A001612 -id:A001612 - cp's OEIS frontend

A214984 Array: T(m,n) = (F(m) + F(2m) + ... + F(nm))/F(m), by antidiagonals, where F = A000045 (Fibonacci numbers).

1, 2, 1, 4, 4, 1, 7, 12, 5, 1, 12, 33, 22, 8, 1, 20, 88, 94, 56, 12, 1, 33, 232, 399, 385, 134, 19, 1, 54, 609, 1691, 2640, 1487, 342, 30, 1, 88, 1596, 7164, 18096, 16492, 6138, 872, 48, 1, 143, 4180, 30348, 124033, 182900, 110143, 25319, 2256, 77, 1

Offset: 1

Clark Kimberling, Oct 28 2012

col 1: A001612 (except for initial term)
row 1: A000071
row 2: A027941
row 3: A049652
row 4: A092521
row 6: A049664
row 8: A156093 without minus signs

			Northwest corner:
1...2....4.....7......12......20
1...4....12....33.....88......232
1...5....22....94.....399.....1691
1...8....56....385....2640....18096
1...12...134...1487...16492...182900

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A214978, A214985, A214986.

F[n_] := Fibonacci[n]; L[n_] := LucasL[n];
t[m_, n_] := (1/F[m])*Sum[F[m*k], {k, 1, n}]
TableForm[Table[t[m, n], {m, 1, 10}, {n, 1, 10}]]
Flatten[Table[t[k, n + 1 - k], {n, 1, 12}, {k, 1, n}]]

For odd-numbered rows (m odd):
T(m,n) = (F(m*n+m) + F(m*n) - F(m))/(F(m)*L(m)).
For even-numbered rows (m even):
T(m,n) = (F(m*n+m) - F(m*n) - F(m))/(F(m)*(L(m)-2)).

A187890 a(1) = 0, a(2) = 4, a(n) = a(n-1) + a(n-2) - 1.

Original entry on oeis.org

0, 4, 3, 6, 8, 13, 20, 32, 51, 82, 132, 213, 344, 556, 899, 1454, 2352, 3805, 6156, 9960, 16115, 26074, 42188, 68261, 110448, 178708, 289155, 467862, 757016, 1224877, 1981892, 3206768, 5188659, 8395426, 13584084, 21979509, 35563592, 57543100

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 15 2011

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

Cf. A000071, A001611, A001612, A374438.

Join[{a = 0, b = 4}, Table[c = a+b-1; a=b; b=c, {n, 100}]]
LinearRecurrence[{2, 0, -1}, {0, 4, 3}, 40] (* Harvey P. Dale, Sep 25 2013 *)
CoefficientList[Series[(-x (-4 + 5 x))/((x - 1) (x^2 + x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 26 2013 *)

G.f.: -x^2*(-4+5*x) / ( (x-1)*(x^2+x-1) ). - R. J. Mathar, Mar 15 2011
a(n) = 1+A001060(n-2), n>2. - R. J. Mathar, Mar 15 2011
a(n) - a(n-1) = A013655(n-4). - R. J. Mathar, Jun 19 2021
If we start the sequence 1, 3, 6, ... and set the offset to 0, then the sequence has the generating function (1 + x - 3*x^3)/(x^3 - 2*x + 1) and gives the row sums of A374438. - Peter Luschny, Jul 22 2024

Definition adapted to offset by Georg Fischer, Jun 19 2021

A274017 Number of n-bead binary necklaces (no turning over allowed) that avoid the subsequence 110.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 6, 6, 9, 11, 16, 20, 32, 42, 65, 95, 144, 212, 330, 494, 767, 1171, 1812, 2788, 4342, 6714, 10463, 16275, 25416, 39652, 62076, 97110, 152289, 238839, 375168, 589528, 927556, 1459962, 2300349, 3626243, 5721046, 9030452, 14264310, 22542398, 35646313, 56393863, 89264836, 141358276, 223959712

Offset: 0

Marko Riedel, Jun 06 2016

nonn

The pattern in this enumeration must be contiguous (all three values next to each other in one sequence of three letters).

The proofs of all my formulas below become evident once it is realized that A001612(n) gives the number of cyclic sequences of length n consisting of zeros and ones that avoid the pattern 001 (or equivalently, the pattern 110) provided the positions of zeros and ones on a circle are fixed. Everything else follows from the material that can be found in A001612. - Petros Hadjicostas, Jan 11 2017

			The following necklace
     1-1
    /   \
   0     0
   |     |
   1     1
    \   /
     0-0
contains one instance of the subsequence starting in the upper left corner. Unlike a bracelet, the necklace is oriented.
a(8) = 9: 00000000, 00000001, 00000101, 00001001, 00010001, 00010101, 00100101, 01010101, 11111111.
a(9) = 11: 000000000, 000000001, 000000101, 000001001, 000010001, 000010101, 000100101, 000101001, 001001001, 001010101, 111111111.

