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A055830 Triangle T read by rows: diagonal differences of triangle A037027. 31
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

C. Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 3D.

FORMULA

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{R(i-2, k)+R(i-1, k): k=0, 1, ..., j} for i >= 1, j >= 1.

Sum_{k, 0<=k<=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<=k<=[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<=k<=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

EXAMPLE

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

  ...

MATHEMATICA

T[0, 0] := 1; T[1, 1] := 0; T[n_, 0] := Fibonacci[n + 1]; T[n_, k_] := T[n, k] = If[k < 0 , 0, If[k > n, 0, T[n - 1, k - 1] + T[n - 1, k] + T[n - 2, k]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 19 2017 *)

CROSSREFS

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]

Sequence in context: A253829 A107238 A258170 * A293109 A233530 A079123

Adjacent sequences:  A055827 A055828 A055829 * A055831 A055832 A055833

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, May 28 2000

EXTENSIONS

Edited by Ralf Stephan, Jan 12 2005

STATUS

approved

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Last modified November 20 20:40 EST 2019. Contains 329347 sequences. (Running on oeis4.)