imported>Gfis at 18:10, 6 March 2022

2022-03-06T18:10:50Z

imported>Gfis at 13:49, 5 March 2022

2022-03-05T13:49:29Z

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imported>Gfis: squares and cubes (non)dist

2021-08-20T06:29:21Z

squares and cubes (non)dist

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imported>Gfis: Current, without A0033xx

2021-08-19T18:40:12Z

Current, without A0033xx

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imported>Gfis at 20:51, 2 December 2020

2020-12-02T20:51:30Z

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imported>Gfis: Looks good

2020-08-07T15:08:20Z

Looks good

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imported>Gfis: Created page with "==Various Topics== * I recently got an [https://www.rechnen-ohne-strom.de/rechner-galerie/zahlenschieber/addimult/ ADDIMULT]."

2018-04-18T06:48:14Z

Created page with "==Various Topics== * I recently got an [https://www.rechnen-ohne-strom.de/rechner-galerie/zahlenschieber/addimult/ ADDIMULT]."

New page

==Various Topics==
* I recently got an [https://www.rechnen-ohne-strom.de/rechner-galerie/zahlenschieber/addimult/ ADDIMULT].

← Older revision		Revision as of 18:10, 6 March 2022
Line 22:		Line 22:
	A055278 (column 3): x^4(x^3+x^2+1) / ((1-x^2)(1-x^3)*(1-x)^3)		A055278 (column 3): x^4(x^3+x^2+1) / ((1-x^2)(1-x^3)*(1-x)^3)
	A055279 (column 4): x^5(1+x+3x^2+5x^3+7x^4+5x^5+5x^6+2x^7+x^8) / ((1-x)^3(1-x^2)^2(1-x^3)(1-x^4))		A055279 (column 4): x^5(1+x+3x^2+5x^3+7x^4+5x^5+5x^6+2x^7+x^8) / ((1-x)^3(1-x^2)^2(1-x^3)(1-x^4))
	~~When~~ o.g.f.s ~~for higher columns are "guessed"~~ (with Maple's <code>guessgf</code>)~~, it turns out~~ that ~~the denominators~~ are ~~all~~ products of cyclotomic polynomials. The following factorizations of the denominators are obtained:		"Guessed" o.g.f.s (with Maple's <code>guessgf</code>) show denominators that are products of cyclotomic polynomials. The following factorizations of the denominators are obtained:
	deo[2 ] := (x^2-1) *(x-1)^2;		deo[2 ] := (x^2-1) *(x-1)^2;
	deo[3 ] := (x^3-1) (x^2-1) (x-1)^3;		deo[3 ] := (x^3-1) (x^2-1) (x-1)^3;
Line 35:		Line 35:
	deo[12] := (x^12-1)(x^11-1)(x^10-1)(x^9-1)(x^8-1)(x^7-1)(x^6-1)^2(x^5-1)^2(x^4-1)^3(x^3-1)^3(x^2-1)^3*(x-1)^4;		deo[12] := (x^12-1)(x^11-1)(x^10-1)(x^9-1)(x^8-1)(x^7-1)(x^6-1)^2(x^5-1)^2(x^4-1)^3(x^3-1)^3(x^2-1)^3*(x-1)^4;
	Some pattern can be seen, but the higher exponents are not systematic. The problem is that some factors may have been cancelled out in the o.g.f.s polynomial fraction.		Some pattern can be seen, but the higher exponents are not systematic. The problem is that some factors may have been cancelled out in the o.g.f.s polynomial fraction.
	However, by multiplying both the nominator and the denominator polynomials with ~~some factor~~, the following form of the denominators can be obtained:		However, by multiplying both the nominator and the denominator polynomials with such factors, the following form of the denominators can be obtained:
	den[2 ] := (x^2-1) *(x-1)^2 ;		den[2 ] := (x^2-1) *(x-1)^2 ;
	den[3 ] := (x^3-1) (x^2-1) (x-1)^3 ;		den[3 ] := (x^3-1) (x^2-1) (x-1)^3 ;
	den[4 ] := (x^4-1) (x^3-1) (x^2-1)^2*(x-1)^4 ;		den[4 ] := (x^4-1) (x^3-1) (x^2-1)^2*(x-1)^'''<span style="color:red">4</span>''' ;
	den[5 ] := (x^5-1) (x^4-1) (x^3-1) (x^2-1)^2(x-1)^5 ;		den[5 ] := (x^5-1) (x^4-1) (x^3-1) (x^2-1)^2(x-1)^5 ;
	den[6 ] := (x^6-1) (x^5-1) (x^4-1) (x^3-1)^2(x^2-1)^3*(x-1)^6 ;		den[6 ] := (x^6-1) (x^5-1) (x^4-1) (x^3-1)^2(x^2-1)^3*(x-1)^'''<span style="color:red">6</span>''' ;
	den[7 ] := (x^7-1)(x^6-1) (x^5-1) (x^4-1) (x^3-1)^2(x^2-1)^3(x-1)^7 ;		den[7 ] := (x^7-1)(x^6-1) (x^5-1) (x^4-1) (x^3-1)^2(x^2-1)^3(x-1)^'''<span style="color:red">7</span>''' ;
	den[8 ] := (x^8-1)(x^7-1)(x^6-1) (x^5-1) (x^4-1)^2(x^3-1)^2(x^2-1)^4*(x-1)^8 ;		den[8 ] := (x^8-1)(x^7-1)(x^6-1) (x^5-1) (x^4-1)^2(x^3-1)^'''<span style="color:red">2</span>'''(x^2-1)^'''<span style="color:red">4</span>'''*(x-1)^'''<span style="color:red">8</span>''' ;
	den[9 ] := (x^9-1)(x^8-1)(x^7-1)(x^6-1) (x^5-1) (x^4-1)^2(x^3-1)^3(x^2-1)^4(x-1)^9 ;		den[9 ] := (x^9-1)(x^8-1)(x^7-1)(x^6-1) (x^5-1) (x^4-1)^2(x^3-1)^3(x^2-1)^'''<span style="color:red">4</span>'''(x-1)^'''<span style="color:red">9</span>''' ;
	den[10] := (x^10-1)(x^9-1)(x^8-1)(x^7-1)(x^6-1) (x^5-1)^2(x^4-1)^2(x^3-1)^3(x^2-1)^5*(x-1)^10;		den[10] := (x^10-1)(x^9-1)(x^8-1)(x^7-1)(x^6-1) (x^5-1)^2(x^4-1)^2(x^3-1)^3(x^2-1)^'''<span style="color:red">5</span>'''*(x-1)^'''<span style="color:red">10</span>''';
	den[11] := (x^11-1)(x^10-1)(x^9-1)(x^8-1)(x^7-1)(x^6-1) (x^5-1)^2(x^4-1)^2(x^3-1)^3(x^2-1)^5(x-1)^11;		den[11] := (x^11-1)(x^10-1)(x^9-1)(x^8-1)(x^7-1)(x^6-1) (x^5-1)^2(x^4-1)^2(x^3-1)^'''<span style="color:red">3</span>'''(x^2-1)^'''<span style="color:red">5</span>'''(x-1)^'''<span style="color:red">11</span>''';
	den[12] := (x^12-1)(x^11-1)(x^10-1)(x^9-1)(x^8-1)(x^7-1)(x^6-1)^2(x^5-1)^2(x^4-1)^3(x^3-1)^4(x^2-1)^6*(x-1)^12;		den[12] := (x^12-1)(x^11-1)(x^10-1)(x^9-1)(x^8-1)(x^7-1)(x^6-1)^2(x^5-1)^2(x^4-1)^3(x^3-1)^'''<span style="color:red">4</span>'''(x^2-1)^'''<span style="color:red">6</span>'''*(x-1)^'''<span style="color:red">12</span>''';
	Here, the exponents increase in groups of 1 for <code>(x-1)</code>, in groups of 2 for <code>(x^2-1)</code>, in groups of 3 for <code>(x^3-1)</code>, etc.		Here, the exponents increase in groups of 1 for <code>(x-1)</code>, in groups of 2 for <code>(x^2-1)</code>, in groups of 3 for <code>(x^3-1)</code>, etc.
	The ~~triangle~~ of the exponents ~~turns~~ out to be A010766 ~~(with rows reversed)~~. The latter has a very simple formula <code>T(n,k) = floor(n/k)</code> and starts:		Please note that this widening of the g.f.'s denominator polynomials must also be done for the numerator polynomials, but that the initial terms of the sequence remain unaffected.
			The advantage of this deliberate increase of degree in both polynomials is the simple and easy formula for the exponents of <code>(x^m-1)</code> in the denominators.

