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A192062 Square Array T(ij) read by antidiagonals (from NE to SW) with columns 2j being the denominators of continued fraction convergents to square root of (j^2 + 2j). 3
0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 4, 5, 3, 0, 1, 1, 5, 5, 11, 8, 1, 0, 1, 1, 6, 6, 19, 15, 13, 4, 0, 1, 1, 7, 7, 29, 24, 41, 21, 1, 0, 1, 1, 8, 8, 41, 35, 91, 56, 34, 5, 0, 1, 1, 9, 9, 55, 48, 169, 115, 153, 55, 1, 0, 1, 1, 10, 10, 71, 63, 281, 204, 436, 209, 89, 6 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,14

COMMENTS

Column j=1 is the Fibonacci sequence A000045.  Column 2 is A002530; column 4 is A041011; column 6 is A041023; column 8 is A041039, column 10 is A041059, column 12 is A041083, column 14 is A041111 corresponding the denominators of continued fraction convergents to square root of 3,8,15,24,35,48 and 63.

  T(2*i-1,j)*T(2*i,j)^2*T(2*i+1,j)*j/2 appears to be always a triangular number, T(j*T(2*i,j)^2).

  T(2*i,j)*T(2*i+1,j)^2*T(2*i+2)*j/2 appears to always equal a triangular number, T(j*T(2*i,j)*T(2*i+2,j)).

Conjecture re relation of A192062 to the sequence of primes: T(2*n,j) = A(n,j)*T(n,j) where A(n,j) is from the square array A191971. There, A(3*n,j) = A(n,j)*B(n,j) where B(n,j) are integers. It appears further that B(5*n,j)=B(n,j)*C(n,j); C(7*n,j)= C(n,j)*D(n,j); D(11*n,j) = D(n,j)*E(n,j); E(13*n,j) = E(n,j)*F(n,j) and F(17*n,j) = F(n,j)*G(n,j) where C(n,j), D(n,j) etc. are all integers. My conjecture is that this property continues indefinitely and follows the sequence of primes.

LINKS

Table of n, a(n) for n=0..90.

Kenneth J Ramsey,Triangular and Fibonacci Numbers

FORMULA

Each column j is a recursive sequence defined by T(0,j)=0, T(1,j) = 1, T(2i,j)= T(2i-2,j)+T(2i-1,j) and T(2i+1,j) = T(2i-1,j)+j*T(2i,j). Also, T(n+2,j) = (j+2)*T(n,j)-T(n-2,j).

T(2n,j) = Sum(k=1 to n) C(k)*T(2*k,j-1) where the C(k) are the n-th row of the triangle A191579.

T(2*i,j) = T(i,j)*A(i,j) where A(i,j) is from the table A(i,j) of A191971.

T(4*i,j) = (T(2*i+1)^2 - T(2*i-1)^2)/j

T(4*i+2,j) = T(2*i+2,j)^2 - T(2*i,j)^2

EXAMPLE

Array as meant by the definition

First column has index j=0

0  0  0   0   0   0   0 ...

1  1  1   1   1   1   1 ...

1  1  1   1   1   1   1 ...

1  2  3   4   5   6   7 ...

2  3  4   5   6   7   8 ...

1  5 11  19  29  41  55 ...

3  8 15  24  35  48  63 ...

1 13 41  91 169 281 433 ...

4 21 56 115 204 329 496 ...

.

.

.

CROSSREFS

Cf. A191579, A191971.

Sequence in context: A293112 A306910 A112185 * A172371 A279006 A112555

Adjacent sequences:  A192059 A192060 A192061 * A192063 A192064 A192065

KEYWORD

nonn,tabl

AUTHOR

Kenneth J Ramsey, Jun 21 2011

EXTENSIONS

Corrected and edited by Olivier Gérard, Jul 05 2011

STATUS

approved

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Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)