A211266 -id:A211266 - cp's OEIS frontend

A211272 Number of integer pairs (x,y) such that 0

0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 2, 2, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 2, 2, 2, 2, 1, 1, 4, 4, 2, 2, 2, 2, 2, 2, 3, 3, 1, 1, 4, 4, 1, 1, 3, 3, 2, 2, 2, 2, 2, 2, 5, 5, 1, 1, 2, 2, 2, 2, 4, 4, 1, 1, 4, 4, 1, 1, 3, 3, 3, 3, 2, 2, 1, 1, 5, 5, 2, 2

Offset: 1

Clark Kimberling, Apr 07 2012

nonn

For a guide to related sequences, see A211266.

			a(24) counts these pairs: (1,12), (2,6), (3,4).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A211266.

[0] cat [Ceiling(#Divisors( Floor(n/2))/2):n in [2..100]]; // Marius A. Burtea, Feb 07 2020

[seq(ceil(numtheory:-tau(floor(n/2))/2),n=1..100)]; - Robert Israel, Feb 07 2020

a = 1; b = n; z1 = 120;
t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A038548 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211270 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Print
Table[c1[n, n], {n, 1, z1}]          (* A094820 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)

a(n) = ceiling(A000005(floor(n/2))/2). - Robert Israel, Feb 07 2020

A211273 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

1, 3, 5, 7, 10, 13, 15, 19, 22, 25, 28, 32, 35, 39, 43, 46, 49, 55, 57, 62, 66, 69, 73, 78, 82, 86, 90, 95, 98, 104, 106, 112, 117, 120, 125, 131, 133, 138, 143, 148, 152, 158, 161, 166, 172, 176, 179, 186, 189, 196, 200, 204, 209, 215, 219, 225, 229, 233

Offset: 1

Clark Kimberling, Apr 07 2012

nonn

For a guide to related sequences, see A211266.

			a(5) counts these pairs: (1,1), (1,2), (1,3), (1,4), (1,5), (2,2), (2,3), (2,4), (2,5), (3,3)

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A038548 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211270 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Print
Table[c1[n, n], {n, 1, z1}]          (* A094820 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)

a(1)-a(2) corrected by Sean A. Irvine, Jan 22 2025

A211274 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y <= 3n.

Original entry on oeis.org

1, 3, 6, 9, 12, 16, 20, 24, 28, 33, 37, 43, 46, 52, 57, 62, 67, 72, 78, 84, 88, 95, 99, 107, 111, 117, 124, 130, 134, 142, 147, 154, 159, 166, 173, 179, 184, 191, 197, 206, 210, 218, 223, 231, 237, 243, 250, 259, 264, 271, 277, 286, 289, 299, 305, 313

Offset: 1

Clark Kimberling, Apr 07 2012

nonn

For a guide to related sequences, see A211266.

			a(4) counts these pairs: (1,1), (1,2), (1,3), (1,4), (2,3), (2,4), (3,3,), (3,4), (4,4).

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A038548 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211270 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Print
Table[c1[n, n], {n, 1, z1}]          (* A094820 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)

a(1)-a(3) corrected by Sean A. Irvine, Jan 22 2025

A211275 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y <= floor(n/2).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 5, 5, 6, 6, 8, 8, 9, 9, 11, 11, 13, 13, 15, 15, 16, 16, 19, 19, 20, 20, 22, 22, 24, 24, 27, 27, 28, 28, 31, 31, 32, 32, 35, 35, 37, 37, 39, 39, 40, 40, 44, 44, 46, 46, 48, 48, 50, 50, 53, 53, 54, 54, 58, 58, 59, 59, 62, 62, 64, 64, 66, 66, 68, 68

Offset: 1

Clark Kimberling, Apr 07 2012

nonn

For a guide to related sequences, see A211266.

