Powers accumulate by addition

The way powers accumulate in sieved series is found on p. 158 of Islands of Truth: A Mathematical Mystery Cruise, by Ivars Peterson. It is easy to extend the pattern to more than the powers of 2 and 3 depicted there. Isn't it surprising to not be able to find this rather straight-forward approach in another place?

(Click here for explanation of formulas of the sieved addition to powers [and, incidentally, binomial theorem]. Click here for a commentary on the pattern.)

This is a pattern that works for all powers. Here x^11 is demonstrated.

	Start with "1" and continue to add "1" to each previous number across the top line.
	Make every nth number nothing for x^n (in this case every 11th number).
	For all subsequent lines: add 0 to the first number of the previous line
	then to that, add the second number from the line above; after that, add the third, etc.
	Do Not Add Grey Numbers because their value is declared nothing.
	On subsequent lines, grey numbers occur at a more frequent rate than previous lines by one ―
	for example, the line with every 5th number deleted is followed by the line with every 4th-number deleted.
	Repeat the procedure until the rate is every other value made nothing.
	The last line is x^11 values.

1	1	1	1	1	1	1	1	1	1	1	¹	1	1	1	1	1	1	1	1	1	1	1	¹	1
1	2	3	4	5	6	7	8	9	10	¹¹		12	13	14	15	16	17	18	19	20	21	²²		23
1	3	6	10	15	21	28	36	45	⁵⁵			67	80	94	109	125	142	160	179	199	²²⁰			243
1	4	10	20	35	56	84	120	¹⁶⁵				232	312	406	515	640	782	942	1121	¹³²⁰				1563
1	5	15	35	70	126	210	³³⁰					562	874	1280	1795	2435	3217	4159	⁵²⁸⁰					6843
1	6	21	56	126	252	⁴⁶²						1024	1898	3178	4973	7408	10625	¹⁴⁷⁸⁴						21627
1	7	28	84	210	⁴⁶²							1486	3384	6562	11535	18943	²⁹⁵⁶⁸							51195
1	8	36	120	³³⁰								1816	5200	11762	23297	⁴²²⁴⁰								93435
1	9	45	¹⁶⁵									1981	7181	18943	⁴²²⁴⁰									135675
1	10	⁵⁵										2036	9217	²⁸¹⁶⁰										163835
1	¹¹											2047	¹¹²⁶⁴											¹⁷⁵⁰⁹⁹
1												2048												177147

The x^11 integers result in the bottom row.

The orange numbers have a special characteristic: they accumulate to x^n values in a shell-like manner. Math World identifies them as nexus numbers.

The gray numbers plus 1, when added to the left-ward blue x^11, equal the value of the power of 11. This diagrams the binomial definition that is (x+1)^n.

Very interestingly: gray numbers plus 1 add to the shell--the amount to add to previous x^11. However, a first gray value, moving diagonally up from a blue value, less the next, plus the third, less the next, etc. until a the last (here, invisible) value of "1" sums up to the previous shell value! In other words, gray numbers swing both ways defining both a shell that would form (x+1)^n when added to x^n, as well as the shell that adds to (x-1)^n to form x^n. One gray, diagonally upward sequence above defines both the (x-1)th shell value AND the xth shell value!

11 + 55 + 165 + 330 + 462 + 462 + 330 + 165 + 55 + 11 + 1 = 2047 --next shell value
11 - 55 + 165 - 330 + 462 - 462 + 330 - 165 + 55 - 11 + 1 = 1 --previous shell value
11264 + 28160 + 42240 + 42240 + 29568 + 14784 + 5280 + 1320 + 220 + 22 + 1 = 175099
11264 - 28160 + 42240 - 42240 + 29568 - 14784 + 5280 - 1320 + 220 - 22 + 1 = 2047


2048 + 11264 + 28160 + 42240 + 42240 + 29568 + 14784 + 5280 + 1320 + 220 + 22 + 1 = 177147
- 2048 + 11264 - 28160 + 42240 - 42240 + 29568 - 14784 + 5280 - 1320 + 220 - 22 + 1 = - 1

Click for explanations of formulas of the sieved addition to powers.

Start with "1" and continue to add "1" to each previous number across the top line.

Make every nth number nothing for x^n (in this case every 11th number).

For all subsequent lines: add 0 to the first number of the previous line

then to that, add the second number from the line above; after that, add the third, etc.

Do Not Add Grey Numbers because their value is declared nothing.

On subsequent lines, grey numbers occur at a more frequent rate than previous lines by one ―

for example, the line with every 5th number deleted is followed by the line with every 4th-number deleted.

Repeat the procedure until the rate is every other value made nothing.

The last line is x^11 values.

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