A022112 -id:A022112 - cp's OEIS frontend

A210728 a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=1, a(1)=2.

1, 2, 7, 14, 27, 48, 83, 140, 233, 384, 629, 1026, 1669, 2710, 4395, 7122, 11535, 18676, 30231, 48928, 79181, 128132, 207337, 335494, 542857, 878378, 1421263, 2299670, 3720963, 6020664, 9741659, 15762356, 25504049, 41266440, 66770525, 108037002, 174807565

Offset: 0

Alex Ratushnyak, May 10 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A065220: a(n)=a(n-1)+a(n-2)+n-5, a(0)=1,a(1)=2 (except first 2 terms).
Cf. A168043: a(n)=a(n-1)+a(n-2)+n-3, a(0)=1,a(1)=2 (except first 2 terms).
Cf. A131269: a(n)=a(n-1)+a(n-2)+n-2, a(0)=1,a(1)=2.
Cf. A000126: a(n)=a(n-1)+a(n-2)+n-1, a(0)=1,a(1)=2.
Cf. A104161: a(n)=a(n-1)+a(n-2)+n,   a(0)=1,a(1)=2 (except the first term).
Cf. A192969: a(n)=a(n-1)+a(n-2)+n+1, a(0)=1,a(1)=2.
Cf. A210729: a(n)=a(n-1)+a(n-2)+n+3, a(0)=1,a(1)=2.

RecurrenceTable[{a[0] == 1, a[1] == 2, a[n] == a[n - 1] + a[n - 2] + n + 2}, a, {n, 36}] (* Bruno Berselli, Jun 27 2012 *)
nxt[{n_,a_,b_}]:={n+1,b,a+b+n+3}; NestList[nxt,{1,1,2},40][[;;,2]] (* Harvey P. Dale, Aug 26 2024 *)

G.f.: (1-x+3*x^2-2*x^3)/((1-x)^2*(1-x-x^2)). - Bruno Berselli, Jun 27 2012
a(n) = ((5+sqrt(5))*(1+sqrt(5))^(n+1)-(5-sqrt(5))*(1-sqrt(5))^(n+1))/(2^(n+1)*sqrt(5))-n-5. - Bruno Berselli, Jun 27 2012
a(n) = -n-5+A022112(n+1). R. J. Mathar, Jul 03 2012

A024831 a(n) = least m such that if r and s in {F(h)/F(2*h): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers).

Original entry on oeis.org

2, 7, 10, 10, 15, 23, 37, 59, 95, 153, 247, 399, 645, 1043, 1687, 2729, 4415, 7143, 11557, 18699, 30255, 48953, 79207, 128159, 207365, 335523, 542887, 878409, 1421295, 2299703, 3720997, 6020699, 9741695, 15762393, 25504087, 41266479, 66770565, 108037043, 174807607, 282844649

Offset: 2

Clark Kimberling

nonn

Note that F(2*h)/F(h) = Lucas(h) for h > 0. - Editors.

For a guide to related sequences, see A001000. - Clark Kimberling, Aug 07 2012

W. Kuszmaul, Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations, arXiv preprint arXiv:1509.08216 [cs.DM], 2015-2017.

Cf. A001000, A001622, A022112.

leastSeparator[seq_] := Module[{n = 1},
Table[While[Or @@ (Ceiling[n #1[[1]]] <
2 + Floor[n #1[[2]]] &) /@ (Sort[#1, Greater] &) /@
Partition[Take[seq, k], 2, 1], n++]; n, {k, 2, Length[seq]}]];
t = Table[N[Fibonacci[h]/Fibonacci[2 h]], {h, 1, 30}]
t1 = leastSeparator[t]
(* Peter J. C. Moses, Aug 01 2012 *)

From Philippe Deléham, Feb 06 2024: (Start)
a(n) = a(n-1) + a(n-2) - 1 for n >= 8.
a(n) = 2*a(n-1) - a(n-3) for n >= 9.
a(n) = 1 + A022112(n-3) for n >= 6.
a(n) = floor(((1 + sqrt(5))/2)*a(n-1)) for n >= 8.
G.f.: x^2*(x^6+3*x^5+2*x^4-8*x^3-4*x^2+3*x+2)/((x-1)*(x^2+x-1)).
(End)

All the terms were corrected by Clark Kimberling, Aug 07 2012

More terms from Sean A. Irvine, Jul 25 2019

A164413 Number of binary strings of length n with no substrings equal to 0000, 0001 or 1001.

