A000071 -id:A000071 - cp's OEIS frontend

A194055 Natural fractal sequence of A000071 (Fibonacci numbers minus 1).

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29

Offset: 1

Clark Kimberling, Aug 13 2011

nonn

See A194029 for definitions of natural fractal sequence and natural interspersion.

Cf. A194029, A000071, A194056, A194057.

z = 50;
c[k_] := -1 + Fibonacci[k + 2]
c = Table[c[k], {k, 1, z}] (* A000071, F(n+2)-1 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194055 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 11}, {k, 1, n}]] (* A194056 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194057 *)

A202876 Symmetric matrix based on A000071, by antidiagonals.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 7, 10, 10, 7, 12, 18, 21, 18, 12, 20, 31, 38, 38, 31, 20, 33, 52, 66, 70, 66, 52, 33, 54, 86, 111, 122, 122, 111, 86, 54, 88, 141, 184, 206, 214, 206, 184, 141, 88, 143, 230, 302, 342, 362, 362, 342, 302, 230, 143, 232, 374, 493, 562, 602

Offset: 1

Clark Kimberling, Dec 26 2011

Let s=A000071 (Fibonacci numbers -1), and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A202876 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202877 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

			Northwest corner:
1....2....4....7....12....20
2....5....10...18...31....52
4....10...21...38...66....111
7....18...38...70...122...206
12...31...66...122..214...362

Cf. A202877, A202876.

s[k_] := -1 + Fibonacci[k + 2];
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* A001924 *)
Table[m[1, j], {j, 1, 12}]     (* A000071 *)
Table[m[j, j], {j, 1, 12}]     (* A202462 *)

A194056 Natural interspersion of A000071(Fibonacci numbers minus 1), a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 11, 20, 21, 22, 23, 16, 17, 33, 34, 35, 36, 24, 25, 18, 54, 55, 56, 57, 37, 38, 26, 19, 88, 89, 90, 91, 58, 59, 39, 27, 28, 143, 144, 145, 146, 92, 93, 60, 40, 41, 29, 232, 233, 234, 235, 147, 148, 94, 61, 62, 42, 30

Offset: 1

Clark Kimberling, Aug 13 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194056 is a permutation of the positive integers; its inverse is A194057.

			Northwest corner:
1...2...4...7...12
3...5...8...13..21
6...9...14..22..35
10..15..23..36..57
11..16..24..37..58

Cf. A194029, A194055, A000071, A000045, A194057.

z = 50;
c[k_] := -1 + Fibonacci[k + 2]
c = Table[c[k], {k, 1, z}] (* A000071, F(n+2)-1 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194055 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 11}, {k, 1, n}]] (* A194056 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194057 *)

A226649 Fibonacci shuffles: a(2n) = A000071(n) and a(2n+1) = A001611(n+2).

Original entry on oeis.org

0, 2, 0, 3, 1, 4, 2, 6, 4, 9, 7, 14, 12, 22, 20, 35, 33, 56, 54, 90, 88, 145, 143, 234, 232, 378, 376, 611, 609, 988, 986, 1598, 1596, 2585, 2583, 4182, 4180, 6766, 6764, 10947, 10945, 17712, 17710, 28658, 28656, 46369, 46367, 75026, 75024, 121394, 121392, 196419, 196417, 317812, 317810

Offset: 0

V. T. Jayabalaji, Jun 14 2013

a(2*n+1) = a(2*n) + A157725(n); a(2*n) = a(2*n-1) - 2 for n > 0. - Reinhard Zumkeller, Jul 30 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (-1,1,1,1,1).

Cf. A000071, A001611, A096748.

import Data.List (transpose)
a226649 n = a226649_list !! n
a226649_list = concat $ transpose [a000071_list, drop 2 a001611_list]
-- Reinhard Zumkeller, Jul 30 2013

LinearRecurrence[{-1,1,1,1,1},{0,2,0,3,1},60] (* Harvey P. Dale, Sep 12 2018 *)

G.f. -x*(2+x^2+2*x^3+2*x) / ( (1+x)*(x^4+x^2-1) ). - R. J. Mathar, Jul 15 2013
a(n) + a(n+1) = A096748(n+2). - R. J. Mathar, Jul 15 2013
a(2n-1) - 1 = a(2n) + 1 = fib(n+1) = A000045(n+1) for n > 0. - T. D. Noe, Jul 23 2013

A263161 Positive values of n such that A000071(n+2) is divisible by A000217(n).

