A101646 -id:A101646 - cp's OEIS frontend

A101711 Main diagonal of A101646.

1, 3, 8, 15, 21, 32, 40, 55, 72, 84, 105, 128, 144, 171, 189, 220, 253, 275, 312, 336, 377, 420, 448, 495, 544, 576, 629, 663, 720, 779, 817, 880, 945, 987, 1056, 1100, 1173, 1248, 1296, 1375, 1425, 1508, 1593, 1647, 1736, 1827, 1885, 1980, 2040, 2139, 2240

Offset: 1

David Applegate and N. J. A. Sloane, Jan 26 2005

nonn

with(combinat) : A014417 := proc(n) local a,i,nredu,j; i := 1 ; while fibonacci(i) <= n do i := i+1 ; od ; nredu := n ; a := [seq(0,j=1..i-1)] ; while i >= 2 do if fibonacci(i) <= nredu then nredu := nredu-fibonacci(i) ; a := subsop(i=1,a) ; fi ; i := i-1 ; od ; RETURN(a) ; end: A101646 := proc(n,k) local a,nZeck,kZeck,i,j ; a := 0 ; nZeck := A014417(n) ; kZeck := A014417(k) ; for i from 2 to nops(nZeck) do for j from 2 to nops(kZeck) do a := a+op(i,nZeck)*op(j,kZeck)*fibonacci(i+j-2) ; od: od: RETURN(a) ; end: A101711 := proc(n) A101646(n,n) ; end: seq(A101711(n),n=1..80) ; # R. J. Mathar, Aug 07 2007

a[n_] := n^2 - Floor[(n+1)/GoldenRatio^2]^2;
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Mar 30 2024, after Fred Lunnon in A101646 *)

More terms from R. J. Mathar, Aug 07 2007

A101741 4th row of A101646.

Original entry on oeis.org

4, 7, 11, 15, 18, 22, 25, 29, 33, 36, 40, 44, 47, 51, 54, 58, 62, 65, 69, 72, 76, 80, 83, 87, 91, 94, 98, 101, 105, 109, 112, 116, 120, 123, 127, 130, 134, 138, 141, 145, 148, 152, 156, 159, 163, 167, 170, 174, 177, 181, 185, 188, 192, 195, 199, 203, 206, 210, 214

Offset: 1

David Applegate and N. J. A. Sloane, Jan 27 2005

nonn

Cf. A101646.

with(combinat) : A014417 := proc(n) local a,i,nredu,j; i := 1 ; while fibonacci(i) <= n do i := i+1 ; od ; nredu := n ; a := [seq(0,j=1..i-1)] ; while i >= 2 do if fibonacci(i) <= nredu then nredu := nredu-fibonacci(i) ; a := subsop(i=1,a) ; fi ; i := i-1 ; od ; RETURN(a) ; end: A101646 := proc(n,k) local a,nZeck,kZeck,i,j ; a := 0 ; nZeck := A014417(n) ; kZeck := A014417(k) ; for i from 2 to nops(nZeck) do for j from 2 to nops(kZeck) do a := a+op(i,nZeck)*op(j,kZeck)*fibonacci(i+j-2) ; od: od: RETURN(a) ; end: A101741 := proc(n) A101646(4,n) ; end: seq(A101741(n),n=1..80) ; # R. J. Mathar, Aug 07 2007

T[n_, k_] := With[{phi2 = GoldenRatio^2}, n k - Floor[(k + 1)/phi2] Floor[ (n + 1)/phi2]];
a[k_] := T[4, k];
Array[a, 60] (* Jean-François Alcover, Mar 31 2020 *)

More terms from R. J. Mathar, Aug 07 2007

A101330 Array read by antidiagonals: T(n, k) = Knuth's Fibonacci (or circle) product of n and k ("n o k"), n >= 1, k >= 1.

