A001611 -id:A001611 - cp's OEIS frontend

A168514 Number of prime divisors (counted with multiplicity) of Fibonacci(n)+1.

1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 2, 4, 5, 2, 4, 3, 3, 4, 3, 3, 8, 4, 2, 4, 5, 3, 5, 5, 5, 4, 4, 2, 6, 6, 2, 7, 9, 4, 6, 5, 4, 6, 5, 4, 12, 4, 4, 6, 5, 5, 7, 6, 6, 7, 6, 4, 10, 6, 2, 7, 9, 4, 6, 5, 4, 6, 7, 6, 13, 7, 4, 7, 7, 5, 8, 5, 8, 6, 4, 5, 10, 8, 4, 7, 11, 5, 8, 9, 7, 8, 6, 4, 15, 5, 3, 7, 10, 7, 8, 7, 8

Offset: 1

Jason Earls, Nov 28 2009

nonn

Always greater than 1 for any n >= 4.

Jason Earls, "Fibonacci," Mathematical Bliss, Pleroma Publications, 2009, pages 60-64. ASIN: B002ACVZ6O

Amiram Eldar, Table of n, a(n) for n = 1..2803

Cf. A000045, A001222, A001611.

Table[PrimeOmega[Fibonacci[n] + 1], {n, 100}] (* Wesley Ivan Hurt, Feb 04 2014 *)

A168514(n) = bigomega(fibonacci(n)+1) \\ Michael B. Porter, Mar 02 2010

a(n) = Omega(Fibonacci(n) + 1) = A001222(A001611(n)). - Wesley Ivan Hurt, Feb 04 2014

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

A373889 Square array read by ascending antidiagonals: T(k,n) is the cardinality of {(E is a proper finite subset of the natural numbers) such that E = {} or w_k(E) < min(E) <= max(E) <= n}, where w_k(E) = Sum_{i in E, i <> k} 1, with n, k >= 1.

Original entry on oeis.org

2, 1, 3, 1, 2, 4, 1, 2, 4, 6, 1, 2, 4, 7, 9, 1, 2, 3, 6, 11, 14, 1, 2, 3, 6, 10, 17, 22, 1, 2, 3, 5, 10, 17, 26, 35, 1, 2, 3, 5, 10, 16, 28, 40, 56, 1, 2, 3, 5, 8, 16, 26, 45, 62, 90, 1, 2, 3, 5, 8, 16, 26, 43, 71, 97, 145, 1, 2, 3, 5, 8, 13, 26, 42, 71, 111, 153, 234

Offset: 1

Paolo Xausa, Jun 21 2024

			The array begins:
  k\n|  1  2  3  4   5   6   7   8   9   10  ...
  ----------------------------------------------
   1 |  2, 3, 4, 6,  9, 14, 22, 35, 56,  90, ... = A001611 (from n = 2).
   2 |  1, 2, 4, 7, 11, 17, 26, 40, 62,  97, ...
   3 |  1, 2, 4, 6, 10, 17, 28, 45, 71, 111, ...
   4 |  1, 2, 3, 6, 10, 16, 26, 43, 71, 116, ...
   5 |  1, 2, 3, 5, 10, 16, 26, 42, 68, 111, ...
   6 |  1, 2, 3, 5,  8, 16, 26, 42, 68, 110, ...
   7 |  1, 2, 3, 5,  8, 13, 26, 42, 68, 110, ...
   8 |  1, 2, 3, 5,  8, 13, 21, 42, 68, 110, ...
   9 |  1, 2, 3, 5,  8, 13, 21, 34, 68, 110, ...
  10 |  1, 2, 3, 5,  8, 13, 21, 34, 55, 110, ...
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).
Hung Viet Chu and Zachary Louis Vasseur, Weighted Schreier-type Sets and the Fibonacci Sequence, arXiv:2405.19352 [math.CO], 2024. See p. 2, Table 1 and Theorem 1.2.

Cf. A000045, A001611, A373345, A373556, A373557.

A373889[k_, n_] := Which[n < k, Fibonacci[n+1], k == 1, Fibonacci[n-k+2] + 1, True, 2*Sum[Binomial[n-k, i]*Fibonacci[k-i], {i, 0, k-2}] + 2*Binomial[n-k, k-1] + Sum[Binomial[j, n-j], {j, n-k}]];
Table[A373889[k-n+1, n], {k, 15}, {n, k}]

T(k,n) = A000045(n-k+2) + 1, for k = 1 and n >= k;
T(k,n) = 2*(Sum_{i=0..k-2} binomial(n-k,i)*A000045(k-i)) + 2*binomial(n-k,k-1) + Sum_{j=1..n-k} binomial(j,n-j), for k >= 2 and n >= k;
T(k,n) = A000045(n+1) otherwise.
T(n,n) = 2*A000045(n).

