G.f.: (2-x)*(1+x^2)/(1-3*x+x^2).
a(n) = 3*F(2n+1) = 3*A001519(n+1) = A022086(2n+1) for n>=2; F(n) = A000045(n) are the Fibonacci numbers.
a(n) = F(n-1)^2 + F(n)^2 + F(n+1)^2 + F(n+2)^2 for n > 1, where F(n) is the n-th Fibonacci number (A000045). - Amiram Eldar, Jan 11 2022

cp's OEIS Frontend

A194000 Triangular array: the self-fission of (p(n,x)), where sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

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A286311 a(n) = 2*a(n-1) - a(n-2) + a(n-4), n>3, a(0)=0, a(1)=a(2)=1, a(3)=3.

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A036605 a(n) = a(n-2) + 2*a(n-3) + a(n-4).

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Maple

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A036999 Restricted permutations.

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Magma

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A099255 Expansion of g.f. (7+6*x-6*x^2-3*x^3)/((x^2+x-1)*(x^2-x-1)).

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A099256 Expansion of g.f. (3-x)*(1+3*x+x^2)/((1-x-x^2)*(1+x-x^2)).

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Mathematica

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A199535 Clark Kimberling's even first column Stolarsky array read by antidiagonals.

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A286350 a(n) = 2*a(n-1) - a(n-2) + a(n-4) for n>3, a(0)=0, a(1)=a(2)=2, a(3)=3.

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A099255 Expansion of g.f. (7+6x-6x^2-3x^3)/((x^2+x-1)(x^2-x-1)).

A099256 Expansion of g.f. (3-x)(1+3x+x^2)/((1-x-x^2)*(1+x-x^2)).