A102426 -id:A102426 - cp's OEIS frontend

A169803 Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 3, 0, 0, 1, 5, 6, 1, 0, 0, 1, 6, 10, 4, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 0, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0, 1, 12, 55, 120, 126, 56, 7, 0, 0, 0, 0, 0, 0

Offset: 0

Nadia Heninger and N. J. A. Sloane, May 21 2010

T(n,k) = 0 if k <0 or k > n+1-k.
T(n,k) is the number of binary vectors of length n and weight k containing no pair of adjacent 1's.
Take Pascal's triangle A007318 and push the k-th column downwards by 2k-1 places (k>=1).
Row sums are A000045.
From Emanuele Munarini, May 24 2011: (Start)
Diagonal sums are A000930(n+1).
A sparse subset (or scattered subset) of {1,2,...,n} is a subset never containing two consecutive elements. T(n,k) is the number of sparse subsets of {1,2,...,n} having size k. For instance, for n=4 and k=2 we have the 3 sparse 2-subsets of {1,2,3,4}: 13, 14, 24. (End)
As a triangle, row 2*n-1 consists of the coefficients of Morgan-Voyce polynomial B(n,x), A172431, and row 2*n to the coefficients of Morgan-Voyce polynomial b(n,x), A054142.
Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link p. 19). - Tom Copeland, Oct 11 2014
Antidiagonals of the Pascal matrix A007318 read bottom to top, omitting the first antidiagonal. These are also the antidiagonals (omitting the first antidiagonal) read from top to bottom of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Reverse is embedded in A102426. - Tom Copeland, Jul 02 2018

			Triangle begins:
  [1]
  [1, 1]
  [1, 2, 0]
  [1, 3, 1, 0]
  [1, 4, 3, 0, 0]
  [1, 5, 6, 1, 0, 0]
  [1, 6, 10, 4, 0, 0, 0]
  [1, 7, 15, 10, 1, 0, 0, 0]
  [1, 8, 21, 20, 5, 0, 0, 0, 0]
  [1, 9, 28, 35, 15, 1, 0, 0, 0, 0]
  [1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0]
  [1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0]
  [1, 12, 55, 120, 126, 56, 7, 0, 0, 0, 0, 0, 0]
  [1, 13, 66, 165, 210, 126, 28, 1, 0, 0, 0, 0, 0, 0]
  [1, 14, 78, 220, 330, 252, 84, 8, 0, 0, 0, 0, 0, 0, 0]
  [1, 15, 91, 286, 495, 462, 210, 36, 1, 0, 0, 0, 0, 0, 0, 0]
  [1, 16, 105, 364, 715, 792, 462, 120, 9, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0]
  [1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
  ...

Indranil Ghosh, Rows 0..125, flattened
Pantelis A. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Emanuele Munarini and Norma Zagaglia Salvi, Scattered Subsets, Discrete Mathematics 267 (2003), 213-228.
Emanuele. Munarini and Norma Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
Emanuele Munarini, A combinatorial interpretation of the Chebyshev polynomials, SIAM Journal on Discrete Mathematics, Volume 20, Issue 3 (2006), 649-655.
Peter J. Olver, The canonical contact form.
James J. Y. Zhao, Infinite log-concavity and higher order Turán inequality for Speyer's g-polynomial of uniform matroids, arXiv:2409.08085 [math.CO], 2024. See p. 11.

Cf. A000045, A000930, A007318, A011973 (another version), A218272.
All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - N. J. A. Sloane, May 29 2011
A172431 and A054142 describe the odd and even lines of the triangle.

T[n_,k_]:= Binomial[n+1-k,k]; Table[T[n,k],{n,0,12},{k,0,n}]//Flatten (* Stefano Spezia, Sep 16 2024 *)

create_list(binomial(n-k+1,k),n,0,20,k,0,n); /* Emanuele Munarini, May 24 2011 */

T(n,k)=binomial(n+1-k,k) \\ Charles R Greathouse IV, Oct 24 2012

A092865 Nonzero elements in Klee's identity Sum[(-1)^k binomial[n,k]binomial[n+k,m],{k,0,n}] == (-1)^n binomial[n,m-n].

