MagicNKZ Formulas Depicted

More "Magic NKZ" FORMULAS

WITHIN THE EULER/PASCAL CUBE

The MagicNKZ definition of n

The MagicNKZ definition of k

The MagicNKZ definition of z depicted The MagicNKZ definition of z by tables of equations

Yellow sequences are power values. Blue sequences are n! Green values are Euler's Triangle.

(in formats from the Mathematica program, from Wolfram Research, Inc.)

<<DiscreteMath`Combinatorica`

MagicNKZ =

Table[ MagicNKZ, {n,1,7},{k,0,11},{z,0,5}]//MatrixForm

The MagicNKZ definition of n

Table[ MagicNKZ, {z,0,5},{k,0,4},{n,n,n}]//TableForm

Click for fully simplified versions.

Connections of the sequences generated by above equations to Sloane integer series numbers:

solving for n	z=0	z=1	z=2	z=3	z=4	z=5
k=0	ones	ones	ones	ones	ones	ones
k=1	A000295 Column 2 of Euler's triangle	A000325 2^n - n.
k=2	A000460 Column 3 of Euler's triangle
k=3	A000498 Column 4 of Euler's triangle
k=4	A000505 Column 5 of Euler's triangle
k=5	A000514 Column 6 of Euler's triangle
k=6	A001243 Column 7 of Euler's triangle
k=7	A001244 Column 8 of Euler's triangle

Read equations across above rows per sequence: Read equations down above columns per sequence:

z=0; k = 0 to 4 k=0; z = 0 to 4

k=0 1		z=0 1
k=1 A000295 Column 2 of Euler's triangle		z=1 1
k=2 A000460 Column 3 of Euler's triangle		z=2 1
k=3 A000460 Column 4 of Euler's triangle		z=3 1
k=4 A000505 Column 5 of Euler's triangle		z=4 1

z=1; k = 0 to 6 k=1; z = 0 to 5

k=0 n=1		z=0 A000295 Column 2 of Euler's triangle
k=1 A000325 2^n - n.		z=1 A000325 2^n - n.
k=2		z=2
k=3		z=3
k=4		z=4
k=5		z=5
k=6

z=2; k = 0 to 5 k=2; z = 0 to 5

k=0 1		z=0 A000460 Column 3 of Euler's triangle
k=1		z=1
k=2		z=2
k=3		z=3
k=4		z=4
		z=5

z=3; k = 0 to 5 k=3; z = 0 to 5

k=0 1		z=0 A000460 Column 4 of Euler's triangle
k=2		z=1
k=2		z=2
k=3		z=3
k=4		z=4
k=5		z=5

z=4; k=4;

k=0 1		z=0 A000505 Column 5 of Euler's triangle
k=1		z=1
k=2		z=2
k=3		z=3
k=4		z=4
k=5		z=5

z=5;

k=0 1
k=1
k=2
k=3
k=4
k=5

The MagicNKZ definition of z depicted

n=1; k=0;

k=0 1		n=1 1
k=1 z integers or the power of 1, from 0		n=2 1
k=2 A000217 3rd Pascal Triangle Figurate numbers or binomial coefficients C(n,2).		n=3 1
k=3 A000292 4th Pascal Triangle Figurate numbers or binomial coefficients C(n,3).		n=4 1
k=4 A000332 5th Pascal Triangle Figurate numbers or binomial coefficients C(n,4).		n=5 1
k=5 A000389 6th Pascal Triangle Figurate numbers or binomial coefficients C(n,5).		n=6 1

n=2; k=1

k=0 1		n=1 z integers or the power of 1, from 0
k=1 z + 1 integers or the power of 1, from 1		n=2 z + 1 integers or the power of 1, from 1
k=2 A000096		n=3 z + 4 integers or the power of 1, from 4
k=3 A005581		n=4 z + 11 integers or the power of 1, from 11
k=4 A005582		n=5 z + 26 integers or the power of 1, from 26
k=5 A005583 Coefficients of Chebyshev polynomials.		n=6 z + 57 integers or the power of 1, from 57
k=6 A005584 Coefficients of Chebyshev polynomials.		n=7 z + 120 integers or the power of 1, from 120
k=7		n=8 z + 247 integers or the power of 1, from 247

