A120743 -id:A120743 - cp's OEIS frontend

A190491 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),3,1) and []=floor.

1, 2, 1, 2, 0, 1, 3, 1, 2, 0, 2, 3, 1, 2, 1, 2, 0, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 2, 0, 1, 2, 1, 2, 0, 1, 3, 1, 2, 0, 2, 3, 1, 2, 1, 2, 0, 1, 3, 1, 2, 0, 2, 3, 1, 2, 1, 2, 0, 1, 2, 1, 2, 0, 1, 3, 1, 2, 0, 2, 3, 1, 2, 1, 2, 0, 1, 3, 1, 2, 0, 2, 3, 1, 2, 1, 2, 0, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 2, 0, 1, 2, 1, 2, 0, 1, 3, 1, 2, 0, 2, 3, 1, 2, 1, 2, 0, 1, 3, 1, 2, 0, 2, 3, 1, 2, 0, 2, 0

Offset: 1

Clark Kimberling, May 11 2011

nonn

Write a(n)=[(bn+c)r]-b[nr]-[cr].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,1):  A190427-A190430
(sqrt(2),2,0):  A190480, A120743, A170749
(sqrt(2),2,1):  A190483-A190486
(sqrt(2),3,0):  A190487-A190490
(sqrt(2),3,1):  A190491-A190495
(sqrt(2),3,2):  A190496-A190500

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A190492, A190493, A190494, A190495.

r = Sqrt[2]; b = 3; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 200}]  (* A190491 *)
Flatten[Position[t, 0]]   (* A190492 *)
Flatten[Position[t, 1]]   (* A190493 *)
Flatten[Position[t, 2]]   (* A190494 *)
Flatten[Position[t, 3]]   (* A190495 *)

A221131 Table, T, read by antidiagonals where T(-j,k) = ((1+sqrt(j))^k + (1-sqrt(j))^k)/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -4, 1, 1, 1, -3, -8, -7, -4, 1, 1, 1, -4, -11, -8, 1, 0, 1, 1, 1, -5, -14, -7, 16, 23, 8, 1, 1, 1, -6, -17, -4, 41, 64, 43, 16, 1, 1, 1, -7, -20, 1, 76, 117, 64, 17, 16, 1, 1, 1, -8, -23, 8, 121, 176, 29, -128, -95, 0, 1

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com) and Robert G. Wilson v, Jan 02 2013

.j\k.........0..1...2....3...4....5....6......7.......8......9......10
.0: A000012..1..1...1....1...1....1....1......1.......1......1.......1
-1: A146559..1..1...0...-2..-4...-4....0......8......16.....16.......0
-2: A087455..1..1..-1...-5..-7....1...23.....43......17....-95....-241
-3: A138230..1..1..-2...-8..-8...16...64.....64....-128...-512....-512
-4: A006495..1..1..-3..-11..-7...41..117.....29....-527..-1199.....237
-5: A138229..1..1..-4..-14..-4...76..176...-104...-1264..-1904....3776
-6: A090592..1..1..-5..-17...1..121..235...-377...-2399..-2159...12475
-7: A090590..1..1..-6..-20...8..176..288...-832...-3968..-1280...29184
-8: A025172..1..1..-7..-23..17..241..329..-1511...-5983...1633...57113
-9: A120743..1..1..-8..-26..28..316..352..-2456...-8432...7696...99712
-10: ........1..1..-9..-29..41..401..351..-3709..-11279..18241..160551

Cf. A084097,
Rows: A000012, A146559, A087455, A138230, A006495, A138229, A090592, A090590, A025172, A120743.
Columns: A000012, A000012, A024000, 1-3n, A028884, A134593.

T[j_, k_] := Expand[((1 + Sqrt[j])^k + (1 - Sqrt[j])^k)/2]; Table[ T[ -j + k, k], {j, 0, 11}, {k, 0, j}] // Flatten

cp's OEIS Frontend

A190491 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(sqrt(2),3,1) and []=floor.

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