A103128 -id:A103128 - cp's OEIS frontend

A101688 Once 1, once 0, repeat, twice 1, twice 0, repeat, thrice 1, thrice 0, ... and so on.

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1

Offset: 0

Ralf Stephan, Dec 19 2004

The definition is that of a linear sequence. Equivalently, define a (0,1) infinite lower triangular matrix T(n,k) (0 <= k <= n) by T(n,k) = 1 if k >= n/2, 0 otherwise, and read it by rows. The triangle T begins:
  1
  0 1
  0 1 1
  0 0 1 1
  0 0 1 1 1
  0 0 0 1 1 1
...  The matrix T is used in A168508. [Comment revised by N. J. A. Sloane, Dec 05 2020]
Also, square array A read by antidiagonals upwards: A(n,k) = 1 if k >= n, 0 otherwise.
For n >= 1, T(n,k) = number of partitions of n into k parts of sizes 1 or 2. - Nicolae Boicu, Aug 23 2018
T(n, k) is the number of ways to distribute n balls to k unlabeled urns in such a way that no urn receives more than one ball (see Beeler). - Stefano Spezia, Jun 16 2023

			The array A (on the left) and the triangle T of its antidiagonals (on the right):
  1 1 1 1 1 1 1 1 1 ......... 1
  0 1 1 1 1 1 1 1 1 ........ 0 1
  0 0 1 1 1 1 1 1 1 ....... 0 1 1
  0 0 0 1 1 1 1 1 1 ...... 0 0 1 1
  0 0 0 0 1 1 1 1 1 ..... 0 0 1 1 1
  0 0 0 0 0 1 1 1 1 .... 0 0 0 1 1 1
  0 0 0 0 0 0 1 1 1 ... 0 0 0 1 1 1 1
  0 0 0 0 0 0 0 1 1 .. 0 0 0 0 1 1 1 1
  0 0 0 0 0 0 0 0 1 . 0 0 0 0 1 1 1 1 1

Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015. See Proposition 4.2.1 at p. 98.

Boris Putievskiy, Transformations (of) Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Row sums of T (and antidiagonal sums of A) are A008619.
Cf. A079813, A168508.
Cf. A103128, A003056.

rows = 15; A = Array[If[#1 <= #2, 1, 0]&, {rows, rows}]; Table[A[[i-j+1, j]], {i, 1, rows}, {j, 1, i}] // Flatten (* Jean-François Alcover, May 04 2017 *)

from math import isqrt
def A101688(n): return isqrt((m:=n<<1)+1)-(isqrt((m<<2)+8)+1>>1)+1 # Chai Wah Wu, Feb 10 2023

G.f.: 1/((1 - x*y)*(1 - y)).
G.f. of k-th row of the array: x^(k-1)/(1 - x).
T(n, k) = 1 if binomial(k, n-k) > 0, otherwise 0. - Paul Barry, Aug 23 2005
From Boris Putievskiy, Jan 09 2013: (Start)
a(n) = floor((2*A002260(n)+1)/A003056(n)+3).
a(n) = floor((2*n-t*(t+1)+1)/(t+3)), where
t = floor((-1+sqrt(8*n-7))/2). (End)
a(n) = floor(sqrt(2*n+1)) - floor(sqrt(2*n+1) - 1/2). - Ridouane Oudra, Jul 16 2020
a(n) = A103128(n+1) - A003056(n). - Ridouane Oudra, Apr 09 2022
E.g.f. of k-th column of the array: exp(x)*Gamma(1+k, x)/k!. - Stefano Spezia, Jun 16 2023

Edited by N. J. A. Sloane, Dec 05 2020

A172471 a(n) = floor(sqrt(2*n)).

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12

Offset: 0

Vincenzo Librandi, Feb 04 2010

If k is even, k appears k+1 times; if k is odd, k appears k times. - N. J. A. Sloane, Jan 25 2021

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 [First 10000 terms from Vincenzo Librandi]

Cf. A000196, A003056, A103128, A005843.

a172471 = a000196 . (* 2)  -- Reinhard Zumkeller, Feb 12 2012

Magma
```
[Floor(Sqrt(2*n)) : n in [0..80]];
```
Mathematica
```
Table[Floor[Sqrt[2n]],{n,0,80}]
```

a(n) = sqrtint(2*n); \\ Michel Marcus, May 23 2025

a(n) = A000196(2*n). - R. J. Mathar, Jul 06 2010

A176615 Number of edges in the graph on n vertices, labeled 1 to n, where two vertices are joined just if their labels sum to a perfect square.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 16, 17, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 49, 52, 55, 57, 59, 61, 63, 65, 68, 71, 74, 77, 80, 83, 86, 89, 91, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 127, 131, 135, 138, 141, 144, 147, 150

Offset: 1

Franklin T. Adams-Watters, Apr 21 2010

Equivalently, number of pairs of integers 0 < i < j <= n such that i + j is a square.

