A081349 -id:A081349 - cp's OEIS frontend

A081352 Main diagonal of square maze arrangement of natural numbers A081349.

1, 7, 11, 21, 29, 43, 55, 73, 89, 111, 131, 157, 181, 211, 239, 273, 305, 343, 379, 421, 461, 507, 551, 601, 649, 703, 755, 813, 869, 931, 991, 1057, 1121, 1191, 1259, 1333, 1405, 1483, 1559, 1641, 1721, 1807, 1891, 1981, 2069, 2163, 2255, 2353, 2449, 2551

Offset: 0

Paul Barry, Mar 19 2003

Conjecture: let a and b be integers such that 0 < a < b so that 0 < a/b is a proper fraction. Define the map f(a,b,D) = a/b + gcd(a,b)/D. Of course, all such a/b can be partially ordered by value, i.e., 1/2 = 0.5 < 2/3 = 4/6 = 6/9 = 0.6666... < 3/4 = 6/8 = 0.75 < 4/5 = 0.8 etc. The map f appears to specify a total strict order on the co-domain for all a/b that is consistent with the given partial order of the domain, i.e., the partial order remains intact, while equivalent fractions are given a total strict order themselves. Moreover, equivalent fractions are strictly ordered by numerator (or denominator), e.g., 1/2 < 2/4 < 3/6 etc. The conditions are that for n >= 4 all of the fractions with denominator b <= n are listed and the minimum integer value of D to achieve the total strict order of the co-domain is 2*C(n-1,2) - (-1)^(n-1). So, a(n-3) = D for n >= 4. Example: given n = 4, we have D = 2*(4-1,2) - (-1)^(4-1) = 2*3 + 1 = 7 = a(4-3) = a(1). Partial order of domain. 1/4 < 1/3 < 1/2 = 2/4 < 2/3 < 3/4. Total order of co-domain. f(1,4,7) = 1/4 + 1/7 = 33/84 < f(1,3,7) = 1/3 + 1/7 = 40/84 < f(1,2,7) = 1/2 + 1/7 = 54/84 < f(2,4,7) = 2/4 + 2/7 = 66/84 < f(2,3,7) = 2/3 + 2/7 = 68/84 < f(3,4,7) = 3/4 + 1/7 = 75/84. Observe that if D = 6, then f(2,4,6) = 2/4 + 2/6 = 10/12 = f(2,3,6) = 2/3 + 1/6. Computation shows the same failure to achieve total strict order of the co-domain for D = 2..5. (As a >= 1, then b >=2, from the above). Computation also shows that the conjecture holds for n = 4..17. - Ross La Haye, Oct 02 2016

B. Berselli, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A081350, A081351, A054569.

I:=[1,7,11,21]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Aug 08 2013

A081352:=n->(n + 1)*(n + 2) - (-1)^n; seq(A081352(n), n=0..50); # Wesley Ivan Hurt, Feb 26 2014

CoefficientList[Series[(1 + 5 x - 3 x^2 + x^3) / ((1 + x) (1 - x)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Aug 08 2013 *)

x='x+O('x^99); Vec((1+5*x-3*x^2+x^3)/((1+x)*(1-x)^3)) \\ Altug Alkan, Mar 26 2016

a(n) = (n + 1)*(n + 2) - (-1)^n = 2*C(n+2, 2) - (-1)^n.
G.f.: (1 +5*x -3*x^2 +x^3) / ((1+x)*(1-x)^3). [Bruno Berselli, Aug 01 2010]
a(n) -2*a(n-1) +2*a(n-3) -a(n-4) = 0 with n>3. [Bruno Berselli, Aug 01 2010]
a(n) = 3*A000982(n + 2) - A000982(n + 3). - Miko Labalan, Mar 26 2016
a(n) = A116940(n) + A236283(n + 1). - Miko Labalan, Dec 04 2016
a(n) = (2*n^2 + 6*n - 2*(-1)^n + (-1)^(2*n) + 3)/2. - Kritsada Moomuang, Oct 24 2019

A081350 First column in maze array of natural numbers A081349.

Original entry on oeis.org

1, 2, 3, 4, 15, 16, 35, 36, 63, 64, 99, 100, 143, 144, 195, 196, 255, 256, 323, 324, 399, 400, 483, 484, 575, 576, 675, 676, 783, 784, 899, 900, 1023, 1024, 1155, 1156, 1295, 1296, 1443, 1444, 1599, 1600, 1763, 1764, 1935, 1936, 2115, 2116, 2303, 2304, 2499

Offset: 0

Paul Barry, Mar 19 2003

Interleaves A000466 with even squares A016742.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A081351, A081352.

a(0)=1, a(1)=2, a(n)=(n-1)*(n+(-1)^n), n>1.

G.f.: -(2*x^6-10*x^4+x^3+x^2-x-1)/((1-x)^3*(1+x)^2). [Colin Barker, Sep 03 2012]

A081351 First row in square maze array of natural numbers A081349.

Original entry on oeis.org

1, 8, 9, 24, 25, 48, 49, 80, 81, 120, 121, 168, 169, 224, 225, 288, 289, 360, 361, 440, 441, 528, 529, 624, 625, 728, 729, 840, 841, 960, 961, 1088, 1089, 1224, 1225, 1368, 1369, 1520, 1521, 1680, 1681, 1848, 1849, 2024, 2025, 2208, 2209, 2400, 2401, 2600

Offset: 0

Paul Barry, Mar 19 2003

Interleaves the odd squares A016754 with 8 times the triangular numbers A000217.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A081350, A081352.

