A058071 -id:A058071 - cp's OEIS frontend

A098357 a(n) = A061017(n) - pi(n+1).

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3

Offset: 1

David Applegate and N. J. A. Sloane, Oct 22 2008

nonn

a(n) >= 0 for all n >= 1.

The old entry with this sequence number was a duplicate of A058071.

A companion to A052511. Cf. A061017, A006218, A088526.

A100059 First differences of A052911.

Original entry on oeis.org

1, 5, 14, 45, 139, 434, 1351, 4209, 13110, 40837, 127203, 396226, 1234207, 3844441, 11975078, 37301261, 116189979, 361921042, 1127350583, 3511592833, 10938286998, 34071752661, 106130359315, 330586256610

Offset: 1

Gary W. Adamson, Oct 31 2004

nonn

a(n)/a(n-1) tends to 3.11490754148...an eigenvalue of M and a root of the characteristic polynomial x^3 - 3x^2 - x + 2.

			a(5) = 139 = rightmost term in M^5 * [1 1 1] which is [434 205 139]. 434 = a(6), while 205 = A052911(5).
a(6) = 434 = 3*a(5) + a(4) - 2*a(3) = 3*139 + 45 - 2*14.

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Index entries for linear recurrences with constant coefficients, signature (3,1,-2).
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 27.

Cf. A019481, A052550, A052939, A100058, A058071.

LinearRecurrence[{3,1,-2},{1,5,14},30] (* Harvey P. Dale, Apr 21 2016 *)

G.f.: (2*x^2-2*x-1)*x / (-2*x^3+x^2+3*x-1).
Recurrence: a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = rightmost term in M^5 * [1 1 1], where M = the 3 X 3 upper triangular matrix [2 1 2 / 1 1 0 / 1 0 0].
INVERT transform of (1, 4, 5, 6, 7, 8, 9, ...) with offset 0.

Edited by Ralf Stephan, Nov 02 2004

A250111 Number of orbits of size 2 in vertices of Fibonacci cube Gamma_n under the action of its automorphism group.

Original entry on oeis.org

1, 1, 1, 3, 4, 9, 13, 25, 38, 68, 106, 182, 288, 483, 771, 1275, 2046, 3355, 5401, 8811, 14212, 23112, 37324, 60580, 97904, 158717, 256621, 415715, 672336, 1088661, 1760997, 2850645, 4611642, 7463884, 12075526, 19541994, 31617520, 51163695, 82781215

Offset: 1

N. J. A. Sloane, Nov 19 2014

Colin Barker, Table of n, a(n) for n = 1..1000
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, Vertex and edge orbits of Fibonacci and Lucas cubes, 2014; See Table 1.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,0,-1,-1).

Cf. A000045, A058071.

[n eq 1 select 1 else (1/2)*(Fibonacci(n+2)-Fibonacci(Floor((n-(-1)^n)/2)+2)): n in [1..40]]; // Vincenzo Librandi, Nov 22 2014

LinearRecurrence[{1,2,-1,0,-1,-1},{1,1,1,3,4,9,13},40] (* Harvey P. Dale, Feb 10 2018 *)

a(n)=if(n==1,1,(fibonacci(n+2) - fibonacci((n-(-1)^n)\2+2))/2); \\ Joerg Arndt, Nov 22 2014

Vec(x*(1-2*x^2+x^3+x^5+x^6)/((1-x-x^2)*(1-x^2-x^4)) + O(x^100)) \\ Colin Barker, Dec 01 2014

def A250111(n): return bool(n==1) + sum( fibonacci(j+1)*fibonacci(n-2*j-1) for j in (0..((n-1)//2)) )
[A250111(n) for n in (1..50)] # G. C. Greubel, Apr 06 2022