Original entry on oeis.org

3, 5, 5, 8, 8, 8, 11, 13, 13, 11, 13, 18, 21, 18, 13, 16, 21, 29, 29, 21, 16, 18, 26, 34, 40, 34, 26, 18, 21, 29, 42, 47, 47, 42, 29, 21, 24, 34, 47, 58, 55, 58, 47, 34, 24, 26, 39, 55, 65, 68, 68, 65, 55, 39, 26, 29, 42, 63, 76, 76, 84, 76, 76, 63, 42, 29, 32, 47

Offset: 1

N. J. A. Sloane, Jan 25 2005

Let n = Sum_{i >= 2} eps(i) Fib_i and k = Sum_{j >= 2} eps(j) Fib_j be the Zeckendorf expansions of n and k, respectively (cf. A035517, A014417). (The eps(i) are 0 or 1 and no two consecutive eps(i) are both 1.) Then the Fibonacci (or circle) product of n and k is n o k = Sum_{i,j} eps(i)*eps(j) Fib_{i+j} (= T(n,k)).
The Zeckendorf expansion can be written n = Sum_{i=1..k} F(a_i), where a_{i+1} >= a_i + 2. In this formulation, the product becomes: if n = Sum_{i=1..k} F(a_i) and m = Sum_{j=1..l} F(b_j) then n o m = Sum_{i=1..k} Sum_{j=1..l} F(a_i + b_j).
Knuth shows that this multiplication is associative. This is not true if we change the product to n X k = Sum_{i,j} eps(i)*eps(j) Fib_{i+j-2}, see A101646. Of course 1 is not a multiplicative identity here, whereas it is in A101646.
The papers by Arnoux, Grabner et al. and Messaoudi discuss this sequence and generalizations.

			Array begins:
   3   5   8  11   13   16   18   21   24 ...
   5   8  13  18   21   26   29   34   39 ...
   8  13  21  29   34   42   47   55   63 ...
  11  18  29  40   47   58   65   76   87 ...
  13  21  34  47   55   68   76   89  102 ...
  16  26  42  58   68   84   94  110  126 ...
  18  29  47  65   76   94  105  123  141 ...
  21  34  55  76   89  110  123  144  165 ...
  24  39  63  87  102  126  141  165  189 ...
  ...........................................

T. D. Noe, First 100 antidiagonals, flattened
P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (1989), 319-320.
P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (No. 4, 1989), 319-320. [Annotated scanned copy]
Vincent Canterini and Anne Siegel, Geometric representation of substitutions of Pisot type, Trans. Amer. Math. Soc. 353 (2001), 5121-5144.
P. Grabner et al., Associativity of recurrence multiplication, Appl. Math. Lett. 7 (1994), 85-90.
D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60.
W. F. Lunnon, Proof of formula
A. Messaoudi, Propriétés arithmétiques et dynamiques du fractal de Rauzy, Journal de théorie des nombres de Bordeaux, 10 no. 1 (1998), p. 135-162.
A. Messaoudi, Propriétés arithmétiques et dynamiques du fractal de Rauzy [alternative copy]
A. Messaoudi, Généralisation de la multiplication de Fibonacci, Math. Slovaca, 50 (2) (2000), 135-148.
A. Messaoudi, Tribonacci multiplication, Appl. Math. Lett. 15 (2002), 981-985.

See A101646 and A135090 for other versions.
Main diagonal is A101332.
Rows: A026274 (row 1), A101345 (row 2), A101642 (row 3).
Cf. A035517, A014417, A060144.
Cf. A101385, A101633, A101858 for related definitions of product.

