A371382 -id:A371382 - cp's OEIS frontend

A371381 Main diagonal of A219875.

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.

A295573 Array read by upwards antidiagonals: T(n,k) = nk + floor(phi n) ceiling(phi k) where phi = (1 + sqrt(5))/2.

Original entry on oeis.org

3, 8, 6, 11, 16, 8, 16, 22, 21, 11, 21, 32, 29, 29, 14, 24, 42, 42, 40, 37, 16, 29, 48, 55, 58, 51, 42, 19, 32, 58, 63, 76, 74, 58, 50, 21, 37, 64, 76, 87, 97, 84, 69, 55, 24, 42, 74, 84, 105, 111, 110, 100, 76, 63, 27, 45, 84, 97, 116, 134, 126, 131, 110, 87, 71, 29, 50, 90, 110, 134, 148, 152, 150, 144, 126, 98, 76, 32

Offset: 1

N. J. A. Sloane, Dec 03 2017

This is a hybrid of the Porta-Stolarsky star product (A101858) and the Arnoux product (A101866)

			The array begins:
3, 6, 8, 11, 14, 16, 19, 21, 24, 27, 29, 32, ...
8, 16, 21, 29, 37, 42, 50, 55, 63, 71, 76, 84, ...
11, 22, 29, 40, 51, 58, 69, 76, 87, 98, 105, 116, ...
16, 32, 42, 58, 74, 84, 100, 110, 126, 142, 152, 168, ...
21, 42, 55, 76, 97, 110, 131, 144, 165, 186, 199, 220, ...
24, 48, 63, 87, 111, 126, 150, 165, 189, 213, 228, 252, ...
29, 58, 76, 105, 134, 152, 181, 199, 228, 257, 275, 304, ...
32, 64, 84, 116, 148, 168, 200, 220, 252, 284, 304, 336, ...
...

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 (No. 4, 1989), 319-320.
P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (No. 4, 1989), 319-320. [Annotated scanned copy]

Cf. A001622, A101858, A101866, A371382 (main diagonal).

T := proc(n, k) local phi;
        phi := (1+sqrt(5))/2 ;
        n*k+floor(n*phi)*ceil(phi*k) ;
end proc:
for n from 1 to 12 do
lprint([seq(T(n-i+1,i),i=1..n)]);
od: # by antidiagonals
for n from 1 to 12 do
lprint([seq(T(n,i),i=1..12)]);
od: # by rows

A295573[n_, k_] := n*k + Floor[n * GoldenRatio] * Ceiling[k * GoldenRatio];
Table[A295573[n-k+1,k], {n, 15}, {k, n}] (* Paolo Xausa, Mar 20 2024 *)

cp's OEIS Frontend

A371381 Main diagonal of A219875.

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