stroniken: matemàtiques-números-figurats-y-particions

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lunes, 20 de enero de 2020

números hexagonals

n=2
0110
1111
0110

n=3
0011100
0111110
1111111
0111110
0011100

P(n) = ( n+2·(n+(-1)) )·( 1+2·(n+(-1)) )+(-2)·n(n+(-1))
P(n) = ( n+(n+1)·2·(n+(-1))+4·(n+(-1))^{2} )+(-2)·n(n+(-1))
P(n) = ( n+2·(n^{2}+(-1))+4·(n^{2}+(-2)n+1) )+(-2)·n(n+(-1))
P(n) = 4n^{2}+(-5)n+2

P(2) = 8
P(3) = 23

números triangulars

n=1
10

n=2
100
110

n=3
1000
1100
1110

P(n) = (1/2)·(n+1)·n

números mitx-hexagonals

n=2
0110
1111

n=3
0011100
0111110
1111111

n=4
0001111000
0011111100
0111111110
1111111111

n=5
0000111110000
0001111111000
0011111111100
0111111111110
1111111111111

P(n) = ( n+2(n+(-1)) )·n+(-1)(n+(-1))·n
P(n) = n( 2n+(-1) ) = 2n^{2}+(-n)

P(0) = 0
P(1/2) = 0

P(2) = 6
P(3) = 15
P(4) = 28
P(5) = 45

miércoles, 1 de enero de 2020

partición de números enteros en base dos-seis

pm+8 = (6+2)+(m+...(p)...+m)

pm+10 = (6+2+2)+(m+...(p)...+m)

pm+12 = (6+2+2+2)+(m+...(p)...+m)
pm+12 = (6+6)+(m+...(p)...+m)

pm+14 = (6+2+2+2+2)+(m+...(p)...+m)
pm+14 = (6+6+2)+(m+...(p)...+m)

pm+16 = (6+2+2+2+2+2)+(m+...(p)...+m)
pm+16 = (6+6+2+2)+(m+...(p)...+m)

pm+18 = (6+2+2+2+2+2+2)+(m+...(p)...+m)
pm+18 = (6+6+2+2+2)+(m+...(p)...+m)
pm+18 = (6+6+6)+(m+...(p)...+m)

P( pm+(6k) ) = ( (6k)/6 )
P( pm+(6k+2) ) = ( ((6k+2)+(-2))/6 )
P( pm+(6k+4) ) = ( ((6k+4)+(-4))/6 )

martes, 31 de diciembre de 2019

partición de números enteros en base tres-seis

pm+9 = (6+3)+(m+...(p)...+m)

pm+12 = (6+3+3)+(m+...(p)...+m)
pm+12 = (6+6)+(m+...(p)...+m)

pm+15 = (6+3+3+3)+(m+...(p)...+m)
pm+15 = (6+6+3)+(m+...(p)...+m)

pm+18 = (6+3+3+3+3)+(m+...(p)...+m)
pm+18 = (6+6+3+3)+(m+...(p)...+m)
pm+18 = (6+6+6)+(m+...(p)...+m)

P( pm+(6k) ) = ( (6k)/6 )
P( pm+(6k+3) ) = ( ((6k+3)+(-3))/6 )

partición de números enteros en base dos-cuatro

pm+6 = (4+2)+(m+...(p)...+m)

pm+8 = (4+2+2)+(m+...(p)...+m)
pm+8 = (4+4)+(m+...(p)...+m)

pm+10 = (4+2+2+2)+(m+...(p)...+m)
pm+10 = (4+4+2)+(m+...(p)...+m)

pm+12 = (4+2+2+2+2)+(m+...(p)...+m)
pm+12 = (4+4+2+2)+(m+...(p)...+m)
pm+12 = (4+4+4)+(m+...(p)...+m)