Petros Hadjicostas and L. Zhang, On cyclic strings avoiding a pattern, Discrete Mathematics, 341 (2018), 1662-1674.
Petros Hadjicostas, Proof of two formulas for a(n).
Marko Riedel, Maple code for this sequence.
Math Stackexchange, Marko Riedel et al. Counting circular sequences.

Cf. A000010, A000031, A000045, A000204, A000358, A001612, A093305, A274018, A274019, A274020.

From Petros Hadjicostas, Jan 11 2017: (Start)
For all the formulas below, assume n>=1.
a(n) = 1 + A000358(n). (Notice the different offsets.)
a(n) = 1 + (1/n) * Sum_{d|n} totient(n/d)*(Fibonacci(d-1)+Fibonacci(d+1)).
a(n) = (1/n) * Sum_{d divides n} totient(n/d)*A001612(d).
G.f.: 1/(1-x) + Sum_{k>=1} (phi(k)/k) * log(1/(1-B(x^k))) where B(x) = x*(1+x). (This is a modification of a formula due to Joerg Arndt.)
G.f.: 1 + Sum_{k>=1} (phi(k)/k) * log(1/C(x^k)) where C(x) = (1-x)*(1-B(x)). (End)
a(n) = 1 + (1/n) * Sum_{d|n} A000010(n/d)*A000204(d). [After the second formula above given by Hadjicostas]. - Antti Karttunen, Jan 12 2017

A166471 a(n) = 2^L(n+1)*3^L(n), where L(n) is the n-th Lucas number (A000032(n)).

Original entry on oeis.org

18, 24, 432, 10368, 4478976, 46438023168, 207994791256915968, 9658866935211987562213146624, 2008994011967745042140303999261186371230892032

Offset: 0

Matthew Vandermast, Nov 05 2009, Nov 07 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..14

Cf. A000032, A001612, A002110, A166469, A166470, A166472, A166473.
All terms but the first belong to A025487.
Subsequence of A003586.

[2^Lucas(n+1)*3^Lucas(n): n in [0..10]]; // G. C. Greubel, Jul 30 2024

Table[2^LucasL[n+1]*3^LucasL[n], {n,0,10}] (* G. C. Greubel, May 15 2016 *)

def l(n): return lucas_number2(n,1,-1);
[2^l(n+1)*3^l(n) for n in range(11)] # G. C. Greubel, Jul 30 2024

a(n) = a(n-1)*a(n-2), for n > 1, with a(0) = 18, a(1) = 24.
For m>1, n>0, A166469(A002110(m)*(a(n)^k)/12) = k*Lucas(m+n).
A166469(a(n)) = Lucas(n+2) + 1 = A001612(n+2).

A187893 a(0)=1, a(1)=4, a(n) = a(n-1) + a(n-2) - 1.

Original entry on oeis.org

1, 4, 4, 7, 10, 16, 25, 40, 64, 103, 166, 268, 433, 700, 1132, 1831, 2962, 4792, 7753, 12544, 20296, 32839, 53134, 85972, 139105, 225076, 364180, 589255, 953434, 1542688, 2496121, 4038808, 6534928, 10573735, 17108662, 27682396, 44791057, 72473452, 117264508

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 15 2011

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

Cf. A000045, A000071, A001611, A001612, A039834, A187890.

Join[{a=1,b=4},Table[c=a+b-1;a=b;b=c,{n,100}]]
LinearRecurrence[{2,0,-1},{1,4,4},40] (* Harvey P. Dale, Jun 06 2020 *)

a(n)=3*fibonacci(n)+1 \\ Charles R Greathouse IV, Oct 29 2016

G.f.: -x*(-1-2*x+4*x^2) / ( (x-1)*(x^2+x-1) ). - R. J. Mathar, Mar 15 2011

a(n) = 1+3*|A039834(n)| = 1+3*A000045(n). - R. J. Mathar, Mar 15 2011

A214985 Array: T(m,n) = (F(n) + F(2n) + ... + F(nm))/F(n), by antidiagonals; transpose of A214984.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 5, 12, 7, 1, 8, 22, 33, 12, 1, 12, 56, 94, 88, 20, 1, 19, 134, 385, 399, 232, 33, 1, 30, 342, 1487, 2640, 1691, 609, 54, 1, 48, 872, 6138, 16492, 18096, 7164, 1596, 88, 1, 77, 2256, 25319, 110143, 182900, 124033, 30348, 4180, 143

Offset: 1

Clark Kimberling, Oct 28 2012

row 1: A001612 (except for initial term)
col 1: A000071
col 2: A027941
col 3: A049652
col 4: A092521
col 6: A049664
col 8: A156093 without minus signs

			Northwest corner:
1....1.....1......1.......1
2....4.....5......8.......12
4....12....22.....56......134
7....33....94.....385.....1487
12...88....399....2640....16492
20...232...1691...18096...182900

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A214978, A214984, A214986.