Original entry on oeis.org

1, 240, 600, 768, 1008, 1200, 1320, 1800, 2160, 2688, 2736, 3000, 3360, 3888, 4800, 5280, 5520, 6120, 6479, 6480, 6720, 6840, 7320, 7680, 8208, 8640, 9000, 9600, 9720, 10368, 11160, 11663, 12240, 12288, 13200, 13248, 13440, 13680, 14400, 15120, 15360, 15456, 16560, 18048

Offset: 1

Altug Alkan, Oct 11 2015

nonn

Interestingly, the minimum value of a(n) - a(n-1) is 1. Is there a maximum value of a(n) - a(n-1)?
From Robert Israel, Oct 19 2015: (Start)
n is in the sequence if either n is odd and A001175(n) and A001175((n+1)/2) both divide n+1, or n is even and A001175(n/2) and A001175(n+1) both divide n.
Most of the terms of the sequence appear to fall in these categories. The first two that do not are 15456 and 41640.
In particular, if n = 2^j * 3^k * 5^m with j >= 4, k >= 1 and m >= 1, and n+1 is prime, then n is in the sequence.  There are believed to be infinitely many numbers of this form.  The first few are 240, 1200, 2160, 4800, 6480, 7680, 8640, 9600, 14400, 15360. (End)

			For n = 1, A000071(1+2) = 1 is divisible by A000217(1) = 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000045, A000071, A000217, A001175, A263225.

[n: n in [1..20000] | IsDivisibleBy(Fibonacci(n+2)-1, n*(n+1) div 2)]; // Bruno Berselli, Oct 19 2015

fmod:= proc(a, b) local A, t;
  uses LinearAlgebra[Modular];
  if b < 4295022903 then t:= integer[8] else t:= integer fi;
  A:= Mod(b, <<1, 1>|<1, 0>>, t);
  MatrixPower(b, A, a)[1, 2];
end proc:
filter:= n -> (fmod(n+2, n*(n+1)/2) = 1):
filter(1):= true:
select(filter, [$1..10^5]); # Robert Israel, Oct 19 2015

fQ[n_] := Mod[Fibonacci[n + 2] - 1, n (n + 1)/2] == 0; Select[Range@20000, fQ] (* Bruno Berselli, Oct 19 2015 - after Robert G. Wilson v in A263225 *)

for(n=1, 20000, if((fibonacci(n+2)-1) % (n*(n+1)/2) == 0, print1(n", ")));

is(n)=((Mod([1,1;1,0],n*(n+1)/2))^(n+2))[1,2]==1 \\ Charles R Greathouse IV, Oct 19 2015

A129705 Triangle T(n,m) = A000071(n+2)-m*(m+1)/2 read by rows.

Original entry on oeis.org

0, 1, 0, 2, 1, -1, 4, 3, 1, -2, 7, 6, 4, 1, -3, 12, 11, 9, 6, 2, -3, 20, 19, 17, 14, 10, 5, -1, 33, 32, 30, 27, 23, 18, 12, 5, 54, 53, 51, 48, 44, 39, 33, 26, 18, 88, 87, 85, 82, 78, 73, 67, 60, 52, 43, 143, 142, 140, 137, 133, 128, 122, 115, 107, 98, 88

Offset: 0

Roger L. Bagula, Jun 08 2007

The row sums are 0, 1, 2, 6, 15, 37, 84, 180, 366, 715, 1353, 2498, 4524 = (n+1)*(A000071(n+2) -(n+2)*n/6). - R. J. Mathar, Sep 09 2011

			0;
1, 0;
2, 1, -1;
4, 3, 1, -2;
7, 6, 4, 1, -3;
12, 11, 9, 6, 2, -3;
20, 19, 17, 14, 10, 5, -1;
33, 32, 30, 27, 23, 18, 12, 5;

Eric Weisstein's World of Mathematics, Schubert Variety

Cf. A000045.

