A371381 -id:A371381 - cp's OEIS frontend

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

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

A219875 Multiplication table of the operation "n o m" = nm + ceiling(n/phi) ceiling(m/phi), with phi = (1+sqrt(5))/2, read by antidiagonals.

Original entry on oeis.org

2, 4, 4, 5, 8, 5, 7, 10, 10, 7, 9, 14, 13, 14, 9, 10, 18, 18, 18, 18, 10, 12, 20, 23, 25, 23, 20, 12, 13, 24, 26, 32, 32, 26, 24, 13, 15, 26, 31, 36, 41, 36, 31, 26, 15, 17, 30, 34, 43, 46, 46, 43, 34, 30, 17, 18, 34, 39, 47, 55, 52, 55, 47, 39, 34, 18

Offset: 1

Michel Marcus, Dec 01 2012

Like A101866, this operation is associative.
First rows of the table are:
1: 2,  4,  5,  7,  9, 10, 12,  13,  15,  17, ...
2: 4,  8, 10, 14, 18, 20, 24,  26,  30,  34, ...
3: 5, 10, 13, 18, 23, 26, 31,  34,  39,  44, ...
4: 7, 14, 18, 25, 32, 36, 43,  47,  54,  61, ...
5: 9, 18, 23, 32, 41, 46, 55,  60,  69,  78, ...
6:10, 20, 26, 36, 46, 52, 62,  68,  78,  88, ...
7:12, 24, 31, 43, 55, 62, 74,  81,  93, 105, ...
8:13, 26, 34, 47, 60, 68, 81,  89, 102, 115, ...
9:15, 30, 39, 54, 69, 78, 93, 102, 117, 132, ...
Row 1 is A004956.
Row 3 is A101868.

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).
P. Arnoux, Some remarks about Fibonacci multiplication, Applied Mathematics Letters, Volume 2, Issue 4, 1989, Pages 319-320.

Cf. A001622, A004956, A101385, A101858, A101866, A101868, A371381 (main diagonal).

A219875[n_, m_] := n*m + Ceiling[n / GoldenRatio] * Ceiling[m / GoldenRatio];
Table[A219875[n-m+1, m], {n, 15}, {m, n}] (* Paolo Xausa, Mar 20 2024 *)

prod(m,n) = {phi = (1+sqrt(5))/2; return (m*n + ceil(m/phi)*ceil(n/phi));}

cp's OEIS Frontend

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

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A219875 Multiplication table of the operation "n o m" = nm + ceiling(n/phi) ceiling(m/phi), with phi = (1+sqrt(5))/2, read by antidiagonals.

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cp's OEIS Frontend

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

A219875 Multiplication table of the operation "n o m" = n*m + ceiling(n/phi)* ceiling(m/phi), with phi = (1+sqrt(5))/2, read by antidiagonals.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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

A219875 Multiplication table of the operation "n o m" = nm + ceiling(n/phi) ceiling(m/phi), with phi = (1+sqrt(5))/2, read by antidiagonals.