A214857 -id:A214857 - cp's OEIS frontend

A214858 Natural numbers missing from A214857.

5, 8, 12, 15, 19, 22, 26, 29, 32, 36, 39, 43, 46, 49, 53, 56, 60, 63, 67, 70, 73, 77, 80, 84, 87, 90, 94, 97, 101, 104, 108, 111, 114, 118, 121, 125, 128, 131, 135, 138, 142, 145, 149, 152, 155, 159, 162, 166, 169, 172, 176, 179, 183, 186, 189, 193, 196, 200

Offset: 1

Philippe Deléham, Mar 09 2013

Cf. A063957, A214857.

tri = Table[n (n + 1)/2, {n, 0, 1000}]; t = Table[Count[tri, ?(# <= n^2 &)], {n, 0, Sqrt[tri[[-1]]]}]; Complement[Range[t[[-1]]], t] (* _T. D. Noe, Mar 11 2013 *)

a(n+1) - a(n) = 3 or 4.

Extended by T. D. Noe, Mar 11 2013

A281871 Number T(n,k) of k-element subsets of [n] having a square element sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 2, 0, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 4, 5, 5, 2, 1, 0, 1, 2, 5, 8, 8, 6, 3, 0, 1, 1, 3, 6, 11, 14, 13, 7, 4, 1, 0, 1, 3, 7, 15, 23, 24, 19, 10, 3, 1, 0, 1, 3, 8, 20, 34, 43, 39, 25, 13, 3, 1, 0, 1, 3, 9, 26, 49, 71, 74, 60, 34, 14, 5, 0, 0

Offset: 0

Alois P. Heinz, Jan 31 2017

			T(7,0) = 1: {}.
T(7,1) = 2: {1}, {4}.
T(7,2) = 4: {1,3}, {2,7}, {3,6}, {4,5}.
T(7,3) = 5: {1,2,6}, {1,3,5}, {2,3,4}, {3,6,7}, {4,5,7}.
T(7,4) = 5: {1,2,6,7}, {1,3,5,7}, {1,4,5,6}, {2,3,4,7}, {2,3,5,6}.
T(7,5) = 2: {1,2,3,4,6}, {3,4,5,6,7}.
T(7,6) = 1: {1,2,4,5,6,7}.
T(7,7) = 0.
T(8,8) = 1: {1,2,3,4,5,6,7,8}.
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 1, 0;
  1, 1, 1,  0;
  1, 2, 1,  1,  0;
  1, 2, 2,  2,  0,  0;
  1, 2, 3,  3,  2,  1,  0;
  1, 2, 4,  5,  5,  2,  1,  0;
  1, 2, 5,  8,  8,  6,  3,  0,  1;
  1, 3, 6, 11, 14, 13,  7,  4,  1,  0;
  1, 3, 7, 15, 23, 24, 19, 10,  3,  1, 0;
  1, 3, 8, 20, 34, 43, 39, 25, 13,  3, 1, 0;
  1, 3, 9, 26, 49, 71, 74, 60, 34, 14, 5, 0, 0;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000012, A000196, A176615, A281706, A281864, A281865, A281866, A281867, A281868, A281869, A281870.
Main diagonal is characteristic function of A001108.
Diagonals T(n+k,n) for k=2-10 give: A281965, A281966, A281967, A281968, A281969, A281970, A281971, A281972, A281973.
Row sums give A126024.
T(2n,n) gives A281872.
Cf. A000217, A000290, A007318, A214857, A214858, A278339, A281994, A284249, A377572.

b:= proc(n, s) option remember; expand(`if`(n=0,
      `if`(issqr(s), 1, 0), b(n-1, s)+x*b(n-1, s+n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..16);

b[n_, s_] := b[n, s] = Expand[If[n == 0, If[IntegerQ @ Sqrt[s], 1, 0], b[n - 1, s] + x*b[n - 1, s + n]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

T(n,n) = 1 for n in { A001108 }, T(n,n) = 0 otherwise.
T(n,n-1) = 1 for n in { A214857 }, T(n,n-1) = 0 for n in { A214858 }.
Sum_{k=0..n} k * T(n,k) = A377572(n).

A214856 Number of triangular numbers in interval ](n-1)^2, n^2] for n>0, a(0)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1

Offset: 0

Philippe Deléham, Mar 08 2013

			10, 15 are in interval ]9, 16] , a(4) = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000217, A000290, A214848, A214857.

a(n) = if (n, sum(i=(n-1)^2+1, n^2, ispolygonal(i, 3)), 1); \\ Michel Marcus, Nov 12 2022

from math import isqrt
def A214856(n): return (isqrt((m:=n**2<<3)+8)+1>>1)-(isqrt(m-(n-1<<4))+1>>1) if n else 1 # Chai Wah Wu, Dec 09 2024

A214859 Triangle read by rows, T(n,k) = n^2 - k(k+1)/2 if k(k+1)/2 <= n^2.