F[n_] := Fibonacci[n]; L[n_] := LucasL[n];
t[m_, n_] := (1/F[n])*Sum[F[k*n], {k, 1, m}]
TableForm[Table[t[m, n], {m, 1, 10}, {n, 1, 10}]]
Flatten[Table[t[k, n + 1 - k], {n, 1, 12}, {k, 1, n}]]

For odd-numbered columns (m odd):
T(m,n) = (F(m*n+m) + F(m*n) - F(m))/(F(m)*L(m)).
For even-numbered columns (m even):
T(m,n) = (F(m*n+m) - F(m*n) - F(m))/(F(m)*(L(m)-1)).

A187891 a(0)=0, a(1)=5, a(n)=a(n-1)+a(n-2)-1.

Original entry on oeis.org

0, 5, 4, 8, 11, 18, 28, 45, 72, 116, 187, 302, 488, 789, 1276, 2064, 3339, 5402, 8740, 14141, 22880, 37020, 59899, 96918, 156816, 253733, 410548, 664280, 1074827, 1739106, 2813932, 4553037, 7366968, 11920004, 19286971, 31206974, 50493944, 81700917

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 15 2011

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

Cf. A000071, A001611, A001612, A022120, A187890.

Join[{a=0,b=5},Table[c=a+b-1;a=b;b=c,{n,100}]]
nxt[{a_,b_}]:={b,a+b-1}; NestList[nxt,{0,5},40][[All,1]] (* Harvey P. Dale, Nov 03 2022 *)

a(n) = 1+A022120(n-2), n>2. - R. J. Mathar, Mar 15 2011

G.f.: -x^2*(-5+6*x) / ( (x-1)*(x^2+x-1) ). - R. J. Mathar, Mar 15 2011

A187892 a(0)=0, a(1)=6, a(n)=a(n-1)+a(n-2)-1.

Original entry on oeis.org

0, 6, 5, 10, 14, 23, 36, 58, 93, 150, 242, 391, 632, 1022, 1653, 2674, 4326, 6999, 11324, 18322, 29645, 47966, 77610, 125575, 203184, 328758, 531941, 860698, 1392638, 2253335, 3645972, 5899306, 9545277, 15444582, 24989858, 40434439, 65424296, 105858734

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 15 2011

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

Cf. A000071, A001611, A001612, A187890.

Join[{a=0,b=6},Table[c=a+b-1;a=b;b=c,{n,100}]]
LinearRecurrence[{2,0,-1},{0,6,5},40] (* Harvey P. Dale, Aug 17 2019 *)

G.f.: -x^2*(-6+7*x) / ( (x-1)*(x^2+x-1) ). - R. J. Mathar, Mar 15 2011

a(n) = 1+A022130(n-3), n>2. - R. J. Mathar, Mar 15 2011

A377184 Numbers k such that L(k)+1 is prime, where L(k) is the k-th Lucas number.

Original entry on oeis.org

0, 1, 3, 6, 18

Offset: 1

Harsh Aggarwal, Oct 18 2024

Positive terms must be a power of 3 or twice a power of 3. All other terms have trivial factors. No other terms under 10^7 (a(6) > 10^7).

			18 is a term because L(18)+1 = 5779 which is prime. Also 18 = 2*3^2.

Indices of primes in A001612.

Cf. A000032, A000204, A001606.

cp's OEIS Frontend

A214984 Array: T(m,n) = (F(m) + F(2*m) + ... + F(n*m))/F(m), by antidiagonals, where F = A000045 (Fibonacci numbers).

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A187890 a(1) = 0, a(2) = 4, a(n) = a(n-1) + a(n-2) - 1.

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A274017 Number of n-bead binary necklaces (no turning over allowed) that avoid the subsequence 110.

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A166471 a(n) = 2^L(n+1)*3^L(n), where L(n) is the n-th Lucas number (A000032(n)).

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A187893 a(0)=1, a(1)=4, a(n) = a(n-1) + a(n-2) - 1.

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A214985 Array: T(m,n) = (F(n) + F(2*n) + ... + F(n*m))/F(n), by antidiagonals; transpose of A214984.

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A187891 a(0)=0, a(1)=5, a(n)=a(n-1)+a(n-2)-1.

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A187892 a(0)=0, a(1)=6, a(n)=a(n-1)+a(n-2)-1.

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A214984 Array: T(m,n) = (F(m) + F(2m) + ... + F(nm))/F(m), by antidiagonals, where F = A000045 (Fibonacci numbers).

A214985 Array: T(m,n) = (F(n) + F(2n) + ... + F(nm))/F(n), by antidiagonals; transpose of A214984.