Original entry on oeis.org

1, -1, -1, 1, 2, -1, 1, -3, 1, -3, 4, -1, -1, 6, -5, 1, 4, -10, 6, -1, 1, -10, 15, -7, 1, -5, 20, -21, 8, -1, -1, 15, -35, 28, -9, 1, 6, -35, 56, -36, 10, -1, 1, -21, 70, -84, 45, -11, 1, -7, 56, -126, 120, -55, 12, -1, -1, 28, -126, 210, -165, 66, -13, 1, 8, -84, 252, -330, 220, -78, 14, -1, 1, -36, 210, -462, 495

Offset: 0

Eric W. Weisstein, Mar 07 2004

Triangle, with zeros omitted, given by (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 26 2011

Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link p. 19). - Tom Copeland, Oct 11 2014

			1;
-1;
-1, 1;
2, -1;
1, -3, 1;
-3, 4, -1;
-1, 6, -5, 1;
4, -10, 6, -1;
Triangle (0, 1, -1, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, ...) begins:
1
0, -1
0, -1, 1
0, 0, 2, -1
0, 0, 1, -3, 1
0, 0, 0, -3, 4, -1
0, 0, 0, -1, 6, -5, 1 ... - _Philippe Deléham_, Dec 26 2011

H.-H. Chern, H.-K. Hwang, T.-H. Tsai, Random unfriendly seating arrangement in a dining table, arXiv preprint arXiv:1406.0614 [math.PR], 2014
T. Copeland, Addendum to Elliptic Lie Triad
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Eric Weisstein's World of Mathematics, Klee's Identity

All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - N. J. A. Sloane, May 29 2011

Flatten[Table[(-1)^n Binomial[n, m-n], {m, 0, 20}, {n, Ceiling[m/2], m}]]

G.f.: 1/(1+y*x+y*x^2). - Philippe Deléham, Feb 08 2012

A248752 Decimal expansion of limit of the imaginary part of f(1-i,n)/f(1-i,n+1), where f(x,n) is the n-th Fibonacci polynomial.

Original entry on oeis.org

2, 5, 7, 0, 6, 5, 8, 6, 4, 1, 2, 1, 6, 7, 7, 1, 6, 0, 9, 0, 8, 5, 6, 8, 0, 6, 2, 0, 5, 2, 7, 3, 3, 7, 1, 8, 9, 0, 3, 7, 5, 7, 0, 0, 7, 9, 9, 8, 8, 1, 3, 4, 9, 4, 5, 2, 4, 1, 3, 0, 7, 9, 3, 7, 8, 0, 9, 4, 2, 2, 3, 6, 0, 4, 3, 1, 2, 1, 4, 5, 0, 9, 4, 0, 7, 6

Offset: 0

Clark Kimberling, Oct 13 2014

The analogous limit of f(1,n)/f(1,n+1) is the golden ratio (A001622).

			limit = 0.2570658641216771609085680620527337189037570...
Let q(x,n) = f(x,n)/f(x,n+1) and c = 1-i.
n   f(x,n)            Re(q(c,n))   Im(q(c,n))
1   1                 1/2          1/2
2   x                 3/5          1/5
3   1 + x^2           1/2          1/4
4   2*x + x^3         8/15         4/15
5   1 + 3*x^2 + x^4   69/130       33/130
Re(q(1-i,11)) = 5021/9490 = 0.5290832...
Im(q(1-i,11)) = 4879/18980 = 0.257060...