P( pm+(4k) ) = ( (4k)/4 )
P( pm+(4k+2) ) = ( ((4k+2)+(-2))/4 )

partición de números enteros en base cuatro

pm+5 = (4+1)+(m+...(p)...+m)

pm+6 = (4+1+1)+(m+...(p)...+m)

pm+7 = (4+1+1+1)+(m+...(p)...+m)

pm+8 = (4+1+1+1+1)+(m+...(p)...+m)
pm+8 = (4+4)+(m+...(p)...+m)

pm+9 = (4+1+1+1+1+1)+(m+...(p)...+m)
pm+9 = (4+4+1)+(m+...(p)...+m)

pm+10 = (4+1+1+1+1+1+1)+(m+...(p)...+m)
pm+10 = (4+4+1+1)+(m+...(p)...+m)

pm+11 = (4+1+1+1+1+1+1+1)+(m+...(p)...+m)
pm+11 = (4+4+1+1+1)+(m+...(p)...+m)

pm+12 = (4+1+1+1+1+1+1+1+1)+(m+...(p)...+m)
pm+12 = (4+4+1+1+1+1)+(m+...(p)...+m)
pm+12 = (4+4+4)+(m+...(p)...+m)

P( pm+(4k) ) = ( (4k)/4 )
P( pm+(4k+1) ) = ( ((4k+1)+(-1))/4 )
P( pm+(4k+2) ) = ( ((4k+2)+(-2))/4 )
P( pm+(4k+3) ) = ( ((4k+3)+(-3))/4 )

partición de números enteros en base tres

pm+4 = (3+1)+(m+...(p)...+m)

pm+5 = (3+1+1)+(m+...(p)...+m)

pm+6 = (3+1+1+1)+(m+...(p)...+m)
pm+6 = (3+3)+(m+...(p)...+m)

pm+7 = (3+1+1+1+1)+(m+...(p)...+m)
pm+7 = (3+3+1)+(m+...(p)...+m)

pm+8 = (3+1+1+1+1+1)+(m+...(p)...+m)
pm+8 = (3+3+1+1)+(m+...(p)...+m)

pm+9 = (3+1+1+1+1+1+1)+(m+...(p)...+m)
pm+9 = (3+3+1+1+1)+(m+...(p)...+m)
pm+9 = (3+3+3)+(m+...(p)...+m)

P( pm+(3k) ) = ( (3k)/3 )
P( pm+(3k+1) ) = ( ((3k+1)+(-1))/3 )
P( pm+(3k+2) ) = ( ((3k+2)+(-2))/3 )

partición de números enteros en base dos

pm+3 = (2+1)+(m+...(p)...+m)

pm+4 = (2+1+1)+(m+...(p)...+m)
pm+4 = (2+2)+(m+...(p)...+m)

pm+5 = (2+1+1+1)+(m+...(p)...+m)
pm+5 = (2+2+1)+(m+...(p)...+m)

pm+6 = (2+1+1+1+1)+(m+...(p)...+m)
pm+6 = (2+2+1+1)+(m+...(p)...+m)
pm+6 = (2+2+2)+(m+...(p)...+m)

pm+7 = (2+1+1+1+1+1)+(m+...(p)...+m)
pm+7 = (2+2+1+1+1)+(m+...(p)...+m)
pm+7 = (2+2+2+1)+(m+...(p)...+m)

pm+8 = (2+1+1+1+1+1+1)+(m+...(p)...+m)
pm+8 = (2+2+1+1+1+1)+(m+...(p)...+m)
pm+8 = (2+2+2+1+1)+(m+...(p)...+m)
pm+8 = (2+2+2+2)+(m+...(p)...+m)

P( pm+(2k) ) = ( (2k)/2 )
P( pm+(2k+1) ) = ( ((2k+1)+(-1))/2 )

Problema:

Sigui pm+7 = 13 ==>

2·3+7 = (2+1+1+1+1+1)+(3+3)
2·3+7 = (2+2+1+1+1)+(3+3)
2·3+7 = (2+2+2+1)+(3+3)

2·3+7 = (2+1+1+1+1+1)+(2+2+2)
2·3+7 = (2+2+1+1+1)+(2+2+2)
2·3+7 = (2+2+2+1)+(2+2+2)