Original entry on oeis.org

1, 2, 4, 8, 13, 22, 36, 58, 94, 152, 246, 398, 644, 1042, 1686, 2728, 4414, 7142, 11556, 18698, 30254, 48952, 79206, 128158, 207364, 335522, 542886, 878408, 1421294, 2299702, 3720996, 6020698, 9741694, 15762392, 25504086, 41266478, 66770564, 108037042

Offset: 0

R. H. Hardin, Aug 14 2009

Essentially the same as A022112. - R. J. Mathar, Nov 30 2011

David A. Corneth, Table of n, a(n) for n = 0..1999 (terms n = 4..500 from R. H. Hardin)
Index entries for linear recurrences with constant coefficients, signature (1,1).

Vec(-(x^2+1)*(x^2-x+1)*(x+1)^2/(x^2+x-1) + O(x^50)) \\ Colin Barker, Oct 27 2017

first(n) = {my(start = [1, 2, 4, 8, 13, 22, 36]); if(n <= 7, return(vector(n+1, i, start[i]))); res = concat(start, vector(n-7)); for(i=8, n,
res[i] = res[i-1] + res[i-2]); res} \\ David A. Corneth, Oct 27 2017

From Colin Barker, Oct 27 2017: (Start)
G.f.: -(x^2+1)*(x^2-x+1)*(x+1)^2/(x^2+x-1).
a(n) = 2*(((1 - sqrt(5))/2)^n + ((1 + sqrt(5))/2)^n) for n>4.
a(n) = a(n-1) + a(n-2) for n>6.
(End)

A244213 Inverse binomial transform of -2 followed by A000032(n+1).

Original entry on oeis.org

-2, 3, -1, 0, 3, -7, 14, -25, 43, -72, 119, -195, 318, -517, 839, -1360, 2203, -3567, 5774, -9345, 15123, -24472, 39599, -64075, 103678, -167757, 271439, -439200, 710643, -1149847, 1860494, -3010345, 4870843, -7881192

Offset: 0

Paul Curtz, Jun 23 2014

A simple transform of a(n) is a(n) with -a(0) instead of nonzero a(0) (or -a(0) followed by a(n+1)). Example: -1 followed by A198631(n+1)/A006519(n+2). Its inverse binomial transform is -1, 3/2, -2, 9/4, -2, 3/2, -2,... = -(-1)^n*A143074(n).
Difference table of -2 followed by A000032(n+1):
-2,  1,  3,  4,  7, 11, 18,...
3,   2,  1,  3,  4,  7, 11,...
-1, -1,  2,  1,  3,  4,  7,...
0,   3, -1,  2,  1,  3,  4,...
3,  -4,  3, -1,  2,  1,  3,...
-7,  7, -4,  3, -1,  2,  1,...
14, -11, 7, -4,  3, -1,  2,...
etc.
a(n) is the first column.

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

Cf. A000045, A000032, A000204.

Vec(-(5*x^2-x-2)/((x+1)*(x^2-x-1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

a(n) = -2, 3, -1, followed by -(-1)^n*A206417(n).
a(n) = (-1)^n* (A000032(n) - 4).
a(n+3) = -a(n) -(-1)^n*A022112(n).
a(n) = -2*a(n-1) + a(n-3). - Colin Barker, Jun 23 2014
G.f.: -(5*x^2-x-2) / ((x+1)*(x^2-x-1)). - Colin Barker, Jun 23 2014

A247526 a(n) = L(n+1) * L(n) * L(n-1) * L(n-2) / 6, where L(n) = Lucas numbers (A000032).