A000071 := proc(n) if n = 0 then 0; else combinat[fibonacci](n)-1 ; end if; end proc:
A129705 := proc(n,m) A000071(n+2)-m*(m+1)/2 ; end proc: # R. J. Mathar, Sep 09 2011

fib[n_Integer?Positive] := fib[n] = fib[n - 1] + fib[n - 2]; fib[0] = 0; fib[1] = fib[2] = 1; t[n_, m_] = Sum[fib[i], {i, 0, n}] - Sum[i, {i, 0, m}]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]

t(n,m) = sum_{i=0..n} A000045(i) - sum_{i=0..m} i, 0<=m<=n.

A138859 Smallest prime factor of Fibonacci(3n)-1, i.e., A020639(A000071(3n)).

Original entry on oeis.org

7, 3, 11, 3, 3, 5, 199, 13, 13, 3, 37, 3, 3, 89, 5, 11, 7, 3, 59, 3, 3, 11, 11, 37, 37, 3, 370248451, 3, 3, 139, 13, 5, 5, 3, 5, 3, 3, 59, 709, 557, 127, 3, 11, 3, 3, 5, 9375829, 73, 7, 3, 29, 3, 3, 2789, 5, 11, 47, 3, 13, 3, 3, 11, 11, 13, 7, 3, 809, 3, 3, 953, 47927441, 5, 5

Offset: 2

M. F. Hasler, Apr 05 2008

nonn

F(n)-1 is even if n is not a multiple of 3.

Amiram Eldar, Table of n, a(n) for n = 2..934 (terms 2..379 from Chai Wah Wu)

Cf. A000071, A014445, A020639.

Table[FactorInteger[Fibonacci[3n]-1][[1,1]],{n,2,80}] (* Harvey P. Dale, Aug 08 2017 *)

A138859(n)=factor(fibonacci(3*n)-1)[1,1] /* A138859 is defined only for n>1 ! */

a(n) = A020639(A000071(3n)).

A140992 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-2) + a(n-1) + A000071(n+1).

Original entry on oeis.org

0, 1, 2, 5, 11, 23, 46, 89, 168, 311, 567, 1021, 1820, 3217, 5646, 9849, 17091, 29523, 50794, 87081, 148820, 253611, 431087, 731065, 1237176, 2089633, 3523226, 5930669, 9968123, 16730831, 28045222, 46954361, 78524160, 131181407

Offset: 0

Juri-Stepan Gerasimov, Jul 08 2008

			If n = 4, then a(4) = a(4-2) + a(4-1) + A000071(4+1) = a(2) + a(3) + A000071(5) = 2 + 5 + 4 = 11.

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

Cf. A000071, A000045, A140993, A140998.

LinearRecurrence[{3,-1,-3,1,1},{0,1,2,5,11},40] (* Harvey P. Dale, Jun 12 2014 *)

From R. J. Mathar, Apr 27 2010: (Start)
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) + a(n-5).
G.f.: -x*(1 - x + x^3) / ( (x - 1)*(x^2 + x - 1)^2 ). (End)
a(n) = A140998(n+1, k = 2) = A140993(n+2, n) for n >= 1. - Petros Hadjicostas, Jun 10 2019

Corrected (5980669 replaced by 5930669) by R. J. Mathar, Apr 27 2010

A158950 Triangle read by rows, A158948 * (an infinite lower triangular matrix with A000071 prefaced with a 1 as the right border; and the rest zeros).