A374440 Triangle read by rows: T(n, k) = T(n - 1, k) + T(n - 2, k - 2), with boundary conditions: if k = 0 or k = 2 then T = 1; if k = 1 then T = n - 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 3, 1, 1, 1, 1, 4, 1, 3, 2, 0, 1, 5, 1, 6, 3, 1, 1, 1, 6, 1, 10, 4, 4, 3, 0, 1, 7, 1, 15, 5, 10, 6, 1, 1, 1, 8, 1, 21, 6, 20, 10, 5, 4, 0, 1, 9, 1, 28, 7, 35, 15, 15, 10, 1, 1, 1, 10, 1, 36, 8, 56, 21, 35, 20, 6, 5, 0

Offset: 0

Peter Luschny, Jul 21 2024

Member of the family of Lucas-Fibonacci polynomials.

			Triangle starts:
  [ 0]  1;
  [ 1]  1,  0;
  [ 2]  1,  1,  1;
  [ 3]  1,  2,  1,  0;
  [ 4]  1,  3,  1,  1,  1;
  [ 5]  1,  4,  1,  3,  2,  0;
  [ 6]  1,  5,  1,  6,  3,  1,  1;
  [ 7]  1,  6,  1, 10,  4,  4,  3,  0;
  [ 8]  1,  7,  1, 15,  5, 10,  6,  1,  1;
  [ 9]  1,  8,  1, 21,  6, 20, 10,  5,  4,  0;
  [10]  1,  9,  1, 28,  7, 35, 15, 15, 10,  1, 1;

Cf. A374441.

Cf. A000032 (Lucas), A001611 (even sums, Fibonacci + 1), A000071 (odd sums, Fibonacci - 1), A001911 (alternating sums, Fibonacci(n+3) - 2), A025560 (row lcm), A073028 (row max), A117671 & A025174 (central terms), A057979 (subdiagonal), A000217 (column 3).

T := proc(n, k) option remember; if k = 0 or k = 2 then 1 elif k > n then 0
elif k = 1 then n - 1 else T(n - 1, k) + T(n - 2, k - 2) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..9);
T := (n, k) -> ifelse(k = 0, 1, binomial(n - floor(k/2), ceil(k/2)) -
binomial(n - ceil((k + irem(k + 1, 2))/2), floor(k/2))):

T(n, k) = binomial(n - floor(k/2), ceiling(k/2)) - binomial(n - ceiling((k + even(k))/2), floor(k/2)) if k > 0, T(n, 0) = 1, where even(k) = 1 if k is even, otherwise 0.

Columns with odd index agree with the odd indexed columns of A374441.

A052660 E.g.f. (2-2x-x^2)/((1-x)(1-x-x^2)).

Original entry on oeis.org

2, 2, 6, 24, 144, 1080, 10080, 110880, 1411200, 20321280, 326592000, 5787936000, 112086374400, 2353813862400, 53265935923200, 1291982275584000, 33434618241024000, 919452001628160000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 607

spec := [S,{S=Union(Sequence(Z),Sequence(Union(Z,Prod(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

E.g.f.: -(-2+x^2+2*x)/(-1+x)/(-1+x+x^2)
Recurrence: {a(1)=2, a(2)=6, a(0)=2, (n^3+6*n^2+11*n+6)*a(n)+(-2*n-6)*a(n+2)+a(n+3)=0}
(1+Sum(1/5*(1+2*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+_Z^2)))*n!
a(n) = n!*A001611(n+1). - R. J. Mathar, Nov 27 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

cp's OEIS Frontend

A168514 Number of prime divisors (counted with multiplicity) of Fibonacci(n)+1.

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

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A373889 Square array read by ascending antidiagonals: T(k,n) is the cardinality of {(E is a proper finite subset of the natural numbers) such that E = {} or w_k(E) < min(E) <= max(E) <= n}, where w_k(E) = Sum_{i in E, i <> k} 1, with n, k >= 1.

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A374440 Triangle read by rows: T(n, k) = T(n - 1, k) + T(n - 2, k - 2), with boundary conditions: if k = 0 or k = 2 then T = 1; if k = 1 then T = n - 1.

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A052660 E.g.f. (2-2x-x^2)/((1-x)(1-x-x^2)).

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

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