Original entry on oeis.org

-1, -1, 4, 14, 154, 924, 6699, 44979, 310764, 2123554, 14571974, 99833524, 684385079, 4690541639, 32150245204, 220358978774, 1510368355474, 10352204457804, 70955102255139, 486333408161979, 3333379024971324, 22847319059525674, 156597856242950654

Offset: 0

Michael Somos, Sep 19 2014

			G.f. = -1 - x + 4*x^2 + 14*x^3 + 154*x^4 + 924*x^5 + 6699*x^6 + 44979*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Cf. A000032, A085696.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-(1-4*x-24*x^2+6*x^3-4*x^4)/((1-x)*(1+3*x+x^2)*(1 - 7*x+x^2)))); // G. C. Greubel, Aug 05 2018

CoefficientList[Series[-(1-4*x-24*x^2+6*x^3-4*x^4)/((1-x)*(1+3*x+x^2)*(1 - 7*x+x^2)), {x, 0, 60}], x] (* G. C. Greubel, Aug 05 2018 *)
Times@@#/6&/@Partition[LucasL[Range[-2,30]],4,1] (* or *) LinearRecurrence[{5,15,-15,-5,1},{-1,-1,4,14,154},30] (* Harvey P. Dale, Apr 20 2022 *)

{a(n) = my(u = fibonacci(n), v = fibonacci(n-1)); (3*u + v) * (u + 2*v) * (2*u - v) * (-u + 3*v) / 6};

{a(n) = if( n<1, n=1-n); polcoeff( - (1 - 4*x - 24*x^2 + 6*x^3 - 4*x^4) / ((1 - x) * (1 + 3*x + x^2) * (1 - 7*x +x^2)) + x * O(x^n), n)};

G.f.: -(1 - 4*x - 24*x^2 + 6*x^3 - 4*x^4)/((1-x)*(1 + 3*x + x^2)*(1 - 7*x + x^2)).
a(n) = a(1-n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5) for all n in Z.
0 = a(n)*(+a(n+1) - 2*a(n+2)) + a(n+1)*(-5*a(n+1) + a(n+2)) for all n in Z.
From Klaus Purath, Oct 02 2020: (Start)
a(n) = (L(n-2)*L(n-1)^3 - L(n-1)*L(n-2)^3)/6 where L = Lucas.
a(n) = f(n-3)*f(n-2)*f(n-1)*f(n)/96 where f = A022112.
a(n) = (f(n-2)*f(n-1)^3 - f(n-1)*f(n-2)^3)/96 where f = A022112.
(End)
2*a(n) = A098149(n) +A004187(n+1)-6*A004187(n) -2 . - R. J. Mathar, Sep 24 2021

A372242 a(n) = 10*Fibonacci(n) + (-1)^n.

Original entry on oeis.org

1, 9, 11, 19, 31, 49, 81, 129, 211, 339, 551, 889, 1441, 2329, 3771, 6099, 9871, 15969, 25841, 41809, 67651, 109459, 177111, 286569, 463681, 750249, 1213931, 1964179, 3178111, 5142289, 8320401, 13462689, 21783091, 35245779, 57028871, 92274649, 149303521, 241578169

Offset: 0

Paul Curtz, Apr 23 2024

			a(0) = 10*0 + 1 =  1,
a(1) = 10*1 - 1 =  9,
a(2) = 10*1 + 1 = 11,
a(3) = 10*2 - 1 = 19.

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

Cf. A000045, A022093, A153382, A052952, A371843.

Cf. A000032, A022112, A280154.

LinearRecurrence[{0, 2, 1}, {1, 9, 11}, 50] (* Paolo Xausa, May 20 2024 *)

Floor(a(n+2)/10) = A052952(n).
a(n) = 1 + A153382(n).
a(n) = 2*A371843(n) - (-1)^n.
a(n) = A022093(n) + (-1)^n.
G.f.: -(9*x^2+9*x+1)/((x+1)*(x^2+x-1)).
a(0) = 1. a(n) = -a(n-1) + 10*A000045(n+1) for n >= 1.
a(n) = a(n-4) + (2*A280154(n-2) = 5*A022112(n-3) = 10*A000032(n-2)) for n >= 4.

