A233035 -id:A233035 - cp's OEIS frontend

A242669 a(n) = n*floor(n/3).

0, 0, 0, 3, 4, 5, 12, 14, 16, 27, 30, 33, 48, 52, 56, 75, 80, 85, 108, 114, 120, 147, 154, 161, 192, 200, 208, 243, 252, 261, 300, 310, 320, 363, 374, 385, 432, 444, 456, 507, 520, 533, 588, 602, 616, 675, 690, 705, 768, 784, 800, 867, 884, 901, 972, 990

Offset: 0

Bruno Berselli, Jul 01 2014

For n = 0, 1, 2, 4, 8, 49, 98, 676, 1352, 9409, 18818, 131044, 262088, 1825201, 3650402, ... a(n) is a square.

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

Cf. A000290 (n^2), A010762 (floor(n/2)*floor(n/3)), A093353 (n*floor(n/2)), A213033 (n*floor(n/2)*floor(n/3)), A233035 (n*floor(n/4)).
Cf. A033428, A045944, A049451.
Cf. A002264 (floor(n/3)).

Magma
```
[n*Floor(n/3): n in [0..60]];
```
Mathematica
```
Table[n Floor[n/3], {n, 0, 60}]
```

a(n)=n\3*n \\ Charles R Greathouse IV, Oct 07 2015

Sage
```
[n*floor(n/3) for n in (0..60)];
```

G.f.: x^3*(3 + x + x^2 + x^3)/((1 - x)^3*(1 + x + x^2)^2).
a(3m) = A033428(m), a(3m+1) = A049451(m), a(3m+2) = A045944(m).
Sum_{n>=3} (-1)^(n+1)/a(n) = 9/4 + Pi^2/36 - Pi/(2*sqrt(3)) - 2*log(2). - Amiram Eldar, Mar 30 2023

A233036 The maximum number of I-tetrominoes that can be packed into an n X n array of squares when rotation is allowed.

Original entry on oeis.org

0, 0, 0, 4, 6, 8, 12, 16, 20, 24, 30, 36, 42, 48, 56, 64, 72, 80, 90, 100, 110, 120, 132, 144, 156, 168, 182, 196, 210, 224, 240, 256, 272, 288, 306, 324, 342, 360, 380, 400, 420, 440, 462, 484, 506, 528, 552, 576, 600, 624, 650, 676, 702, 728, 756, 784, 812, 840, 870, 900, 930, 960, 992, 1024, 1056

Offset: 1

Kival Ngaokrajang, Dec 03 2013

By de Bruijn's theorem (see the de Bruijn link), an m X n rectangle can't be tiled with I tetrominoes unless m or n is divisible by 4. - Robert Israel, Oct 15 2015

N. G. de Bruijn, "Filling boxes with bricks", The American Mathematical Monthly 76 (1969), 37-40.
Robert Israel, Illustration of initial terms
Wikipedia, Tetromino
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A233035.

0$3, seq(op([4*k^2, 2*k*(2*k+1),4*k*(k+1),(2*k+1)*(2*k+2)]),k=1..20);# Robert Israel, Oct 15 2015

CoefficientList[Series[2 x^3/((1 + x) (1 + x^2) (1 - x)^3) - 2 x^3, {x, 0, 100}], x] (* Vincenzo Librandi, Oct 15 2015 *)
LinearRecurrence[{2,-1,0,1,-2,1},{0,0,0,4,6,8,12,16,20},70] (* Harvey P. Dale, Dec 16 2018 *)

From Robert Israel, Oct 15 2015: (Start)
a(4*k) = 4*k^2.
a(2*k+1) = k*(k+1) for k >= 2.
a(4*k+2) = 4*k*(k+1).
G.f.: 2*x^3/((1 + x)*(1 + x^2)*(1 - x)^3) - 2*x^3. (End)
Apparently a(n) = A182568(n+2) for n > 3. - Georg Fischer, Oct 14 2018

Corrected by Robert Israel, Oct 15 2015

A233735 G.f.: x^3(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 + x^2 - x + 1) / ((1-x^5) (1-x)^2).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 6, 8, 10, 13, 16, 20, 25, 29, 34, 39, 45, 52, 58, 65, 72, 80, 88, 96, 105, 114, 124, 134, 144, 155, 166, 178, 190, 202, 215, 228, 242, 256, 270, 285, 300, 316, 332, 348, 365, 382, 400, 418, 436, 455, 474, 494, 514, 534

Offset: 0

Kival Ngaokrajang, Dec 15 2013

The second differences repeat with period 1,0,1,0,0 for n >= 20.

a(n) is a lower bound on A085577(n-2). The Ngaokrajang link shows arrangements of a(n) Greek crosses in an n X n grid. Note that a(11)=16, whereas A085577(9)=17, so the bound is not always tight. - N. J. A. Sloane, Apr 19 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Packings of a(n) Greek crosses. [Note that it is possible to pack 17 Greek crosses into an 11 X 11 grid (see A085577), so these arrangements are not always optimal. - _N. J. A. Sloane_, Apr 19 2015]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1)

Cf. A085576, A085577, A233035, A233036.