6·1+7 = (2+1+1+1+1+1)+6
6·1+7 = (2+2+1+1+1)+6
6·1+7 = (2+2+2+1)+6

6·1+7 = (2+1+1+1+1+1)+(1+1+1+1+1+1)
6·1+7 = (2+2+1+1+1)+(1+1+1+1+1+1)
6·1+7 = (2+2+2+1)+(1+1+1+1+1+1)

Problema:

Sigui pm+5 = 11 ==>

2·3+5 = (2+1+1+1)+(3+3)
2·3+5 = (2+2+1)+(3+3)

2·3+5 = (2+1+1+1)+(2+2+2)
2·3+5 = (2+2+1)+(2+2+2)

6·1+5 = (2+1+1+1)+6
6·1+5 = (2+2+1)+6

6·1+5 = (2+1+1+1)+(1+1+1+1+1+1)
6·1+5 = (2+2+1)+(1+1+1+1+1+1)

particiones de un número entero

n = 1+...(n)...+1

m+(1+...(q+1)...+1) = m+(q+1)

P_{1}(n) = (n+(-1))

m+(2+...(q+1)...+2) = m+(q+1)·2

Si n=2k ==> P_{2}(n) = k+(-1) = ( (n+(-2))/2 )
Si n=2k+1 ==> P_{2}(n) = k = ( (n+(-1))/2 )

m+(3+...(q+1)...+3) = m+(q+1)·3

Si n=3k ==> P_{3}(n) = k+(-1) = ( (n+(-3))/3 )
Si n=3k+1 ==> P_{3}(n) = k = ( (n+(-1))/3 )
Si n=3k+2 ==> P_{3}(n) = k = ( (n+(-2))/3 )

m+(4+...(q+1)...+4) = m+(q+1)·4

Si n=4k ==> P_{4}(n) = k+(-1) = ( (n+(-4))/4 )
Si n=4k+1 ==> P_{4}(n) = k = ( (n+(-1))/4 )
Si n=4k+2 ==> P_{4}(n) = k = ( (n+(-2))/4 )
Si n=4k+3 ==> P_{4}(n) = k = ( (n+(-3))/4 )

m+(5+...(q+1)...+5) = m+(q+1)·5

Si n=5k ==> P_{5}(n) = k+(-2) = ( (n+(-5))/5 )
Si n=5k+1 ==> P_{5}(n) = k+(-2) = ( (n+(-1))/5 )
Si n=5k+2 ==> P_{5}(n) = k+(-1) = ( (n+(-2))/5 )
Si n=5k+3 ==> P_{5}(n) = k+(-1) = ( (n+(-3))/5 )
Si n=5k+4 ==> P_{5}(n) = k+(-1) = ( (n+(-4))/5 )

(2+...(p+2)...+2)+(1+...(q+2)...+1) = (p+2)·2+(q+2)

Si n=2k ==> P_{1,2}(n) = k+(-2) = ( (n+(-4))/2 )
Si n=2k+1 ==> P_{1,2}(n) = k+(-2) = ( (n+(-5))/2 )

(3+...(p+2)...+3)+(1+...(q+2)...+1) = (p+2)·3+(q+2)

Si n=3k ==> P_{1,3}(n) = k+(-2) = ( (n+(-6))/3 )
Si n=3k+1 ==> P_{1,3}(n) = k+(-2) = ( (n+(-7))/3 )
Si n=3k+2 ==> P_{1,3}(n) = k+(-1) = ( (n+(-5))/3 )

(4+...(p+2)...+4)+(1+...(q+2)...+1) = (p+2)·4+(q+2)

Si n=4k ==> P_{1,4}(n) = k+(-2) = ( (n+(-8))/4 )
Si n=4k+1 ==> P_{1,4}(n) = k+(-2) = ( (n+(-9))/4 )
Si n=4k+2 ==> P_{1,4}(n) = k+(-1) = ( (n+(-6))/4 )
Si n=4k+3 ==> P_{1,4}(n) = k+(-1) = ( (n+(-7))/4 )