Cf. A248750, A248751, A102426, A001622, A300070.

evalf((1-sqrt(sqrt(5)-2))/2, 120); # Vaclav Kotesovec, Oct 19 2014

z = 300; t = Table[Fibonacci[n, x]/Fibonacci[n + 1, x], {n, 1, z}];
u = t /. x -> 1 - I;
d1 = N[Re[u][[z]], 130]
d2 = N[Im[u][[z]], 130]
r1 = RealDigits[d1]  (* A248751 *)
r2 = RealDigits[d2]  (* A248752 *)

Equals (1-sqrt(sqrt(5)-2))/2. - Vaclav Kotesovec, Oct 19 2014
From Wolfdieter Lang, Mar 02 2018: (Start)
Equals (1 - (2 - phi)*sqrt(phi))/2, with phi = A001622.
Equals (1/10)*y*(1 - (1/50)*y^2) with y = A300070. (End)

A033538 a(0)=1, a(1)=1, a(n) = 3*a(n-1) + a(n-2) + 1.

Original entry on oeis.org

1, 1, 5, 17, 57, 189, 625, 2065, 6821, 22529, 74409, 245757, 811681, 2680801, 8854085, 29243057, 96583257, 318992829, 1053561745, 3479678065, 11492595941, 37957465889, 125364993609, 414052446717, 1367522333761, 4516619448001, 14917380677765, 49268761481297

Offset: 0

Antti Karttunen

Number of times certain simple recursive programs (such as the Lisp program shown) call themselves on an input of length n.

This is the sequence A(1,1;3,1;1) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

E. Hyvönen and J. Seppänen, LISP-kurssi, Osa 6 (Funktionaalinen ohjelmointi), Prosessori 4/1983, pp. 48-50 (in Finnish).

T. D. Noe, Table of n, a(n) for n = 0..200
A. Karttunen, More information
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-1).

Cf. A001595, A006190, A033539, A102426.

a:=[1,1];; for n in [3..40] do a[n]:=3*a[n-1]+a[n-2] +1; od; a; # G. C. Greubel, Jul 10 2019

a033538 n = a033538_list !! n
a033538_list =
   1 : 1 : (map (+ 1) $ zipWith (+) a033538_list
                                    $ map (3 *) $ tail a033538_list)
-- Reinhard Zumkeller, Aug 14 2011

(defun rewerse (lista) (cond ((null (cdr lista)) lista) (t (cons (car (rewerse (cdr lista))) (rewerse (cons (car lista) (rewerse (cdr (rewerse (cdr lista))))))))))

I:=[1,1]; [n le 2 select I[n] else 3*Self(n-1) +Self(n-2) +1: n in [1..40]]; // G. C. Greubel, Jul 10 2019

a := proc(n) option remember; if(n < 2) then RETURN(1); else RETURN(3*a(n-1)+a(n-2)+1); fi; end;

CoefficientList[ Series[(1-3x+3x^2)/(1-4x+2x^2+x^3), {x,0,40}], x](* Jean-François Alcover, Nov 30 2011 *)
RecurrenceTable[{a[0]==a[1]==1,a[n]==3a[n-1]+a[n-2]+1},a,{n,40}] (* or *) LinearRecurrence[{4,-2,-1},{1,1,5},41] (* Harvey P. Dale, Jan 05 2012 *)
Table[(4*(Fibonacci[n,3] +Fibonacci[n-1,3]) -1)/3, {n,0,30}] (* G. C. Greubel, Oct 13 2019 *)

a(n)=([0,1,0; 0,0,1; -1,-2,4]^n*[1;1;5])[1,1] \\ Charles R Greathouse IV, Feb 19 2017

((1-3*x+3*x^2)/((1-x)*(1-3*x-x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 10 2019

From R. J. Mathar, Aug 22 2008: (Start)
O.g.f.: (1-3*x+3*x^2)/((1-x)*(1-3*x-x^2)).
a(n) = (4*A006190(n+1) - 8*A006190(n) - 1)/3. (End)
a(n) = 4*a(n-1) - 2*a(n-2) - a(n-3), a(0)=1=a(1), a(2)=5. Observed by G. Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010
a(n) = (4*(F(n,3) + F(n-1,3)) -1)/3, where F(n,x) is the Fibonacci polynomial (see A102426). - G. C. Greubel, Oct 13 2019

