A015455 -id:A015455 - cp's OEIS frontend

A055830 Triangle T read by rows: diagonal differences of triangle A037027.

1, 1, 0, 2, 1, 0, 3, 3, 1, 0, 5, 7, 4, 1, 0, 8, 15, 12, 5, 1, 0, 13, 30, 31, 18, 6, 1, 0, 21, 58, 73, 54, 25, 7, 1, 0, 34, 109, 162, 145, 85, 33, 8, 1, 0, 55, 201, 344, 361, 255, 125, 42, 9, 1, 0, 89, 365, 707, 850, 701, 413, 175, 52, 10, 1, 0, 144, 655, 1416, 1918, 1806, 1239, 630, 236, 63, 11, 1, 0

Offset: 0

Clark Kimberling, May 28 2000

Or, coefficients of a generalized Lucas-Pell polynomial read by rows. - Philippe Deléham, Nov 05 2006

Equals A046854(shifted) * Pascal's triangle; where A046854 is shifted down one row and "1" inserted at (0,0). - Gary W. Adamson, Dec 24 2008

			Triangle begins:
   1
   1,   0
   2,   1,   0
   3,   3,   1,   0
   5,   7,   4,   1,   0
   8,  15,  12,   5,   1,   0
  13,  30,  31,  18,   6,   1,  0
  21,  58,  73,  54,  25,   7,  1, 0
  34, 109, 162, 145,  85,  33,  8, 1, 0
  55, 201, 344, 361, 255, 125, 42, 9, 1, 0
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 3D.

Left-hand columns include A000045, A023610.
Right-hand columns include A055831, A055832, A055833, A055834, A055835, A055836, A055837, A055838, A055839, A055840.
Row sums: A001333 (numerators of continued fraction convergents to sqrt(2)).
Cf. A122075 (another version).
Cf. A046854. - Gary W. Adamson, Dec 24 2008

function T(n,k)
  if k lt 0 or k gt n then return 0;
  elif k eq 0 then return Fibonacci(n+1);
  elif n eq 1 and k eq 1 then return 0;
  else return T(n-1,k-1) + T(n-1,k) + T(n-2,k);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 21 2020

with(combinat);
T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=0 then fibonacci(n+1)
    elif n=1 and k=1 then 0
    else T(n-1, k-1) + T(n-1, k) + T(n-2, k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Jan 21 2020

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, Fibonacci[n+1], If[n==1 && k==1, 0, T[n-1, k-1] + T[n-1, k] + T[n-2, k]]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 19 2017 *)

T(n,k) = if(k<0 || k>n, 0, if(k==0, fibonacci(n+1), if(n==1 && k==1, 0, T(n-1, k-1) + T(n-1, k) + T(n-2, k) )));
for(n=0,12, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jan 21 2020

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return fibonacci(n+1)
    elif (n==1 and k==1): return 0
    else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 21 2020

G.f.: (1-y*z) / (1-y*(1+y+z)).
T(i, j) = R(i-j, j), where R(0, 0)=1, R(0, j)=0 for j >= 1, R(1, j)=1 for j >= 0, R(i, j) = Sum_{k=0..j} (R(i-2, k) + R(i-1, k)) for i >= 1, j >= 1.
Sum_{k=0..n} x^k*T(n,k) = 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. - Philippe Deléham, Oct 22 2006
Sum_{k=0..floor(n/2)} T(n-k,k) = A011782(n). - Philippe Deléham, Oct 22 2006
Triangle T(n,k), 0 <= k <= n, given by [1, 1, -1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 05 2006
T(n,0) = Fibonacci(n+1) = A000045(n+1). Sum_{k=0..n} T(n,k) = A001333(n). T(n,k)=0 if k > n or if k < 0, T(0,0)=1, T(1,1)=0, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k). - Philippe Deléham, Nov 05 2006

Edited by Ralf Stephan, Jan 12 2005

A213898 Fixed points of a sequence h(n) defined by the minimum number of 9's in the relation n*[n,9,9,...,9,n] = [x,...,x] between simple continued fractions.

Original entry on oeis.org

2, 11, 31, 43, 47, 67, 79, 103, 127, 199, 211, 223, 263, 307, 311, 383, 431, 439, 463, 467, 499, 523, 563, 571, 587, 691, 719, 751, 811, 839, 863, 883, 911, 967, 991, 1051, 1063, 1087, 1091, 1123, 1151, 1231, 1307, 1327, 1399, 1447, 1451, 1459, 1483, 1499

Offset: 1

Art DuPre, Jun 24 2012

nonn

In a variant of A213891, multiply n by a number with simple continued fraction [n,9,9,..,9,n] and increase the number of 9's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are
2 * [2, 9, 9, 2] = [4, 4, 1, 1, 4, 4],
3 * [3, 9, 3] = [9, 3, 9],
4 * [4, 9, 9, 9, 9, 9, 4] = [16, 2, 3, 1, 1, 1, 1, 8, 1, 1, 1, 1, 3, 2, 16] ,
5 * [5, 9, 9, 9, 9, 5] = [25, 1, 1, 4, 1, 1, 1, 1, 1, 1, 4, 1, 1, 25],
6 * [6, 9, 9, 9, 9, 9, 6] = [36, 1, 1, 1, 13, 6, 13, 1, 1, 1, 36],
7 * [7, 9, 9, 9, 9, 9, 7] = [49, 1, 3, 3, 6, 1, 6, 3, 3, 1, 49].
The number of 9's needed defines the sequence h(n) = 2, 1,5, 4, 5, 5, 5, 1, 14,...  (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 sequence (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 sequences satisfying f(n)=9*f(n-1)+f(n-2) like A099371, A015455 etc. This would mean that a prime is in the sequence A213898 if and only if it divides some term in each of the sequences satisfying f(n)=9*f(n-1)+f(n-2).

Cf. A213358; A000057, A213891 - A213897, A213899, A261311.

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[9, #] == # &] (* 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,9), 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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A055830 Triangle T read by rows: diagonal differences of triangle A037027.

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A213898 Fixed points of a sequence h(n) defined by the minimum number of 9's in the relation n*[n,9,9,...,9,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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Magma

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