a(n) = (1/2) * (F(n+2) - F(floor((n-(-1)^n)/2)+2)) for n >= 2, a(1)=1. - Joerg Arndt, Nov 22 2014
a(n) = a(n-1)+2*a(n-2)-a(n-3)-a(n-5)-a(n-6) for n>7. - Colin Barker, Dec 01 2014
G.f.: x*(1-2*x^2+x^3+x^5+x^6)/((1-x-x^2)*(1-x^2-x^4)). - Colin Barker, Dec 01 2014
From G. C. Greubel, Apr 06 2022: (Start)
a(n) = [n=1] + Sum_{k=0..floor((n-1)/2)} Fibonacci(k+1)*Fibonacci(n-2*k-1).
a(2*n) = (1/2)*(Fibonacci(2*n+2) - Fibonacci(n+1)), n >= 1.
a(2*n+1) = (1/2)*(Fibonacci(2*n+3) - Fibonacci(n+3) + 2*[n=0]), n >= 0. (End)

More terms from Vincenzo Librandi, Nov 22 2014

A283845 Square array read by antidiagonals: T(1,1) = T(1,2) = T(2,1) = T(2,2) = 1; thereafter T(m,n) = min {T(m,n-2) + T(m,n-1), T(m-2,n) + T(m-1,n), T(m-2,n-2) + T(m-1,n-1)}.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 5, 3, 2, 3, 5, 8, 5, 3, 3, 5, 8, 13, 8, 5, 3, 5, 8, 13, 21, 13, 8, 5, 5, 8, 13, 21, 34, 21, 13, 8, 5, 8, 13, 21, 34, 55, 34, 21, 13, 8, 8, 13, 21, 34, 55, 89, 55, 34, 21, 13, 8, 13, 21, 34, 55, 89, 144, 89, 55, 34, 21, 13, 13, 21, 34, 55, 89, 144

Offset: 1

N. J. A. Sloane, Mar 31 2017

A naive version of a two-dimensional Fibonacci array.
There should probably be another entry for the array which has offset 0 and starts with T(0,0) = 0, T(0,1) = T(1,0) = T(1,1) = 1.
See A058071 for a more interesting version.
T(n, 1) = T(n, n) = A000045(n) for n > 0. - Indranil Ghosh, Apr 01 2017

			The square array begins:
   1,  1,  2,  3,  5,  8, 13, 21, ...
   1,  1,  2,  3,  5,  8, 13, 21, ...
   2,  2,  2,  3,  5,  8, 13, 21, ...
   3,  3,  3,  3,  5,  8, 13, 21, ...
   5,  5,  5,  5,  5,  8, 13, 21, ...
   8,  8,  8,  8,  8,  8, 13, 21, ...
  13, 13, 13, 13, 13, 13, 13, 21, ...
  ...
The first few antidiagonals are:
   1;
   1, 1;
   2, 1, 2;
   3, 2, 2, 3;
   5, 3, 2, 3, 5;
   8, 5, 3, 3, 5, 8;
  13, 8, 5, 3, 5, 8, 13;
  ...

Indranil Ghosh, Rows 1..120, flattened
Indranil Ghosh, C program to generate the triangle

Cf. A000045, A058071.

Table[Fibonacci[Max[m, n - m + 1]], {n, 20}, {m, n}] // Flatten (* Indranil Ghosh, Apr 01 2017 *)

tabl(nn) = {for(n=1, nn, for(m=1, n, print1(fibonacci(max(m, n - m + 1)),", ");); print(););}
tabl(20) \\ Indranil Ghosh, Apr 01 2017

from sympy import fibonacci
for n in range(1, 21):
    print([fibonacci(max(m, n - m + 1)) for m in range(1, n + 1)]) # Indranil Ghosh, Apr 01 2017

T(m,n) = Fibonacci(k) where k = max(m,n).

Extended by Indranil Ghosh, Apr 01 2017

A304726 a(n) = n^4 + 4*n^2 + 3.