A284938 Triangle read by rows: coefficients of the edge cover polynomial for the n-path graph P_n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 1, 6, 5, 1, 0, 0, 0, 0, 0, 4, 10, 6, 1, 0, 0, 0, 0, 0, 1, 10, 15, 7, 1, 0, 0, 0, 0, 0, 0, 5, 20, 21, 8, 1, 0, 0, 0, 0, 0, 0, 1, 15, 35, 28, 9, 1, 0, 0, 0, 0, 0, 0, 0, 6, 35, 56, 36, 10, 1, 0, 0, 0, 0, 0, 0, 0, 1, 21, 70, 84, 45, 11, 1

Offset: 1

Eric W. Weisstein, Apr 06 2017

			0;
0,1;
0,0,1;
0,0,1,1;
0,0,0,2,1;
0,0,0,1,3,1;
0,0,0,0,3,4,1;
0,0,0,0,1,6,5,1;
0,0,0,0,0,4,10,6,1;
0,0,0,0,0,1,10,15,7,1;
0,0,0,0,0,0,5,20,21,8,1;
0,0,0,0,0,0,1,15,35,28,9,1;
0,0,0,0,0,0,0,6,35,56,36,10,1;
0,0,0,0,0,0,0,1,21,70,84,45,11,1;
...

Eric Weisstein's World of Mathematics, Edge Cover Polynomial
Eric Weisstein's World of Mathematics, Path Graph

Unsigned version of A057094.
Row sums are A000045(n-1).
Cf. A286912, A258993, A030528, A085478, A098925, A102426, A143858, etc.

Prepend[CoefficientList[Table[x^(n/2) Fibonacci[n - 1, Sqrt[x]], {n, 2, 14}], x], {0}] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
Prepend[CoefficientList[LinearRecurrence[{x, x}, {0, x}, {2, 14}], x], {0}] // Flatten (* Eric W. Weisstein, Apr 07 2017 *)

a(n) = abs(A057094(n)).

A102428 Central column of triangle A102427.

Original entry on oeis.org

1, 3, 6, 28, 55, 286, 560, 3060, 5985, 33649, 65780, 376740, 736281, 4272048, 8347680, 48903492, 95548245, 563921995, 1101716330, 6540715896, 12777711870, 76223753060, 148902215280, 891794789340, 1742058970275, 10468434365991

Offset: 0

Russell Walsmith, Jan 11 2005

nonn

Cf. A011973, A102426, A102427, A102429.

Table[Binomial[Floor[(5n+2)/2],Floor[(n+1)/2]],{n,0,30}] (* Harvey P. Dale, Mar 13 2015 *)

a(n) = binomial(floor((5*n+2)/2), floor((n+1)/2)). - David Wasserman, Apr 04 2008

More terms from David Wasserman, Apr 04 2008

A102429 Row sums of A102427.

Original entry on oeis.org

1, 2, 6, 11, 23, 44, 85, 163, 311, 594, 1132, 2158, 4112, 7835, 14928, 28441, 54186, 103234, 196679, 374707, 713880, 1360061, 2591143, 4936559, 9404966, 17918024, 34136814, 65036304, 123904967, 236059552, 449732673, 856815474, 1632375854

Offset: 0

Russell Walsmith, Jan 11 2005

			1=1, 1+1=2, 1+3+2=6; 1+5+4+1=11

Cf. A011973, A102426, A102427, A102428.

G.f.: (1+x^2-x^3)/((1-x)*(1-x-2*x^2+x^4)). - Vladeta Jovovic, Apr 05 2005

More terms from Vladeta Jovovic, Apr 05 2005

A108756 A triangle related to the Jacobsthal polynomials.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 4, 5, 1, 1, 3, 6, 6, 7, 1, 1, 1, 10, 15, 8, 9, 1, 1, 4, 10, 21, 28, 10, 11, 1, 1, 1, 20, 35, 36, 45, 12, 13, 1, 1, 5, 15, 56, 84, 55, 66, 14, 15, 1, 1, 1, 35, 70, 120, 165, 78, 91, 16, 17, 1, 1, 6, 21, 126, 210, 220, 286, 105, 120, 18, 19, 1, 1

Offset: 0

Paul Barry, Jun 22 2005

Riordan array ((1 + x - x^2)/(1 - x^2)^2, x/(1 - x^2)^2). Row sums are A108742. Diagonal sums are Fibonacci(n+1) = A000045(n+1). Corresponding diagonals triangle is A102426.

