A127561 Array T(n,k) = n^2+5*n*k+5*k^2 read downwards antidiagonals, n,k>=0.

0, 5, 1, 20, 11, 4, 45, 31, 19, 9, 80, 61, 44, 29, 16, 125, 101, 79, 59, 41, 25, 180, 151, 124, 99, 76, 55, 36, 245, 211, 179, 149, 121, 95, 71, 49, 320, 281, 244, 209, 176, 145, 116, 89, 64, 405, 361, 319, 279, 241, 205, 171, 139, 109, 81, 500, 451, 404, 359, 316, 275, 236, 199, 164, 131

Offset: 0

Author

Kenneth J Ramsey, Jan 18 2007, Feb 05 2007, Feb 06 2007

Keywords

• nonn
• tabl
• easy
• uned

Comments

Lattice table of Fibonacci characteristic values from Wechsler's J determinant sequence A022344 uniquely position such that the row and column determine starting a,b values of a Fibonacci sequence having the same characteristic value.

A vector from (0,0) to any prime value P in the array does not pass through any other lattice point. If that vector is extended it passes through lattice points having successively the values 0, P*1^2, P*2^2, P*3^2, P*4^2 ... All primes ending in 1, 5 and 9 or the product thereof appear in the array, no prime ending in 3 or 7 appears in the array except in a square product which may be multiplied by a squarefree product of primes ending in 1, 5 or 9.

The table can be expanded by allowing negative arguments in the formula, but any positive value obtained can be expressed with nonnegative arguments.

The second row is the sequence A062786. The term in every succeeding row is 2* the term immediately above minus the next above term plus 2.

If the table is rearranged by shifting each column down by twice the column number, then the terms in second column would be equal to the row number squared plus the row number minus 1 and every succeeding term to the right would be equal to twice the left-hand term minus the next left-hand term minus 2.

It appears that any prime ending in 1,5, or 9 or any such prime times 5 appears only once in the table and that every power of such a prime or product thereof has one and only one nonnegative row and column position such that the row and column positions are coprime. A method for finding a coprime row and column position of the 2^n th power of any prime ending in 1,5,or 9, or of the product thereof, from the coprime row and column position of that prime or product is suggested by the discussion in the link titled "Wythoff Array, Pythagorean Triples, Primes".

It seems that if you stack the row and column positions of two numbers in the array that the determinant gives a column in which the product appears. Thus since the row and column position of 29 and 41 are 3,1 and 4,1 respectively then the product (41*29) appears in column 1*4 - 3*1 or column 1. The same value appears also in column -1 so 3*1-1*4 is a valid answer also. For our purposes however we choose the order that gives a positive value. Once the column number of the product is known it is easy to find the row number. There may be new determinant based math to find the row directly, but I don't know of any. It may happen that the row is negative, in which case the following transformation works a(r,c) = a(-r,c+r). Applied twice this transformation gives the original starting pair. I have yet to find any case in which one starts out with positive values for the row and column of each factor of a number appearing in the table and using the above determinant math cannot find positive values for the row and column of the product. I posted a few interesting results in the Cut-the-knot forum. Use the link given previously.

Examples

			T(0,1) = 5 because (0+2*1)^2 + 1*(0+2*1) - 1^2 = 5 and also because the Fibonacci sequence having the Horadam ID {a,b,1,1} with a = 0+2*1 and b = 1 has the characteristic value a^2 + b*a - b^2.
  0,  5, 20, 45, 80,125,180,245,320,405,500,... A033429
  1, 11, 31, 61,101,151,211,281,361,451,551,..  A062786
  4, 19, 44, 79,124,179,244,319,404,499,604,..  A134538
  9, 29, 59, 99,149,209,279,359,449,549,659,... A143198 (row 9)
 16, 41, 76,121,176,241,316,401,496,601,716,...
 25, 55, 95,145,205,275,355,445,545,655,775,..
 36, 71,116,171,236,311,396,491,596,711,836,...
 49, 89,139,199,269,349,439,539,649,769,899,...
 64,109,164,229,304,389,484,589,704,829,964,...
 81,131,191,261,341,431,531,641,761,891,1031,...
100,155,220,295,380,475,580,695,820,955,1100,...
		

Links

• K. J. Ramsey, Wythoff Array, Pythagorean triples, Primes

Crossrefs

Cf. A035513, A022344.

Formula

T(a,b) = (a+2b)^2 + b(a+2b) - b^2.