A347351 Triangle read by rows: T(n,k) is the number of links of length k in a set of all necklaces A000358 of length n, 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 4, 2, 0, 1, 5, 1, 1, 0, 1, 6, 4, 2, 1, 0, 1, 7, 3, 2, 1, 1, 0, 1, 8, 8, 3, 3, 1, 1, 0, 1, 9, 8, 7, 3, 2, 1, 1, 0, 1, 10, 18, 9, 5, 4, 2, 1, 1, 0, 1, 11, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1, 12, 40, 24, 16, 8, 6, 3, 2, 1, 1, 0, 1, 13, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1

Offset: 0

Maxim Karimov and Vladislav Sulima, Aug 28 2021

Definitions:
1. A link is any 0 in any necklace from A000358 and all 1s following this 0 in this necklace to right until another 0 is encountered.
2. Length of the link is the number of elements in the link.
Sum of all elements n-row is Fibonacci(n-1)+n iff n=1 or n=p (follows from the identity for the sum of the Fibonacci numbers and the formula for the triangle T(n,k)).

			For k > 0:
   n\k |  1   2   3   4   5   6   7   8   9  10  ...
  -----+---------------------------------------
   1   |  1
   2   |  2   1
   3   |  3   0   1
   4   |  4   2   0   1
   5   |  5   1   1   0   1
   6   |  6   4   2   1   0   1
   7   |  7   3   2   1   1   0   1
   8   |  8   8   3   3   1   1   0   1
   9   |  9   8   7   3   2   1   1   0   1
  10   | 10  18   9   5   4   2   1   1   0   1
  ...
If we continue the calculation for nonpositive k, we get a table in which each row is a Fibonacci sequence, in which term(0) = A113166, term(1) = A034748.
For k <= 0:
   n\k |  0   -1   -2   -3   -4   -5   -6   -7   -8   -9 ...
  -----+------------------------------------------------
   1   |  0    1    1    2    3    5    8   13   21   34 ... A000045
   2   |  1    2    3    5    8   13   21   34   55   89 ... A000045
   3   |  1    4    5    9   14   23   37   60   97  157 ... A000285
   4   |  3    6    9   15   24   39   63  102  165  267 ... A022086
   5   |  3    9   12   21   33   54   87  141  228  369 ... A022379
   6   |  8   14   22   36   58   94  152  246  398  644 ... A022112
   7   |  8   19   27   46   73  119  192  311  503  814 ... A206420
   8   | 17   30   47   77  124  201  325  526  851 1377 ... A022132
   9   | 23   44   67  111  178  289  467  756 1223 1979 ... A294116
  10   | 41   68  109  177  286  463  749 1212 1961 3173 ... A022103
  ...

Cf. A000010, A000027, A000045, A000358, A113166, A034748.

function [res] = calcLinks(n,k)
if k==1
    res=n;
else
    d=divisors(n);
    res=0;
    for i=1:length(d)
        if d (i) >= k
            res=res+eulerPhi(n/d(i))*fiboExt(d(i)-k-1);
        end
    end
end
function [s] = fiboExt(m) % extended fibonacci function (including negative arguments)
m=sym(m); % for large fibonacci numbers
if m>=0 || mod(m,2)==1
    s=fibonacci(abs(m));
else
    s=fibonacci(abs(m))*(-1);
end

T(n, k) = if (k==1, n, sumdiv(n, d, if (d>=k, eulerphi(n/d)*fibonacci(d-k-1)))); \\ Michel Marcus, Aug 29 2021

If k=1, T(n,k)=n, otherwise T(n,k) = Sum_{d>=k, d|n} Phi(n/d)*Fibonacci(d-k-1), where Phi=A000010.

cp's OEIS Frontend

A210728 a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=1, a(1)=2.

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A024831 a(n) = least m such that if r and s in {F(h)/F(2*h): h = 1,2,...,n} satisfy r < s, then r < k/m < s for some integer k, where F = A000045 (Fibonacci numbers).

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A164413 Number of binary strings of length n with no substrings equal to 0000, 0001 or 1001.

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A244213 Inverse binomial transform of -2 followed by A000032(n+1).

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A247526 a(n) = L(n+1) * L(n) * L(n-1) * L(n-2) / 6, where L(n) = Lucas numbers (A000032).

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A372242 a(n) = 10*Fibonacci(n) + (-1)^n.

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A347351 Triangle read by rows: T(n,k) is the number of links of length k in a set of all necklaces A000358 of length n, 1 <= k <= n.

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