A022344 -id:A022344 - cp's OEIS frontend

A275068 Squarefree numbers in A022344.

1, 5, 11, 19, 29, 31, 41, 55, 59, 61, 71, 79, 89, 95, 101, 109, 131, 139, 145, 149, 151, 155, 179, 181, 191, 199, 205, 209, 211, 229, 239, 241, 251, 269, 271, 281, 295, 305, 311, 319, 331, 341, 349, 355, 359, 379, 389, 395, 401, 409, 419, 421, 431, 439, 445

Offset: 0

Clark Kimberling, Jul 15 2016

The final digit of every number is 1, 5, or 9. As a set, A022344 consists of the numbers m*F^2, where m is in (1,5,11,19,...) and F is a Fibonacci number.

The restriction here to squarefree numbers excludes any of Wechsler's J determinants that derive from rows of the Wythoff array where all terms share a common factor, but there are also nonsquarefree numbers that are determinants of other rows: for example, 121 is the J determinant of row 45 (..., 3, 13, 16, 29, 45, 74, 119, 193, ...). Compare with A089270, which includes 121 and other such numbers. - Peter Munn, Aug 20 2025

			A022344 = (1,5,4,9,16,11,19,11,20,31,19,31,45,29,... ), and deletion of 4,9,16,20, ... leaves (1,5,11,19,29,31,...).

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A000045, A022344, A035513, A089270.

g = GoldenRatio; a[n_] := Floor[(n + 1)*g]^2 - n*Floor[(n + 1)*g] - n^2;
u = Table[a[n], {n, 0, 200}]  (* A022344 *)
Union[Select[u, SquareFreeQ[#] &]]  (* A275068 *)

A357097 A multiplication table for the rows of the extended Wythoff array. See comments for definition.

Original entry on oeis.org

0, 1, 1, 2, 15, 2, 3, 8, 8, 3, 4, 12, 4, 12, 4, 5, 44, 18, 18, 44, 5, 6, 19, 24, 27, 24, 19, 6, 7, 62, 28, 96, 96, 28, 62, 7, 8, 26, 34, 42, 128, 42, 34, 26, 8, 9, 30, 14, 51, 56, 56, 51, 14, 30, 9, 10, 91, 44, 57, 180, 65, 180, 57, 44, 91, 10, 11, 37, 50, 66, 76, 79, 79, 76, 66, 50, 37, 11

Offset: 0

Peter Munn, Sep 11 2022

Square array A(x,y), x >= 0, y >= 0, defined as follows:
(1) Extend the Wythoff array infinitely to the left, maintaining the Fibonacci recurrence (see A287870 examples). We denote this extended array as eW(n,m), n >= 0, m any integer, indexed such that eW(n,0) = n. From each row n, form the set of pairs S_n = {(eW(n,m+1),eW(n,m)) : integer m)}.
(2) Define addition and multiplication of pairs by (j1,k1) + (j2,k2) = (j1+j2, k1+k2) and (j1,k1) o (j2,k2) = (j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2). (This defines a commutative ring with identity (1,0).)
(3) For nonnegative integers x and y, there is an integer z such that for every pair (j_x,k_x) in S_x and every pair (j_y,k_y) in S_y, (j_x,k_x) o (j_y,k_y) is in S_z. Define A(x,y) = z.
As a binary operation, A(.,.) is analogous to multiplication of coefficients in scientific numeric notation. The column position, m, used to define a pair in (1) above does not affect the eventual outcome, A(x,y), in (3), as no special pairs are selected from the pairs in S_x or S_y. The column position is analogous to the exponent. Notice also A(1,1) = 15 is substantially larger than A(2,2) = 4. This can be seen as analogous to 0.3 * 0.4 = 0.12 requiring more digits than 0.5 * 0.8 = 0.4.

			Calculation for A(1,2). Rows 1 and 2 of A287870 (indexed from 0) start 1, 3, ... and 2, 4, ... . So we may use the pairs (3,1) and (4,2). The defined multiplication gives (3*4 + 1*2, 3*2 + 4*1 - 1*2) = (14,8). 8, 14 , ... is in row 8 of A287870, so A(1,2) = 8.
For A(1,1), we start as above to get (3*3 + 1*1, 3*1 + 3*1 - 1*1) = (10,5). In the more general case, we form a sequence using the Fibonacci recurrence (as ..., 5, 10, ... may be in the extension leftwards of A287870). This starts 5, 10, 5+10=15, 10+15=25, 15+25=40, ... . We observe 15, 25, 40, ... is in row 15. So A(1,1) = 15.
The top left corner of the array:
  0   1   2    3    4    5    6    7    8    9
  1  15   8   12   44   19   62   26   30   91
  2   8   4   18   24   28   34   14   44   50
  3  12  18   27   96   42   51   57   66  198
  4  44  24   96  128   56  180   76   88  264
  5  19  28   42   56   65   79   33  102  116
  6  62  34   51  180   79  253  107  124  371
  7  26  14   57   76   33  107   45  138  157
  8  30  44   66   88  102  124  138  160  182
  9  91  50  198  264  116  371  157  182  544

Peter G. Anderson, More Properties of the Zeckendorf Array, Fib. Quart. 52-5 (2014), 15-21.
P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (1989), 319-320.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 3-8.