Original entry on oeis.org

0, 1, 0, 4, 3, 1, 9, 8, 6, 3, 16, 15, 13, 10, 6, 1, 25, 24, 22, 19, 15, 10, 4, 36, 35, 33, 30, 26, 21, 15, 8, 0, 49, 48, 46, 43, 39, 34, 28, 21, 13, 4, 64, 63, 61, 58, 54, 49, 43, 36, 28, 19, 9, 81, 80, 78, 75, 71, 66, 60, 53, 45, 36, 26, 15, 3, 100, 99, 97

Offset: 0

Philippe Deléham, Mar 09 2013

Row lengths are in A214857.

			Triangle begins:
    0;
    1,   0;
    4,   3,   1;
    9,   8,   6,   3;
   16,  15,  13,  10,   6,   1;
   25,  24,  22,  19,  15,  10,   4;
   36,  35,  33,  30,  26,  21,  15,  8,  0;
   49,  48,  46,  43,  39,  34,  28, 21, 13,  4;
   64,  63,  61,  58,  54,  49,  43, 36, 28, 19,  9;
   81,  80,  78,  75,  71,  66,  60, 53, 45, 36, 26, 15,  3;
  100,  99,  97,  94,  90,  85,  79, 72, 64, 55, 45, 34, 22,  9;
  121, 120, 118, 115, 111, 106, 100, 93, 85, 76, 66, 55, 43, 30, 16, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. Diagonals: A000217, A034856, A055999,

Columns: A000290, A005563, A028872, A028878.

Table[s = {}; k = 0; While[tri = k*(k + 1)/2; tri <= n^2, AppendTo[s, n^2 - tri]; k++]; s, {n, 0, 10}] (* T. D. Noe, Mar 11 2013 *)

T(2*n,n) = A022264(n).

T(n,n) = n*(n-1)/2 = A000217(n-1).

A378301 a(n) is the number of triangular numbers (A000217) in the interval [n^2, (n + 1)^2].

Original entry on oeis.org

2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1

Offset: 0

Ctibor O. Zizka, Nov 22 2024

nonn

After n=22, is the frequency of 1 always > 2? Are terms only 1 or 2? - Bill McEachen, Dec 10 2024

The distinct terms of this sequence are only 1 and 2. The asymptotic densities of the occurrences of 1 and 2 are 2-sqrt(2) = 0.585... and sqrt(2)-1 = 0.414..., respectively. The asymptotic mean of this sequence is sqrt(2). - Amiram Eldar, Dec 11 2024

			n = 0: in the interval [0, 1] are 2 triangular numbers {0, 1}, thus a(0) = 2.
n = 1: in the interval [1, 4] are 2 triangular numbers {1, 3}, thus a(1) = 2.

Cf. A000217, A000290, A001110, A002024, A003056, A214856, A214857.

s[n_] := Floor[(Sqrt[8*n+1]-1)/2]; a[n_] := s[(n + 1)^2] - s[n^2 - 1]; a[0] = 2; Array[a, 100, 0] (* Amiram Eldar, Dec 09 2024 *)

a(n) = sum(k=n^2, (n+1)^2, ispolygonal(k,3)); \\ Michel Marcus, Dec 09 2024

from math import isqrt
def A378301(n): return -(isqrt(m:=n**2<<3)+1>>1)+(isqrt(m+(n+1<<4))+1>>1) # Chai Wah Wu, Dec 09 2024

a(n) = 2 for n from A001110.
a(n) = A003056((n+1)^2) - A003056(n^2-1) for n >= 1. - Amiram Eldar, Dec 09 2024
a(n) = A002024((n+1)^2+1) - A002024(n^2). - Chai Wah Wu, Dec 09 2024

a(53) corrected by Michel Marcus, Dec 09 2024

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A214858 Natural numbers missing from A214857.

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A281871 Number T(n,k) of k-element subsets of [n] having a square element sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A214856 Number of triangular numbers in interval ](n-1)^2, n^2] for n>0, a(0)=1.

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A214859 Triangle read by rows, T(n,k) = n^2 - k*(k+1)/2 if k*(k+1)/2 <= n^2.

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A378301 a(n) is the number of triangular numbers (A000217) in the interval [n^2, (n + 1)^2].

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A214859 Triangle read by rows, T(n,k) = n^2 - k(k+1)/2 if k(k+1)/2 <= n^2.