n=3; k=2;

k=0 1		n=1 A000217 3rd Pascal Triangle Figurate numbers or binomial coefficients C(n,2).
k=1 z + 4 integers or the power of 1, from 4		n=2 A000096
k=2 A051936 Truncated triangular numbers: a(n)=n*(n+1)/2-9.		n=3 A051936 Truncated triangular numbers: a(n)=n*(n+1)/2-9.
k=3		n=4 [Similar to A079664 a(2) = LookAndSay(1) + LookAndSay(2) = 11 (one "1") + 12 (one "2") = 23.]
k=4		n=5
k=5		n=6
k=6		n=7
k=7		n=8

n=4; k=3;

k=0 1		n=1 A000292 4th Pascal Triangle Figurate numbers or binomial coefficients C(n,3).
k=1 z + 11 integers or the power of 1, from 11		n=2 A005581
k=2 [Similar to A079664 a(2) = LookAndSay(1) + LookAndSay(2) = 11 (one "1") + 12 (one "2") = 23.]		n=3
k=3		n=4
k=4		n=5
k=5		n=6
k=6		n=7
k=7		n=8

n=5; k=5;

k=0 1		n=1 A000332 5th Pascal Triangle Figurate numbers or binomial coefficients C(n,4).
k=1 z + 26 integers or the power of 1, from 26		n=2 A005581
k=2		n=3
k=3		n=4
k=4		n=5
k=5		n=6
k=6		n=7
k=7		n=8

n=6;

k=0 1

k=1 z + 57 integers or the power of 1, from 57

k=2

k=3

k=4

k=5

k=6

k=7

The MagicNKZ definition of k

Table[MagicNKZ,{n,1,5},{z,0,5},{k,k,k}]//TableForm

Click for more and expanded equations.

Chart of MagicNKZ definitions of k

Connections of the sequences generated by above equations to Sloane integer series numbers:

(Click to see the logically reciprocal "SeriesAtLevelR" formulas for the same series.)

solving for k	z=0	z=1	z=2	z=3	z=4	z=5	z=6	z=7	z=8	z=9
n=1	zeros for k>0	1	integers	A000217 triangular numbers	A000292 tetrahedral numbers	A000332 5th figurate series	A000389 6th figurate series	A000579 7th figurate series	A000580 8th figurate series	A000581 9th figurate series
n=2	zeros from k>1	2 for k>0	A005408 The odds/ nexus to power of 2. 2k + 1.	A000290 The squares.	A000330 The sums of squares.	A002415 The sums of sums of squares--4D pyramidal numbers	A005585 Sums of Sums of Sums of squares--5D pyramidal numbers	A050486 C(n+6,6)* (2n+7)/7.	A053347 C(n+7,7)* (n+4)/4.	A054333 1/256 of tenth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).
n=3	zeros for k>2	A101101 6 for k>1	A008458 for k>0, 6 k	A003215 Hex numbers/ nexus to power of 3.	A000578 The cubes.	A000537 Sum of first n cubes.	A085437 or A024166 Sums of Sums of cubes	A101094 Sums of Sums of Sums of the 3rd power	A101097 Sums of Sums of Sums of Sums of the 3rd power	A101102 Sums of Sums of Sums of Sums of Sums of the 3rd power
n=4	zeros for k>3	A101104 24 for k>2	A101103 for k>1, 24 k - 12	A005914 Points on surface of hexagonal prism. For k>0,	A005917 Nexus to power of 4.	A000583 The fourth power.	A000538 Sums of the 4th power.	A101089 Sums of Sums of the 4th power	A101090 Sums of Sums of Sums of the 4th power	A101091 Sums of Sums of Sums of Sums of the 4th power
n=5	zeros for k>4	A101100 120 for k>3	A101095 for k>2, 120 k - 120	A101096 for k>1,	A101098 for k>0,	A022521 Nexus to power of 5.	A000584 The 5th power.	A000539 Sums of the 5th power.	A101092 Sums of Sums of the 5th power	A101099 Sums of Sums of Sums of the 5th power
n=6	zeros for k>5	720 for k>4	for k>3, 720 k - 1080	for k>2,	for k>1,	for k>0,	A022522 Nexus to power of 6.	A001014 The 6th power.	A000540 Sums of the 6th power.	A101093 Sums of Sums of the 6th power
n=7	zeros for k>6							A022523 Nexus numbers to the 7th power.	A001015 The 7th power.	A000541 Sums of the 7th power.
n=8	zeros for k>7								Nexus numbers to the 8th power.	A001016 The 8th power.
n=9	zeros for k>8									A022525 Nexus numbers to the 9th power.