Suggested by R. K. Guy

			For n = 7 the graph contains the 4 edges 1-3, 2-7, 3-6, 4-5.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000196, A022554, A103128, A281706.

Column k=2 of A281871.

b:= n-> 1+floor(sqrt(2*n-1))-ceil(sqrt(n+1)):
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+b(n)) end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 30 2017

a[n_] := Sum[Floor[Sqrt[2k-1]] - Floor[Sqrt[k]], {k, 1, n}]; Table[a[n], {n, 1, 68}] (* Jean-François Alcover, Nov 04 2011, after Pari *)

a(n)=sum(k=1,sqrtint(n+1),ceil(k^2/2)-1)+sum(k=sqrtint(n+1)+1,sqrtint(2*n -1),n-floor(k^2/2))

a(n)=sum(k=1,n,sqrtint(2*k-1)-sqrtint(k))

Asymptotically, a(n) ~ (2*sqrt(2) - 2)/3 n^(3/2). The error term is probably O(n^(1/2)); O(n) is easily provable.

A273190 a(n) is the number of nonnegative m < n for which m + n is a perfect square.

Original entry on oeis.org

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

Offset: 0

Alec Jones, May 17 2016

			a(1) = 1 because 1 + 0 is a perfect square.
a(2) = 0 because neither 2 + 0 nor 2 + 1 are perfect squares.
a(5) = 1 because 5 + 4 is a perfect square.
a(9) = 2 because 9 + 0 and 9 + 7 are perfect squares.

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A273185, A273191.

a273190 n = length $ filter (>=n) $ takeWhile (< 2 * n) $ map (^2) [1..] -- Peter Kagey, May 25 2016

int n = 100;
int[] terms = new int[n];
for (int i = 0; i < n; i++) {
     for (int j = 0; j < i; j++) {
          if (Math.sqrt(i+j) == Math.floor(Math.sqrt(i+j))) {
               terms[i]++;
          }
     }
     System.out.print(terms[i] + ", ");
}

Table[Count[Range[0, n - 1], m_ /; IntegerQ@ Sqrt[m + n]], {n, 0, 120}] (* Michael De Vlieger, May 18 2016 *)

a(n) = sum(k=0, n-1, issquare(n+k)); \\ Michel Marcus, May 18 2016

from gmpy2 import isqrt
def A273190(n):
    return isqrt(2*n-1)-isqrt(n-1) if n > 0 else 0 # Chai Wah Wu, May 25 2016

a(n) = floor(sqrt(2*n-1))-floor(sqrt(n-1)) for n > 0. - Chai Wah Wu, May 25 2016
a(n) = A103128(n) - A000196(n-1); after previous formula. - Michel Marcus, May 25 2016
a(n) = Sum_{i=1..n} c(2*n-i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020

A333574 Number of Hamiltonian paths in the n X 2 grid graph which start at any of the n vertices on left side of the graph and terminate at any of the n vertices on the right side.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 20, 26, 34, 42, 52, 62, 74, 86, 100, 114, 130, 146, 164, 182, 202, 222, 244, 266, 290, 314, 340, 366, 394, 422, 452, 482, 514, 546, 580, 614, 650, 686, 724, 762, 802, 842, 884, 926, 970, 1014, 1060, 1106, 1154, 1202, 1252, 1302, 1354, 1406, 1460

Offset: 1

Seiichi Manyama, Mar 27 2020

Conjecture: Numbers k such that A339399(k) = A103128(k). - Wesley Ivan Hurt, Nov 19 2021

			a(1) = 1;
   +--+
a(2) = 2;
   +  +   *--*
   |  |   |  |
   *--*   +  +
a(3) = 4;
   +  +   +--*   *--+   *--*
   |  |      |   |      |  |
   *  *   *--*   *--*   *  *
   |  |   |         |   |  |
   *--*   *--+   +--*   +  +

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Column k=2 of A333571.
Cf. A333510.
Cf. A103128, A339399.