(5+...(p+2)...+5)+(1+...(q+2)...+1) = (p+2)·5+(q+2)

Si n=5k ==> P_{1,5}(n) = k+(-2) = ( (n+(-10))/5 )
Si n=5k+1 ==> P_{1,5}(n) = k+(-2) = ( (n+(-11))/5 )
Si n=5k+2 ==> P_{1,5}(n) = k+(-1) = ( (n+(-7))/5 )
Si n=5k+3 ==> P_{1,5}(n) = k+(-1) = ( (n+(-8))/5 )
Si n=5k+4 ==> P_{1,5}(n) = k+(-1) = ( (n+(-9))/5 )

2 = 1+1

3 = 1+1+1
3 = 2+1

4 = 1+1+1+1
4 = 3+1
4 = 2+(1+1)
4 = 2+2

5 = 1+1+1+1+1
5 = 4+1
5 = 3+(1+1)
5 = 2+(1+1+1)
5 = 3+2
5 = 1+(2+2)

6 = 1+1+1+1+1+1
6 = 5+1
6 = 4+(1+1)
6 = 3+(1+1+1)
6 = 2+(1+1+1+1)
6 = 4+2
6 = 2+(2+2)
6 = 3+3
6 = (1+1)+(2+2)

7 = 1+1+1+1+1+1+1
7 = 6+1
7 = 5+(1+1)
7 = 4+(1+1+1)
7 = 3+(1+1+1+1)
7 = 2+(1+1+1+1+1)
7 = 5+2
7 = 3+(2+2)
7 = 1+(2+2+2)
7 = 4+3
7 = 1+(3+3)
7 = (1+1+1)+(2+2)

8 = 1+1+1+1+1+1+1+1
8 = 7+1
8 = 6+(1+1)
8 = 5+(1+1+1)
8 = 4+(1+1+1+1)
8 = 3+(1+1+1+1+1)
8 = 2+(1+1+1+1+1+1)
8 = 6+2
8 = 4+(2+2)
8 = 2+(2+2+2)
8 = 5+3
8 = 2+(3+3)
8 = 4+4
8 = (1+1+1+1)+(2+2)
8 = (1+1)+(2+2+2)
8 = (1+1)+(3+3)

9= 1+1+1+1+1+1+1+1+1
9 = 8+1
9 = 7+(1+1)
9 = 6+(1+1+1)
9 = 5+(1+1+1+1)
9 = 4+(1+1+1+1+1)
9 = 3+(1+1+1+1+1+1)
9 = 2+(1+1+1+1+1+1+1)
9 = 7+2
9 = 5+(2+2)
9 = 3+(2+2+2)
9 = 1+(2+2+2+2)
9 = 6+3
9 = 3+(3+3)
9 = 5+4
9 = 1+(4+4)
9 = (1+1+1+1+1)+(2+2)
9 = (1+1+1)+(2+2+2)
9 = (1+1+1)+(3+3)

10 = 1+1+1+1+1+1+1+1+1+1
10 = 9+1
10 = 8+(1+1)
10 = 7+(1+1+1)
10 = 6+(1+1+1+1)
10 = 5+(1+1+1+1+1)
10 = 4+(1+1+1+1+1+1)
10 = 3+(1+1+1+1+1+1+1)
10 = 2+(1+1+1+1+1+1+1+1)
10 = 8+2
10 = 6+(2+2)
10 = 4+(2+2+2)
10 = 2+(2+2+2+2)
10 = 7+3
10 = 4+(3+3)
10 = 1+(3+3+3)
10 = 6+4
10 = 2+(4+4)
10 = 5+5
10 = (1+1+1+1+1+1)+(2+2)
10 = (1+1+1+1)+(2+2+2)
10 = (1+1)+(2+2+2+2)
10 = (1+1+1+1)+(3+3)
10 = (1+1)+(4+4)

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