Cf. A022344 (norm), A035513, A120873, A287870, A348853.

See the formula section for the relationships with A000201, A003622, A019586, A035336, A101330.

lowerw(n) = (n+sqrtint(5*n^2))\2 ; \\ A000201
upperw(n) = (sqrtint(n^2*5)+n*3)\2; \\ A001950
compoundw(n) = (sqrtint(n^2*5)+n*3)\2 - 1; \\ A003622
wpair(p) = {my(x=p[2], y = p[1], z); while(1, my(n=1, ok=1); while(ok, my(xx = lowerw(n), yy = upperw(n)); if ((x == xx) && (y == yy), return([xx, yy])); if (xx > x, ok = 0); n++;); z = y; y += x; x = z;);}
row(p) = {my(x=p[1], y=p[2], u); while (1, my(n=1, ok=1); while(ok, my(xx = lowerw(n), yy = compoundw(n)); if ((x == xx) && (y == yy), return(n)); if (xx > x, ok = 0); n++;); u = x; x = y - u; y = u;);} \\ similar to A120873
wrow(p) = row(wpair(p));
prodpair(v1, v2) = my(j1=v1[1], j2 = v2[1], k1 = v1[2], k2 = v2[2]); [j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2];
pair(n) = [lowerw(n+1), n];
T(n, k) = my(pn = pair(n), pk = pair(k), px = prodpair(pn, pk)); wrow(px)-1; \\ Michel Marcus, Sep 18 2022

A(x,y) = g(j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2), where j1 = A035336(x+1), j2 = A035336(y+1), k1 = A003622(x+1), k2 = A003622(y+1) and g(j,k) = (if j = A000201(k+1) then k otherwise g(k,j-k)).
A(x,y) = A(y,x).
A(x,0) = x.
A(x, A(y,z)) = A(A(x,y), z).
A022344(A(x,y)) = A022344(x) * A022344(y).
A(A019586(x), A019586(y)) = A019586(A101330(x,y)). (conjectured)

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

Original entry on oeis.org

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

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

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.

			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,...

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

Cf. A035513, A022344.

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

A125512 Array x read by diagonals, where x(i,j) = floor((T(i,j-1)+T(i,j+1))/2) for i>=0 and j>=0. Here T is Wythoff's array A035513.

Original entry on oeis.org

1, 2, 5, 3, 7, 7, 5, 12, 11, 10, 9, 20, 18, 16, 14, 14, 32, 29, 27, 22, 16, 23, 52, 47, 43, 36, 25, 19, 38, 85, 76, 70, 58, 41, 31, 21, 61, 137, 123, 114, 94, 67, 50, 34, 25, 99, 222, 199, 184, 152, 108, 81, 56, 40, 28, 161, 360, 322, 298, 246, 175, 132, 90, 65, 45

Offset: 0

Kenneth J Ramsey, Dec 28 2006

x(i,j)*(x(i,j) + (T(i,j) mod 2)) = (5*T(i,j)^2 - (T(i,j) mod 2))/4 + A(i)*(-1)^j, where A(i)=A022344(i).

			x(2,4)=floor((T(2,3)+T(2,5))/2)=floor((26+68)/2)=47. Since T(2,4)=42 and A(2)=4, the equation in the first comment becomes 47*(47+0) = (5*42^2-0)/4 + 4*(-1)^4.

Cf. A035513, A022344.

T[i_,j_]:=i*Fibonacci[j+1]+Fibonacci[j+2]*Floor[(i+1)(1+Sqrt[5])/2]; x[i_,j_]:=Floor[(T[i,j-1]+T[i,j+1])/2]

For j>1, x(i,j) = x(i,j-1) + x(i,j-2) + (T(i,j-1)*T(i,j-2) mod 2).

Edited by Dean Hickerson, Jan 14 2007

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A275068 Squarefree numbers in A022344.

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A357097 A multiplication table for the rows of the extended Wythoff array. See comments for definition.

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

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A125512 Array x read by diagonals, where x(i,j) = floor((T(i,j-1)+T(i,j+1))/2) for i>=0 and j>=0. Here T is Wythoff's array A035513.

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