N=66; x='x+O('x^N); Vec(x*(1+2*x*(1-x^2+x^3)/((1+x)*(1-x)^3)))

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A(start, goal, n, k):
    universe = tl.grid(n - 1, k - 1)
    GraphSet.set_universe(universe)
    paths = GraphSet.paths(start, goal, is_hamilton=True)
    return paths.len()
def A333571(n, k):
    if n == 1: return 1
    s = 0
    for i in range(1, n + 1):
        for j in range(k * n - n + 1, k * n + 1):
            s += A(i, j, k, n)
    return s
def A333574(n):
    return A333571(n, 2)
print([A333574(n) for n in range(1, 25)])

G.f.: x*(1+2*x*(1-x^2+x^3)/((1+x)*(1-x)^3)).
From Colin Barker, Mar 27 2020: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>5.
a(n) = (9 + (-1)^(1+n) - 4*n + 2*n^2) / 4 for n>1. (End)
E.g.f.: ((4 - x + x^2)*cosh(x) + (5 - x + x^2)*sinh(x) - 2*(2 + x))/2. - Stefano Spezia, Jun 14 2023

A183572 a(n) = n + floor(sqrt(2*n-1)).

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92

Offset: 1

Clark Kimberling, Jan 05 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Steve Butler and Jia Mao, Problem 11265, Amer. Math. Monthly 114, (2007), p77.

Cf. A074148 (complement), A103128.

[n+Floor(Sqrt(2*n-1)): n in [1..80]]; // Vincenzo Librandi, Feb 18 2017

Table[n + Floor[Sqrt[2 n - 1]], {n, 100}] (* Vincenzo Librandi, Feb 18 2017 *)

a(n) = n + sqrtint(2*n-1); \\ Michel Marcus, Feb 18 2017

A183573 a(n) = n + floor(sqrt(2n+1)).

Original entry on oeis.org

2, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92

Offset: 1

Clark Kimberling, Jan 05 2011

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

Cf. A047838, A103128, A116940 (complement).

seq(n+floor(sqrt(2*n+1)), n=1..100); # Robert Israel, Sep 12 2016

Table[n + Floor@ Sqrt[2 n + 1], {n, 80}] (* Michael De Vlieger, Sep 12 2016 *)

From Robert Israel, Sep 12 2016: (Start)
a(n+1)=a(n)+2 if n is in A047838, otherwise a(n+1) = a(n)+1.
a(n) = n + A103128(n+1).
G.f.: Theta3(x^2)/(2*(1-x)) + Theta2(x^2)/(2*sqrt(x)*(1-x)) - (1-2*x)*(3-x)/(2*(1-x)^2), where Theta2 and Theta3 are Jacobi Theta functions. (End)

A337134 a(n) = Sum_{k=1..n} floor(sqrt(2k-1)).

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 15, 18, 22, 26, 30, 34, 39, 44, 49, 54, 59, 64, 70, 76, 82, 88, 94, 100, 107, 114, 121, 128, 135, 142, 149, 156, 164, 172, 180, 188, 196, 204, 212, 220, 229, 238, 247, 256, 265, 274, 283, 292, 301, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 410, 421, 432

Offset: 1

Wesley Ivan Hurt, Aug 18 2020

Partial sums of A103128.

Cf. A103128.

Table[Sum[Floor[Sqrt[2 i - 1]], {i, n}], {n, 100}]

a(n) = sum(k=1, n, sqrtint(2*k-1)); \\ Michel Marcus, Aug 18 2020

A349489 a(n) = Sum_{k=1..n} k * floor(sqrt(2*k-1)).

Original entry on oeis.org

1, 3, 9, 17, 32, 50, 71, 95, 131, 171, 215, 263, 328, 398, 473, 553, 638, 728, 842, 962, 1088, 1220, 1358, 1502, 1677, 1859, 2048, 2244, 2447, 2657, 2874, 3098, 3362, 3634, 3914, 4202, 4498, 4802, 5114, 5434, 5803, 6181, 6568, 6964, 7369, 7783, 8206, 8638, 9079, 9529, 10039

Offset: 1

Wesley Ivan Hurt, Nov 19 2021

nonn

Cf. A103128, A337134.

Table[Sum[k*Floor[Sqrt[2 k - 1]], {k, n}], {n, 80}]

cp's OEIS Frontend

A101688 Once 1, once 0, repeat, twice 1, twice 0, repeat, thrice 1, thrice 0, ... and so on.

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A172471 a(n) = floor(sqrt(2*n)).

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A176615 Number of edges in the graph on n vertices, labeled 1 to n, where two vertices are joined just if their labels sum to a perfect square.

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A273190 a(n) is the number of nonnegative m < n for which m + n is a perfect square.

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A333574 Number of Hamiltonian paths in the n X 2 grid graph which start at any of the n vertices on left side of the graph and terminate at any of the n vertices on the right side.

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A183572 a(n) = n + floor(sqrt(2*n-1)).

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Mathematica

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A183573 a(n) = n + floor(sqrt(2n+1)).

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