h := n -> floor(2*(n + 1)/(sqrt(5) + 3)):  # A060144(n+1)
T := (n, k) -> 3*n*k - n*h(k) - k*h(n):
seq(print(seq(T(n, k), k = 1..9)), n = 1..7);  # Peter Luschny, Mar 21 2024

zeck[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr]]; kfp[n_, m_] := Block[{y = Reverse[ IntegerDigits[ zeck[ n]]], z = Reverse[ IntegerDigits[ zeck[ m]]]}, Sum[ y[[i]]*z[[j]]*Fibonacci[i + j + 2], {i, Length[y]}, {j, Length[z]}]]; (* Robert G. Wilson v, Feb 09 2005 *)
Flatten[ Table[ kfp[i, n - i], {n, 2, 13}, {i, n - 1, 1, -1}]] (* Robert G. Wilson v, Feb 09 2005 *)
A101330[n_, k_]:=3*n*k-n*Floor[(k+1)/GoldenRatio^2]-k*Floor[(n+1)/GoldenRatio^2];
Table[A101330[n-k+1, k], {n, 15}, {k, n}] (* Paolo Xausa, Mar 20 2024 *)

T(n, k) = 3*n*k - n*floor((k+1)/phi^2) - k*floor((n+1)/phi^2). For proof see link. - Fred Lunnon, May 19 2008

T(n, k) = 3*n*k - n*h(k) - k*h(n) where h(n) = A060144(n + 1). - Peter Luschny, Mar 21 2024

More terms from David Applegate, Jan 26 2005

A348853 Delete any least significant 0's from the Zeckendorf representation of n, leaving its "odd" part.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 4, 1, 9, 6, 4, 12, 1, 14, 9, 6, 17, 4, 19, 12, 1, 22, 14, 9, 25, 6, 27, 17, 4, 30, 19, 12, 33, 1, 35, 22, 14, 38, 9, 40, 25, 6, 43, 27, 17, 46, 4, 48, 30, 19, 51, 12, 53, 33, 1, 56, 35, 22, 59, 14, 61, 38, 9, 64, 40, 25, 67, 6, 69, 43, 27, 72

Offset: 1

Kevin Ryde, Nov 14 2021

Terms are odd Zeckendorfs A003622 and the fixed points are where n is odd already so that a(n) = n iff n is in A003622.
A139764(n) is the least significant "10..00" part of n so Zeckendorf multiplication n = A101646(a(n), A139764(n)).
The equivalent delete least significant 0's in binary is A000265 so that conversion to Fibbinary (A003714) and back gives a(n) = A022290(A000265(A003714(n))).
a(n) = 1 iff n is a Fibonacci number >= 1 (A000045) since they are Zeckendorf 100..00.
a(n) = 4 iff n is a Lucas number >= 4 (A000032) since they are Zeckendorf 10100..00 which reduces to 101.
In the Wythoff array A035513, a(n) is the term in column 0 of the row containing n, and hence the formula below using row number A019586 to select which of the odds (column 0) is a(n).

			n    = 81 = Zeckendorf 101001000.
a(n) = 19 = Zeckendorf 101001.

Cf. A189920 (Zeckendorf digits), A003622 (odds), A003849 (final digit), A005206, A319433 (shift down).
Cf. A000045 (Fibonacci), A000032 (Lucas).
Cf. A035513 (Wythoff array), A019586 (row number).
Cf. A003714 (Fibbinary), A022290 (its inverse).
In other bases: A000265 (binary), A004151 (decimal).
Cf. A001622, A101646, A139764.

my(phi=quadgen(5)); a(n) = my(q,r); while([q,r]=divrem(n+2,phi); r<1, n=q-1); n;

a(n) = n if A003849(n)=1, otherwise a(n) = a(A005206(n)) = a(A319433(n)).
a(n) = A003622(A019586(n) + 1).
Sum_{k=1..n} a(k) ~ n^2/(2*phi), where phi is the golden ratio (A001622). - Amiram Eldar, Feb 17 2024

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A101711 Main diagonal of A101646.

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A101741 4th row of A101646.

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A101330 Array read by antidiagonals: T(n, k) = Knuth's Fibonacci (or circle) product of n and k ("n o k"), n >= 1, k >= 1.

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A348853 Delete any least significant 0's from the Zeckendorf representation of n, leaving its "odd" part.

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