A015449 -id:A015449 - cp's OEIS frontend

A213894 Fixed points of a sequence h(n) defined by the minimum number of 5's in the relation n*[n,5,5,...,5,n] = [x,...,x] between simple continued fractions.

2, 3, 11, 19, 31, 43, 47, 79, 127, 131, 163, 211, 251, 271, 307, 311, 331, 367, 379, 443, 503, 563, 599, 607, 659, 743, 751, 823, 839, 859, 887, 907, 911, 947, 967, 1063, 1087, 1091, 1123, 1163, 1171, 1187, 1259, 1279, 1291, 1303, 1307, 1319, 1423, 1447, 1471, 1487

Offset: 1

Art DuPre, Jun 23 2012

nonn

In a variant of A213891, multiply n by a number with simple continued fraction [n,5,5,...,5,n] and increase the number of 5's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are
2 * [2, 5, 5, 2] = [4, 2, 1, 1, 2, 4],
3 * [3, 5, 5, 5, 3] = [9, 1, 1, 2, 1, 2, 1, 1, 9],
4 * [4, 5, 5, 5, 5, 5, 4] = [16, 1, 3, 2, 1, 4, 1, 2, 3, 1, 16] ,
5 * [5, 5, 5] = [25, 1, 25].
The number of 5's needed defines the sequence h(n) = 2, 3, 5, 1, 11, 5, 5, 3, 5, 11, 11, ... (n >= 2).
The current sequence contains the fixed points of h, i.e., those n where h(n)=n.
We conjecture that this sequence contains prime numbers analogous to the sequence of prime numbers A000057, in the sense that, instead of referring to the Fibonacci sequences (sequences satisfying f(n) = f(n-1) + f(n-2) with arbitrary positive integer values for f(1) and f(2)) it refers to the generalized Fibonacci sequences satisfying f(n) = 5*f(n-1) + f(n-2), A052918, A015449, A164581, etc. This would mean that a prime is in the sequence if and only if it divides some term in each of the sequences satisfying f(n) = 5*f(n-1) + f(n-2).
The above sequence h() is recorded as A262215. - M. F. Hasler, Sep 15 2015

Cf. A000057, A213891 - A213893, A213895 - A213899, A261311; A213358.

Cf. A213648, A262212 - A262220, A213900, A262211.

f[m_, n_] := Block[{c, k = 1}, c[x_, y_] := ContinuedFraction[x FromContinuedFraction[Join[{x}, Table[m, {y}], {x}]]]; While[First@ c[n, k] != Last@ c[n, k], k++]; k]; Select[Range[2, 1000], f[5, #] == # &] (* Michael De Vlieger, Sep 16 2015 *)

{a(n) = local(t, m=1); if( n<2, 0, while( 1,
   t = contfracpnqn( concat([n, vector(m,i,5), n]));
   t = contfrac(n*t[1,1]/t[2,1]);
   if(t[1]

A153764 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Jan 01 2009

A130595*A153342 as infinite lower triangular matrices. Reflected version of A103631. Another version of A046854. Row sums are Fibonacci numbers (A000045).

A055830*A130595 as infinite lower triangular matrices.

			Triangle begins:
  1;
  1, 0;
  1, 1, 0;
  1, 1, 1, 0;
  1, 2, 1, 1, 0;
  1, 2, 3, 1, 1, 0;
  1, 3, 3, 4, 1, 1, 0;
  ...

G. C. Greubel, Table of n, a(n) for the first 45 rows

Cf. A046854, A103631.

/* As triangle */ [[Binomial(Floor((n+k-1)/2),k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 28 2016

Table[Binomial[Floor[(n + k - 1)/2], k], {n, 0, 45}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 27 2016 *)

T(n,k) = binomial(floor((n+k-1)/2),k).
Sum_{k=0..n} T(n,k)*x^k = A122335(n-1), A039834(n-2), A000012(n), A000045(n+1), A001333(n), A003688(n), A015448(n), A015449(n), A015451(n), A015453(n), A015454(n), A015455(n), A015456(n), A015457(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively. - Philippe Deléham, Dec 17 2011
Sum_{k=0..n} T(n,k)*x^(n-k) = A152163(n), A000007(n), A000045(n+1), A026597(n), A122994(n+1), A158608(n), A122995(n+1), A158797(n), A122996(n+1), A158798(n), A158609(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Dec 17 2011
G.f.: (1+(1-y)*x)/(1-y*x-x^2). - Philippe Deléham, Dec 17 2011
T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013

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A213894 Fixed points of a sequence h(n) defined by the minimum number of 5's in the relation n*[n,5,5,...,5,n] = [x,...,x] between simple continued fractions.

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A153764 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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