			Triangle begins (with rows n >= 0 and columns k >= 0) as follows:
  1;
  1,  1;
  1,  1,  1;
  2,  3,  1,  1;
  1,  4,  5,  1,  1;
  3,  6,  6,  7,  1,  1;
  1, 10, 15,  8,  9,  1,  1;
  4, 10, 21, 28, 10, 11,  1,  1;
  1, 20, 35, 36, 45, 12, 13,  1, 1;
  5, 15, 56, 84, 55, 66, 14, 15, 1, 1; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
D. Stutson, V. Kocic, and G. Arora, A Few Identities involving Jacobsthal polynomials, Xavier University of Louisiana, preprint, 2005.

Cf. A000045, A102426, A108742.

[[Binomial(Floor((n+k+1)/2)+k, Floor((n+k)/2)-k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 29 2019

Table[Binomial[Floor[(n+k+1)/2]+k, Floor[(n+k)/2]-k], {n,0,12}, {k,0,n} ]//Flatten (* G. C. Greubel, May 29 2019 *)

{T(n,k) = binomial(floor((n+k+1)/2)+k, floor((n+k)/2)-k)}; \\ G. C. Greubel, May 29 2019

[[binomial(floor((n+k+1)/2)+k, floor((n+k)/2)-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 29 2019

Number triangle: T(n, k) = binomial(floor((n + k + 1)/2) + k, floor((n + k)/2 - k)) for 0 <= k <= n.
From Petros Hadjicostas, May 30 2019: (Start)
Bivariate g.f.: Sum_{n, k > = 0} T(n,k) * x^n * y^k = (1 + x - x^2)/((1 - x^2)^2 - x * y). (Here, we assume T(n, k) = 0 for n < k. Because T(n, k) = A102426(n + 1 + k, k), we may use Tom Copeland's g.f. of the latter array, to get the g.f. of the current triangular array.)
G.f. for column k >= 0: (1 + x - x^2) * x^k/(1 - x^2)^(2*k + 2).
Recurrence: T(n, k) = T(n - 2, k) + T(n - (1 - (-1)^(n + k))/2, k - (1 + (-1)^(n + k))/2), for n >= 3 and 1 <= k <= n - 2, starting with T(n, n) = 1 = T(n + 1, n) for n >= 0, T(n, 0) = 1 when n is even >= 0, and T(n, 0) = (n + 1)/2 when n is odd >= 1.
Another recurrence: T(n, k) = T(n - 1, k - 1) + 2*T(n - 2, k) - T(n - 4, k) for n >= 4 and 1 <= k <= n - 4. (This follows from the fact that the denominator of the bivariate g.f. is x^0 * y^0 - x^1 * y^1 - 2 * x^2 * y^0 - x^4 * y^0.)
(End)

More terms from Petros Hadjicostas, May 29 2019

A175685 Array a(n,m) = Sum_{j=floor((n-1)/2)-m..floor(n-1)/2} binomial(n-j-1,j) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 3, 4, 3, 2, 1, 1, 1, 7, 5, 3, 2, 1, 1, 4, 7, 8, 5, 3, 2, 1, 1, 1, 14, 12, 8, 5, 3, 2, 1, 1

Offset: 1

Roger L. Bagula, Dec 04 2010

A102426 defines an array of binomials in which partial sums of row n yield row a(n,.).

			a(n,m) starts in row n=1 as
  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
  2,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3, ...
  1,  4,  5,  5,  5,  5,  5,  5,  5,  5,  5, ...
  3,  7,  8,  8,  8,  8,  8,  8,  8,  8,  8, ...
  1,  7, 12, 13, 13, 13, 13, 13, 13, 13, 13, ...
  4, 14, 20, 21, 21, 21, 21, 21, 21, 21, 21, ...
  1, 11, 26, 33, 34, 34, 34, 34, 34, 34, 34, ...