Read equations down above columns per sequence: Read equations across above rows per sequence:

z=0; n=1;

n=1 0 for k>0		z=0 0 for k>0
n=2 0 for k>1		z=1 ones
n=3 0 for k>2		z=2 k or integers
n=4 0 for k>3		z=3 A000217 3rd Pascal Triangle Figurate numbers or binomial coefficients C(n,2).
n=5 0 for k>4		z=4 A000292 4th Pascal Triangle Figurate numbers or binomial coefficients C(n,3).
n=6 0 for k>5		z=5 A000332 5th Pascal Triangle Figurate numbers or binomial coefficients C(n,4).
n=7 0 for k>6		z=6 A000389 6th Pascal Triangle Figurate numbers or binomial coefficients C(n,5).
n=8 0 for k>7		z=7 A000579 7th Pascal Triangle Figurate numbers or binomial coefficients C(n,6).

z=1; n=2;

n=1 1		z=0 0 for k>1
n=2 2 for k>0		z=1 2 for k>0
n=3 6 for k>1		z=2 2k + 1 A005408 or odds
n=4 24 for k>2		z=3 k^2 A000290 The squares.
n=5 120 for k>3		z=4 A000330 or summations of squares; Square pyramidal numbers.
n=6 720 for k>4		z=5 A002415 or summation of summations of squares; 4-dimensional pyramidal numbers: n^2*(n^2-1)/12.
n=7 5040 for k>5		z=6 A005585 or summation of summations of summations of x^2; 5-dimensional pyramidal numbers: n(n+1) ... (n+3)(2n+3)/5!.
n=8 40320 for k>6		z=7 A040977 or summation of summations of summations of summations of x^2; C(n+5,5)*(n+3)/3

z=2; n=3;

n=1 integers		z=0 0 for k>2
n=2 2 k + 1 A005408		z=1 6 for k>1
n=3 6 k for k>0 A008458 Coordination sequence for hexagonal lattice.		z=2 6 k A008458 Coordination sequence for hexagonal lattice.
n=4 24 k - 12 for k>1		z=3 A003215 Hex (or centered hexagonal) numbers/nexus numbers for the 3rd power.
n=5 120 k - 120 for k>2		z=4 k^3 A000578 The cubes.
n=6 720 k - 1080 for k>3		z=5 A000537 Sum of first n cubes; or n-th triangular number squared.
n=7 5040 k - 10080 for k>4		z=6 A085437 or A024166 Sum of first n sums of cubes; Sum of (j-i)^3 for 1 <= i < j
n=8 40320 k - 100800 for k>5		z=7 A101094 Sum of first n sums of sums of cubes.
n=9 362880 k - 1088640 for k>6		z=8 A101097 Sum of first n sums of sums of sums of cubes.