[(n+1)*(n+2)-(n+1)*(-1)^n: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

a(n) = (n+1)*(n+2)-(n+1)*(-1)^n = 2*C(n+2,2)-C(n+1,1)*(-1)^n.

G.f.: (x^3-x^2+7*x+1)/((1-x)^3*(1+x)^2). [Colin Barker, Sep 03 2012]

A081353 Diagonal of square maze arrangement of natural numbers A081349.

Original entry on oeis.org

3, 5, 13, 19, 31, 41, 57, 71, 91, 109, 133, 155, 183, 209, 241, 271, 307, 341, 381, 419, 463, 505, 553, 599, 651, 701, 757, 811, 871, 929, 993, 1055, 1123, 1189, 1261, 1331, 1407, 1481, 1561, 1639, 1723, 1805, 1893, 1979, 2071, 2161, 2257, 2351, 2451, 2549

Offset: 0

Paul Barry, Mar 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A081349, A081350, A081351, A081352.
Cf. A109613, A137932.
Bisections are in A054554, A125202.

[(n + 1)*(n + 2) + (-1)^n: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

A081353:=n->(n+1)*(n+2)+(-1)^n: seq(A081353(n), n=0..100); # Wesley Ivan Hurt, Aug 09 2015

Table[(n + 1) (n + 2) + (-1)^n, {n, 0, 70}] (* Wesley Ivan Hurt, Aug 09 2015 *)
LinearRecurrence[{2,0,-2,1},{3,5,13,19},50] (* Harvey P. Dale, Aug 02 2021 *)

a(n) = (n+1)*(n+2)+(-1)^n = 2*binomial(n+2,2)+(-1)^n.
G.f.: (3-x)*(1+x^2)/((1-x)^3*(1+x)). [Colin Barker, Sep 03 2012]
From Wesley Ivan Hurt, Aug 09 2015: (Start)
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4), n>4.
a(n) = n^2+3n+3 if n is even, otherwise n^2+3n+1.
a(n) = A137932(n+3) - A109613(n+1). (End)

A093650 Natural numbers arranged in a square maze beginning 1, 2, 3, then moving right, then up, right, down, left, down, right, etc., and read by antidiagonals upwards.

Original entry on oeis.org

1, 2, 6, 3, 5, 7, 12, 4, 8, 20, 13, 11, 9, 19, 21, 30, 14, 10, 18, 22, 42, 31, 29, 15, 17, 23, 41, 43, 56, 32, 28, 16, 24, 40, 44, 72, 57, 55, 33, 27, 25, 39, 45, 71, 73, 90, 58, 54, 34, 26, 38, 46, 70, 74, 110, 91, 89, 59, 53, 35, 37, 47, 69, 75, 109, 111

Offset: 1

Michael Joseph Halm, May 15 2004

			a(3) = 6 because the maze begins 2 under 1, 3 under 2, 4 right of 3, 5 right of 2 and 6 right of 1.
Array begins:
   1   6---7  20 ...
   |   |   |   |
   2   5   8  19 ...
   |   |   |   |
   3---4   9  18 ...
           |   |
  12--11--10  17 ...
   |           |
  13--14--15--16 ...
  ...

Cf. A080335, A081347, A081348.

Other square mazes: A081344, A081349.

More terms from Jinyuan Wang, Jun 15 2022

A329547 Number of natural numbers k <= n such that k^k is a square.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 40

Offset: 1

Pablo Hueso Merino, Nov 16 2019

For even k, k^k is always a square. For odd k, k^k is a square if and only if k is a square.
It seems the unrepeated terms form A266304 \ {0}. - Ivan N. Ianakiev, Nov 21 2019
Indices of unrepeated terms are A081349. - Rémy Sigrist, Dec 07 2019

			a(5) = 3 because among 1^1, 2^2, ..., 5^5 there are 3 squares: 1^1, 2^2, and 4^4.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A081349, A176693, A266304.

Table[Floor[n/2] + Ceiling[Floor[Sqrt[n]]/2], {n, 1, 100}]

a(n) = sum(k=1, n, issquare(k^k)); \\ Michel Marcus, Nov 17 2019

first(n) = my(res=vector(n), inc); res[1] = 1; for(i=2, n, inc = (1-(i%2)) || issquare(i); res[i] = res[i-1] + inc); res \\ David A. Corneth, Dec 07 2019

a(n) = n\2 + (sqrtint(n)+1)\2 \\ David A. Corneth, Dec 07 2019

from math import isqrt
def A329547(n): return (n>>1)+(isqrt(n)+1>>1) # Chai Wah Wu, Sep 18 2024

a(n) = floor(n/2) + ceiling(floor(sqrt(n))/2).

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A081352 Main diagonal of square maze arrangement of natural numbers A081349.

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A081350 First column in maze array of natural numbers A081349.

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A081351 First row in square maze array of natural numbers A081349.

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A081353 Diagonal of square maze arrangement of natural numbers A081349.

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A093650 Natural numbers arranged in a square maze beginning 1, 2, 3, then moving right, then up, right, down, left, down, right, etc., and read by antidiagonals upwards.

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A329547 Number of natural numbers k <= n such that k^k is a square.

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