Burton, David M., Elementary number theory, McGraw Hill, N.Y., 2002, p. 286.

Cf. A000045, A011973, A102426.

A175685 := proc(n,m) upl := floor( (n-1)/2) ; add( binomial(n-j-1,j),j=upl-m .. upl) ; end proc: # R. J. Mathar, Dec 05 2010

a = Table[Table[Sum[Binomial[n -j - 1, j], {j, Floor[(n - 1)/2] - m, Floor[(n - %t 1)/2]}], {n, 0, 10}], {m, 0, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}];Flatten[%]

A248751 Decimal expansion of limit of the real part of f(1-i,n)/f(1-i,n+1), where f(x,n) is the n-th Fibonacci polynomial.

Original entry on oeis.org

5, 2, 9, 0, 8, 5, 5, 1, 3, 6, 3, 5, 7, 4, 6, 1, 2, 5, 1, 6, 0, 9, 9, 0, 5, 2, 3, 7, 9, 0, 2, 2, 5, 2, 1, 0, 6, 1, 9, 3, 6, 5, 0, 4, 9, 8, 3, 8, 9, 0, 9, 7, 4, 3, 1, 4, 0, 7, 7, 1, 1, 7, 6, 3, 2, 0, 2, 3, 9, 8, 1, 1, 5, 7, 9, 1, 8, 9, 4, 6, 2, 7, 7, 1, 1, 4

Offset: 0

Clark Kimberling, Oct 13 2014

The analogous limit of f(1,n)/f(1,n+1) is the golden ratio (A001622).

Differs from A248749 only in the first digit. - R. J. Mathar, Oct 23 2014

			limit = 0.52908551363574612516099052379022521061936504...
Let q(x,n) = f(x,n)/f(x,n+1) and c = 1-i.
n   f(x,n)            Re(q(c,n))   Im(q(c,n))
1   1                 1/2          1/2
2   x                 3/5          1/5
3   1 + x^2           1/2          1/4
4   2*x + x^3         8/15         4/15
5   1 + 3*x^2 + x^4   69/130       33/130
Re(q(1-i,11)) = 5021/9490 = 0.5290832...
Im(q(1-i,11)) = 4879/18980 = 0.257060...

Index entries for algebraic numbers, degree 4

Cf. A248750, A248752, A102426, A001622.

evalf((sqrt(2+sqrt(5))-1)/2, 120); # Vaclav Kotesovec, Oct 19 2014

z = 300; t = Table[Fibonacci[n, x]/Fibonacci[n + 1, x], {n, 1, z}];
u = t /. x -> 1 - I;
d1 = N[Re[u][[z]], 130]
d2 = N[Im[u][[z]], 130]
r1 = RealDigits[d1]  (* A248751 *)
r2 = RealDigits[d2]  (* A248752 *)

polrootsreal(4*x^4+8*x^3+2*x^2-2*x-1)[2] \\ Charles R Greathouse IV, Nov 26 2024

Equals (sqrt(2+sqrt(5))-1)/2. - Vaclav Kotesovec, Oct 19 2014

cp's OEIS Frontend

A169803 Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).

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A092865 Nonzero elements in Klee's identity Sum[(-1)^k binomial[n,k]binomial[n+k,m],{k,0,n}] == (-1)^n binomial[n,m-n].

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A248752 Decimal expansion of limit of the imaginary part of f(1-i,n)/f(1-i,n+1), where f(x,n) is the n-th Fibonacci polynomial.

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A033538 a(0)=1, a(1)=1, a(n) = 3*a(n-1) + a(n-2) + 1.

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A284938 Triangle read by rows: coefficients of the edge cover polynomial for the n-path graph P_n.

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A102428 Central column of triangle A102427.

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A102429 Row sums of A102427.

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