z=3; n=4;

n=1 A000217 3rd Pascal Triangle Figurate numbers or binomial coefficients C(n,2).		z=0 0 for k>3
n=2 k^2 [or 0 for k=not] A000290 The squares.		z=1 24 for k>2
n=3 A003215 Hex (or centered hexagonal) numbers or nexus numbers for the power of 3.		z=2 24 k - 12 for k>1
n=4 for k>0 A005914 Points on surface of hexagonal prism.		z=3 for k>0 A005914 Points on surface of hexagonal prism.
n=5 for k>1		z=4 A005917 Rhombic dodecahedral numbers/nexus numbers for the fourth power.
n=6 for k>2		z=5 k^4 A000583 The fourth power.
n=7 for k>3		z=6 A000538 Sum of fourth powers: 0^4+1^4+...+n^4.
n=8 for k>4		z=7 A101089 Sum of first n of sums of fourth powers.
n=9 for k>5		z=8 A101090 Sum of first n of sums of sums of fourth powers.
n=10 for k>6		z=9 A101091 Sum of first n of sums of sums of sums of fourth powers.

z=4; n=5;

n=1 A000292 4th Pascal Triangle Figurate numbers or binomial coefficients C(n,3).		z=0 0 for k>4
n=2 A000330 or summations of squares; Square pyramidal numbers.		z=1 120 for k>3
n=3 k^3 [0 for "k is not"] A000578 The cubes.		z=2 120 k - 120 for k>2
n=4 A005917 Rhombic dodecahedral numbers/ nexus numbers for the fourth power.		z=3 for k>1
n=5 for k>0		z=4 for k>0
n=6 for k>1		z=5 A022521 Nexus numbers for the power of 5.
n=7 for k>2		z=6 k^5 A000584 The fifth power.
n=8 for k>3		z=7 A000539 Sum of the first n of the fifth power.
n=9 for k>4		z=8 A101092 Sum of the first n of the sums of the fifth power.
n=10 for k>5		z=9 A101099 Sum of the first n of the sums of sums of the fifth power.

z=5; n=6;

n=1 A000332 5th Pascal Triangle Figurate numbers or binomial coefficients C(n,4).	z=0 0 for k>5
n=2 A002415 or summation of summations of squares; 4-dimensional pyramidal numbers: n^2*(n^2-1)/12.	z=1 720 for k>4
n=3 A000537 Sum of first n cubes; or n-th triangular number squared.	z=2 720 k - 1080 for k>3
n=4 k^4 [0 for "k is not"] A000583 The fourth power	z=3 for k>2
n=5 A022521 Nexus numbers for the power of 5.	z=4 for k>1
n=6 for k>0	z=5 for k>0
n=7 for k>1	z=6 A022522 Nexus numbers for the power of 6.
n=8 for k>2	z=7 k^6 A001014 The sixth power. Numbers both square and cubic
n=9 for k>3	z=8 A000540 Sum of 6th powers: 1^6 + 2^6 + ... + n^6.
n=10 for k>4	z=9 A101093 Sum of first n of sums of 6th powers.

More "Magic NKZ" FORMULAS

WITHIN THE EULER/PASCAL CUBE

MagicNKZ =

The MagicNKZ definition of n

z=0; k = 0 to 4 k=0; z = 0 to 4

z=1; k = 0 to 6 k=1; z = 0 to 5

z=2; k = 0 to 5 k=2; z = 0 to 5

z=3; k = 0 to 5 k=3; z = 0 to 5

z=4; k=4;

z=5;

The MagicNKZ definition of z depicted

n=1; k=0;

n=2; k=1

n=3; k=2;

n=4; k=3;

n=5; k=5;

n=6;

The MagicNKZ definition of k

Chart of MagicNKZ definitions of k

z=0; n=1;

z=1; n=2;

z=2; n=3;

z=3; n=4;

z=4; n=5;

z=5; n=6;

Send mail to Cecilia@noticingnumbers.net with questions or comments about this web site. Copyright © 2004 noticingnumbers.net Last modified: 12/16/05

Send mail to Cecilia@noticingnumbers.net with questions or comments about this web site.
Copyright © 2004 noticingnumbers.net
Last modified: 12/16/05