CoefficientList[Series[x^3*(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 +x^2 - x + 1)/((1 - x^5)*(1 - x)^2), {x, 0, 50}], x] (* G. C. Greubel, Jan 08 2018 *)

x='x+O('x^50); Vec(x^3*(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 +x^2 - x + 1)/((1 - x^5)*(1 - x)^2)) \\ G. C. Greubel, Jan 08 2018

Entry revised by N. J. A. Sloane, Apr 19 2015. The new definition is a g.f. found by Ralf Stephan on Dec 17 2013. The old definition was wrong.

A365686 Numbers k such that there exists a pair of integers (m,h) where 1 <= m < floor(sqrt(k)/2) <= h that satisfy Sum_{j=0..m} (k-j)^2 = Sum_{i=1..m} (h+i)^2.

Original entry on oeis.org

4, 12, 21, 24, 40, 60, 84, 110, 112, 120, 144, 180, 220, 264, 312, 315, 364, 420, 480, 544, 612, 684, 697, 760, 820, 840, 924, 1012, 1080, 1104, 1200, 1265, 1300, 1404, 1512, 1624, 1740, 1860, 1984, 2106, 2112, 2244, 2380, 2520, 2664, 2812, 2964, 3120, 3255

Offset: 1

Darío Clavijo, Sep 15 2023

nonn

The sums are of m+1 consecutive squares ending at k^2, and of m consecutive squares starting somewhere at or beyond (k+1)^2.
If k is a term and h = k then k is in A046092.
All terms are composite numbers.
Also k is a term if there exists a pair of integers (m, h) such that 1 <= m < floor(sqrt(k)/2) <= h and that satisfy k*(m+1)*(k-m)-m*h*(h+m+1)=0.

			k=24 is a term because 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2 with m=3 and h=24.
k=110 is a term because 108^2 + 109^2 + 110^2 = 133^2 + 134^2, with m=2 and h=132.

Project Euler, Problem 261: Pivotal Square Sums.

Cf. A000290, A002378, A046092, A233035.

isok(k) = for (i=1, k-1, my(s1 = sum(j=k-i, k, j^2)); for (m=k+1, oo, my(s2 = sum(j=0, i-1, (m+j)^2)); if (s2 == s1, return(1)); if (s2 > s1, break););); \\ Michel Marcus, Sep 27 2023

from sympy import isprime
from sympy.ntheory.primetest import is_square
from sympy.core.intfunc import isqrt
A002378 = lambda n: n * (n + 1)
A046092 = lambda n: A002378(n) << 1
isA046092 = lambda n: (n & 1 == 0) and is_square((n << 1) + 1)
def isok(k):
    if isprime(k): return False
    if isA046092(k): return True
    k2 = k * k
    for m in range(1, (isqrt(k) >> 1) + 1):
        h, m2, m_2 = k+1, m * m, m << 1
        S = k2 - k * A046092(m)
        while S > 0:
            S -= m2 + (h * m_2)
            h += 1
            if S == 0: return True
print([k for k in range(1, 3256) if isok(k)])

k if Sum_{j=0..m} (k-j)^2 = Sum_{i=1..m} (h+i)^2 where 1 <= m < floor(sqrt(k)/2) <= h.

cp's OEIS Frontend

A242669 a(n) = n*floor(n/3).

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A233036 The maximum number of I-tetrominoes that can be packed into an n X n array of squares when rotation is allowed.

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A233735 G.f.: x^3*(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 + x^2 - x + 1) / ((1-x^5) * (1-x)^2).

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Mathematica

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A365686 Numbers k such that there exists a pair of integers (m,h) where 1 <= m < floor(sqrt(k)/2) <= h that satisfy Sum_{j=0..m} (k-j)^2 = Sum_{i=1..m} (h+i)^2.

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Programs

PARI

Python

Formula

A233735 G.f.: x^3(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 + x^2 - x + 1) / ((1-x^5) (1-x)^2).