A101867 -id:A101867 - cp's OEIS frontend

A101866 Array read by antidiagonals: Arnoux's product T(n,k) = n * k = nk + ceiling(phi n) ceiling(phi k), where phi = (1 + sqrt(5))/2 ; m, n >= 1.

5, 10, 10, 13, 20, 13, 18, 26, 26, 18, 23, 36, 34, 36, 23, 26, 46, 47, 47, 46, 26, 31, 52, 60, 65, 60, 52, 31, 34, 62, 68, 83, 83, 68, 62, 34, 39, 68, 81, 94, 106, 94, 81, 68, 39, 44, 78, 89, 112, 120, 120, 112, 89, 78, 44, 47, 88, 102, 123, 143, 136, 143, 123, 102, 88, 47, 52

Offset: 1

N. J. A. Sloane, Jan 28 2005

Row 1 / column 1 (given in A101868) = positions of 1 in A188009, viz.,
  A188009 = (0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, ...), A101868 = (5, 10, 13, 18, 23, 26, 31, 34, 39, 44, 47, 52, 57, ...). - Clark Kimberling and John W. Layman, Mar 19 2011, corrected and edited by M. F. Hasler, Oct 12 2017
By definition, the array is symmetric, so row n = column n. Row 1 is essentially the same as A188434: T(n,1) = A101868(n) = A188434(n+1). - M. F. Hasler, Oct 12 2017
This product is commutative but is not associative and does not distribute over addition. - Peter Bala, Aug 13 2022

			   5 10 13 18  23 ...
  10 20 26 36  46
  13 26 34 47  60
  18 36 47 65  83
  23 46 60 83 106
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 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]

Cf. A101858, A101330, A101385, A101633 for similarly defined products.
Main diagonal is A101867.
First 3 rows are A101868, A101869, A101870.
Cf. A001622.

A101866[n_, k_] := n*k + Ceiling[n*GoldenRatio]*Ceiling[k*GoldenRatio];
Table[A101866[n-k+1, k], {n, 15}, {k, n}] (* Paolo Xausa, Mar 20 2024 *)

T(n, k) = my(phi = (1+sqrt(5))/2); n*k + ceil(phi*n)*ceil(phi*k); \\ Michel Marcus, Mar 29 2016

A371381 Main diagonal of A219875.

Original entry on oeis.org

2, 8, 13, 25, 41, 52, 74, 89, 117, 149, 170, 208, 250, 277, 325, 356, 410, 468, 505, 569, 610, 680, 754, 801, 881, 965, 1018, 1108, 1165, 1261, 1361, 1424, 1530, 1640, 1709, 1825, 1898, 2020, 2146, 2225, 2357, 2440, 2578, 2720, 2809, 2957, 3109, 3204, 3362, 3461

Offset: 1

Paolo Xausa, Mar 20 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A219875.

Cf. A001622, A019446, A101332, A101863, A101867, A371382.

Array[#^2 + Ceiling[# / GoldenRatio]^2 &, 100]

from math import isqrt
def A371381(n): return (n<<1)*(n-1)+1+(q:=n+isqrt(5*n**2)>>1)*(q-(n-1<<1)) # Chai Wah Wu, Mar 21 2024

a(n) = n^2 + ceiling(n/(1 + sqrt(5))/2)^2 = n^2 + A019446(n)^2.

A371382 a(n) = n^2 + q(q + 1), where q = floor(n(1 + sqrt(5))/2) = A000201(n).

Original entry on oeis.org

3, 16, 29, 58, 97, 126, 181, 220, 291, 372, 427, 524, 631, 702, 825, 906, 1045, 1194, 1291, 1456, 1563, 1744, 1935, 2058, 2265, 2482, 2621, 2854, 3003, 3252, 3511, 3676, 3951, 4236, 4417, 4718, 4909, 5226, 5553, 5760, 6103, 6320, 6679, 7048, 7281, 7666, 8061, 8310

Offset: 1

Paolo Xausa, Mar 20 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000

Main diagonal of A295573.

Cf. A000201, A001622, A101332, A101863, A101867, A371381.

Array[#^2 + Block[{q = Floor[# * GoldenRatio]}, q * (q + 1)] &, 100]

from math import isqrt
def A371382(n): return n**2+(q:=n+isqrt(5*n**2)>>1)*(q+1) # Chai Wah Wu, Mar 21 2024

cp's OEIS Frontend

A101866 Array read by antidiagonals: Arnoux's product T(n,k) = n * k = nk + ceiling(phi n) ceiling(phi k), where phi = (1 + sqrt(5))/2 ; m, n >= 1.

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A371381 Main diagonal of A219875.

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Formula

A371382 a(n) = n^2 + q(q + 1), where q = floor(n(1 + sqrt(5))/2) = A000201(n).

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cp's OEIS Frontend

A101866 Array read by antidiagonals: Arnoux's product T(n,k) = n * k = nk + ceiling(phi n) ceiling(phi k), where phi = (1 + sqrt(5))/2 ; m, n >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A371381 Main diagonal of A219875.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A371382 a(n) = n^2 + q*(q + 1), where q = floor(n*(1 + sqrt(5))/2) = A000201(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

A371382 a(n) = n^2 + q(q + 1), where q = floor(n(1 + sqrt(5))/2) = A000201(n).