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 2, 1, 0, 4, 3, 0, 2, 0, 7, 3, 1, 0, 4, 0, 12, 4, 0, 2, 0, 7, 0, 20, 4, 1, 0, 4, 0, 12, 0, 33, 5, 0, 2, 0, 7, 0, 20, 0, 54, 5, 1, 0, 4, 0, 12, 0, 33, 0, 88, 6, 0, 2, 0, 7, 0, 20, 0, 54, 0, 143

Offset: 1

Gary W. Adamson, Mar 31 2009

Row sums = A000071 starting with nonzero terms: (1, 2, 4, 7, 12,...) As a property of eigentriangles, sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
2, 0, 2;
2, 1, 0, 4;
3, 0, 2, 0, 7;
3, 1, 0, 4, 0, 12;
4, 0, 2, 0, 7, 0, 20;
4, 1, 0, 4, 0, 12, 0, 33;
5, 0, 2, 0, 7, 0, 20, 0, 54;
5, 1, 0, 4, 0, 12, 0, 33, 0, 88;
6, 0, 2, 0, 7, 0, 20, 0, 54, 0, 143;
6, 1, 0, 4, 0, 12, 0, 33, 0, 88, 0, 232;
7, 0, 2, 0, 7, 0, 20, 0, 54, 0, 143, 0, 376;
7, 1, 0, 4, 0, 12, 0, 33, 0, 88, 0, 232, 0, 609;
...
Row 4 = (2, 1, 0, 4) = termwise products of (2, 1, 0, 1) and (1, 1, 2, 4);
where (2, 1, 0, 1) = row 4 of triangle A158948, and (1, 1, 2, 4) = the 3 nonzero terms of A000071 prefaced with a 1.

A158948, A000071

Triangle read by rows, A158948 * M; where M = (an infinite lower triangular matrix with A000071 prefaced with a 1 as the right border, and the rest zeros). M = (1; 0,1; 0,0,2; 0,0,0,4; 0,0,0,7;...).

A263537 Integers k such that A098531(k) is divisible by A000071(k+2).

Original entry on oeis.org

1, 2, 13, 25, 31, 43, 55, 61, 73, 85, 91, 103, 115, 121, 133, 145, 151, 163, 175, 181, 193, 205, 211, 223, 235, 241, 253, 265, 271, 283, 295, 301, 313, 325, 331, 343, 355, 361, 373, 385, 391, 403, 415, 421, 433, 445, 451, 463, 475, 481, 493, 505, 511, 523, 535, 541, 553

Offset: 1

Altug Alkan, Oct 20 2015

Sequence is interesting because of the values of a(n) - a(n-1) that are 12 or 6 for n > 3.

a(2) = 2 is the only even term.

			a(1) = 1 because 1^5 mod 1 = 0.
a(2) = 2 because (1^5 + 1^5) mod (1 + 1) = 0.

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

Cf. A000045, A000071, A098531.

I:=[1,2,13,25,31,43]; [n le 6 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, Nov 20 2015

lim = 560; s = Accumulate[Fibonacci[Range@lim]^5]; t = Fibonacci@ Range[2 lim] - 1; Select[Range@ lim, Divisible[s[[#]], t[[# + 2]]] &] (* Michael De Vlieger, Nov 19 2015, after Harvey P. Dale at A098531 and A000071 *)

for(n=1, 1e3, if(sum(k=1, n, fibonacci(k)^5) % sum(k=1, n, fibonacci(k)) == 0, print1(n", ")));

Vec(x*(x^5+5*x^4+11*x^3+11*x^2+x+1)/((x-1)^2*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Oct 20 2015

From Colin Barker, Oct 20 2015: (Start)
a(n) = a(n-1)+a(n-3)-a(n-4) for n>6.
G.f.: x*(x^5+5*x^4+11*x^3+11*x^2+x+1) / ((x-1)^2*(x^2+x+1)).
(End)

cp's OEIS Frontend

A194055 Natural fractal sequence of A000071 (Fibonacci numbers minus 1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A202876 Symmetric matrix based on A000071, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A194056 Natural interspersion of A000071(Fibonacci numbers minus 1), a rectangular array, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A226649 Fibonacci shuffles: a(2n) = A000071(n) and a(2n+1) = A001611(n+2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A263161 Positive values of n such that A000071(n+2) is divisible by A000217(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

A129705 Triangle T(n,m) = A000071(n+2)-m*(m+1)/2 read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A138859 Smallest prime factor of Fibonacci(3n)-1, i.e., A020639(A000071(3n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A140992 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-2) + a(n-1) + A000071(n+1).

Views

Author

Keywords

Examples