			These exponents turn out to be the reversed rows of A010766. The latter has a very simple formula <code>T(n,k) = floor(n/k)</code> and starts:
	n/k 1 2 3 4 5 6 7 8 9 10 11 12		n/k 1 2 3 4 5 6 7 8 9 10 11 12
	---+------------------------------------		---+------------------------------------
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	# Determine the denomiator polynomials <code>den</code> above, and take their coefficients as the signature for a linear recurrence with constant coefficients.		# Determine the denomiator polynomials <code>den</code> above, and take their coefficients as the signature for a linear recurrence with constant coefficients.
	# Compute a sufficient number of terms (150 - 270) for the column sequences A055278-A055287 with Michale Somos' PARI program in A055277. This took about 1 day.		# Compute a sufficient number of terms (150 - 270) for the column sequences A055278-A055287 with Michale Somos' PARI program in A055277. This took about 1 day.
	# Take <code>degree(den[n])</code> initial terms from step 2, run the recurrence and verify it for all terms.		# Take <code>degree(den[n])</code> initial terms from step 2, run the linear recurrence and verify it for all terms.

	In fact, the initial terms could be prefixed by <code>n</code> zeros.		In fact, the initial terms could have been been prefixed by <code>n</code> zeros, but in the OEIS sequences the leading zeros are omitted.

Talk:Main Page - Revision history

imported>Gfis at 18:10, 6 March 2022

imported>Gfis at 13:49, 5 March 2022

imported>Gfis: squares and cubes (non)dist

imported>Gfis: Current, without A0033xx

imported>Gfis at 20:51, 2 December 2020

imported>Gfis: Looks good

imported>Gfis: Created page with "==Various Topics== * I recently got an [https://www.rechnen-ohne-strom.de/rechner-galerie/zahlenschieber/addimult/ ADDIMULT]."