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A038548 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211270 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Print
Table[c1[n, n], {n, 1, z1}]          (* A094820 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)

A211340 Number of integer pairs (x,y) such that 1

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 17, 23, 30, 38, 45, 53, 64, 74, 86, 97, 110, 123, 138, 154, 168, 186, 203, 220, 241, 261, 282, 302, 324, 348, 370, 396, 421, 448, 476, 501, 531, 558, 591, 622, 651, 684, 717, 753, 788, 821, 858, 894, 933, 973, 1014, 1054, 1093, 1135

Offset: 1

Clark Kimberling, Apr 08 2012

nonn

For a guide to related sequences, see A211266.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A046080, A211266.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for y from 1 to N-1 do
  for x from 1 to y do
    r:= x^2 + y^2;
    if r > N^2 then break fi;
    t:= ceil(sqrt(r));
    V[t]:= V[t]+1
od od:
ListTools:-PartialSums(convert(V,list)); # Robert Israel, Jun 04 2019

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x^2 + y^2, {x, a, b - 1}, {y, x, b}]] (* 1<=x<=y<=n *)
c[n_, k_] := c[n, k] = Count[t[n], k]
TableForm[Table[c[n, k], {n, 1, 7}, {k, 1, n^2}]]
Table[c[n, n^2], {n, 1, z1}]    (* A046080 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n^2], {n, 1, z1/2}] (* A211340 *)

A211339 Number of integer pairs (x,y) such that 1 < x <= y <= n and x^2 + y^2 <= n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 19, 19, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 25, 25, 25, 26, 26, 26, 26, 27, 28, 29

Offset: 1

Clark Kimberling, Apr 08 2012

nonn

Partial sums of A025426.

For a guide to related sequences, see A211266.

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x^2 + y^2, {x, a, b - 1}, {y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
TableForm[Table[c[n, k], {n, 1, 10}, {k, 1, 2 n}]]
Table[c[n, n], {n, 1, z1}]   (* A025426 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]  (* A211339 *)

a(n) = -1/2(-1 + floor(sqrt(n/2)))(floor(sqrt(n/2))) + Sum_{k=1..floor(sqrt(n/2))} floor(sqrt(n - k^2)). - Nicholas Stearns, Apr 03 2017

A335408 Diameter of nearest neighbor interchange distance for free 3-trees.

Original entry on oeis.org

0, 1, 3, 5, 7, 10, 12, 15, 18, 21

Offset: 3

Martin R. Smith, Jun 06 2020

a(n) is the maximum value of the nearest neighbor interchange distance between two unrooted binary trees with n leaves, obtained by evaluating the distance from one tree with each of the unlabeled n-leaf tree shapes (see A000672) to each labeled n-leaf tree (A001147) using the C script described in Li et al. (1996).

The known terms a(3),...,a(12) happen (coincidentally?) to match the first ten terms of A211266. However, it seems unlikely that the sequences will agree for ever.

Ming Li, John Tromp, and Louxin Zhang, Some notes on the nearest neighbour interchange distance, in Goos, G., Hartmanis, J., Leeuwen, J., Cai, J.-Y., and Wong, C. K., eds., "Computing and Combinatorics" 1090, Springer (Berlin, Heidelberg) (1996), 343-351. doi:10.1007/3-540-61332-3_168.

Ming Li, John Tromp, and Louxin Zhang, Some notes on the nearest neighbour interchange distance, on ResearchGate.
Ming Li, John Tromp, and Louxin Zhang, On the Nearest Neighbour Interchange Distance Between Evolutionary Trees, J. Theoretical Biology, 182 (1996), 463-467.

Cf. A211266, which happens to have the same initial terms (offset by two). It is not clear whether this correspondence continues for higher terms.

A000672 gives the number of unrooted tree shapes on n leaves; A001147 gives the number of (labeled) unrooted trees.

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A211272 Number of integer pairs (x,y) such that 0

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A211273 Number of integer pairs (x,y) such that 0

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A211274 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y <= 3n.

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A211275 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y <= floor(n/2).

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A211340 Number of integer pairs (x,y) such that 1

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A211339 Number of integer pairs (x,y) such that 1 < x <= y <= n and x^2 + y^2 <= n.

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A335408 Diameter of nearest neighbor interchange distance for free 3-trees.

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