Original entry on oeis.org

3, 8, 35, 120, 323, 728, 1443, 2600, 4355, 6888, 10403, 15128, 21315, 29240, 39203, 51528, 66563, 84680, 106275, 131768, 161603, 196248, 236195, 281960, 334083, 393128, 459683, 534360, 617795, 710648, 813603, 927368, 1052675, 1190280, 1340963, 1505528, 1684803

Offset: 0

Vincenzo Librandi, May 31 2018

Alternating sum of all points on the fourth row of the Hosoya triangle composed of Fibonacci polynomials, where F_{0}(n) = 1 and F_{1}(n) = n, hence a(n) = F_{5}(n)/F_{1}(n) for n>0 (see Florez et al. reference, page 7, Table 4 and following sum).

Apart from 8, all terms belong to A217554 because a(n) = (n^2+1)^2 + (n+1)^2 + (n-1)^2 = (n^2+2)^2 - 1. - Bruno Berselli, Jun 04 2018

Muniru A Asiru, Table of n, a(n) for n = 0..10000
Rigoberto Florez, Robinson A. Higuita, and Antara Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.5.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A058071, A059100, A217554.

Subsequence of A005563.

List([0..40], n -> (n^2+2)^2-1); # Muniru A Asiru, Jun 03 2018

Magma
```
[n^4+4*n^2+3: n in [0..40]];
```

seq((n^2+2)^2-1,n=0..40); # Muniru A Asiru, Jun 03 2018

Table[n^4 + 4 n^2 + 3, {n, 0, 35}]
LinearRecurrence[{5,-10,10,-5,1},{3,8,35,120,323},40] (* Harvey P. Dale, Mar 04 2021 *)

G.f.: (3 - 7*x + 25*x^2 - 5*x^3 + 8*x^4)/(1-x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A059100(n)^2 - 1.
Sum_{n>=0} 1/a(n) = 1/6 + coth(Pi)*Pi/4 - coth(sqrt(3)*Pi)*Pi/(4*sqrt(3)). - Amiram Eldar, Feb 24 2023

A110361 A triangle of coefficients based on A000931 and A000045: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)a(m + 1)Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 6, 4, 4, 6, 15, 6, 16, 6, 15, 32, 15, 24, 24, 15, 32, 65, 32, 60, 36, 60, 32, 65, 147, 65, 128, 90, 90, 128, 65, 147, 306, 147, 260, 192, 225, 192, 260, 147, 306, 660, 306, 588, 390, 480, 480, 390, 588, 306, 660, 1424, 660, 1224, 882, 975, 1024, 975, 882

Offset: 1

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Row sums are:

{1, 2, 9, 20, 58, 142, 350, 860, 2035, 4848, 11354}.

			{1},
{1, 1},
{4, 1, 4},
{6, 4, 4, 6},
{15, 6, 16, 6, 15},
{32, 15, 24, 24, 15, 32},
{65, 32, 60, 36, 60, 32, 65},
{147, 65, 128, 90, 90, 128, 65, 147},
{306, 147, 260, 192, 225, 192, 260, 147, 306},
{660, 306, 588, 390, 480, 480, 390, 588, 306, 660},
{1424, 660, 1224, 882, 975, 1024, 975, 882, 1224, 660, 1424}

Cf. A141611, A141617, A000931, A000045, A058071.

Clear[t, a, n, m] a[0] = 1; a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 2] + a[n - 3]; t[n_, m_] := a[(n - m + 1)]*a[(m + 1)]*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1)*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].

cp's OEIS Frontend

A098357 a(n) = A061017(n) - pi(n+1).

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A100059 First differences of A052911.

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A250111 Number of orbits of size 2 in vertices of Fibonacci cube Gamma_n under the action of its automorphism group.

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A283845 Square array read by antidiagonals: T(1,1) = T(1,2) = T(2,1) = T(2,2) = 1; thereafter T(m,n) = min {T(m,n-2) + T(m,n-1), T(m-2,n) + T(m-1,n), T(m-2,n-2) + T(m-1,n-1)}.

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A304726 a(n) = n^4 + 4*n^2 + 3.

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A110361 A triangle of coefficients based on A000931 and A000045: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)*a(m + 1)*Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].

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A110361 A triangle of coefficients based on A000931 and A000045: a(n) = a(n - 2) + a(n - 3); t(n,m) := a(n - m + 1)a(m + 1)Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)].