New article: http://www.scipress.org/ /Then click on to No.2 Date: Thu, 7 Jan 2010 20:23:26 -0800 From: qntmpkt@yahoo.com Subject: Toothpick / Tetrahedral "closed walks" - proof - Gary proof of the generalization should be within each reach, where we state that: ... Theorem: (toothpick sequence )/ ("tends to" sequence of triangle rows) = corresponding "closed walks of length n along edges of (tetrahedron, etc) based at vertex. First, by way of example, let's cover the left side of the equation. Given our triangle A160552: 1 1, 3 1, 3, 5, 7 1, 3, 5, 7, 5, 11 17, 15; ... where rows tend to A151548: (1, 3, 5, 7, 5, 11 17, 15, 5, 11, 17, 15, 5, 11, 17, 19,...). Already in your comments section, we have triangle (A160552) as a string * (1, 2, 2, 2, ....) = toothpick, sequence A139250. But (A160552) * (1, 2, 0, 0, 0, ...) = A151548., the "tends to" sequence. So we put A160552 * (1, 2, 2, 2,...) / A160552*(1, 2, 0, 0, 0,...); = (toothpick / "tends to"); then the A160552 cancels., leaving (1, 2, 2, 2,...) / (1, 2, 0, 0, 0,...). ... Now we cover the right side of the equation by listing some examples. Case N=1 turns out to be (1, 0, 1, 0, 1, 0, 1,...) trivial. So next we address A151575: (1, 0, 2, -2, 6, -10, 22, -42, 86,...). But first we access the unsigned version: (1, 0, 2, 2, 6, 10, 22, 42, 86,....); noting that the g.f. is (1-x) / (1 - x - 2x^2). It can be shown that the signed version = (1, 2, 2, 2,....) / (1, 2, 0, 0, 0,...).; so the previous formula should be tweaked to reflect the alternating signs. But (1, 2, 2, 2,...) / (1, 2, 0, 0,....) is what we had before on the left side of the formula. Proof is complete for this case N=2. ... Next, case N=3: Right side of the formula is A054878: (1, 0, 3, 6, 21, 60 183, 546, 1641,...) but with alternating signs. A054878 g.f. is (1 - 2x)/ (1-2x-3x^2) where from the previous g.f. we just add "1" to coeff of x's. it can be shown that the alternating sign version of A054878 = (1, 3, 3, 3,....) / (1, 3, 0, 0, 0); where again this = (toothpick seq N=3) / ("tends to" sequence); where the generating triangle for N=3 = 1 1, 4 1, 4, 7, 13; 1, 4, 7, 13, 7, 19, 34, 40; ... again left side of equation = right side. .... Next, case N=4, right side of equation = A109499: (1, 0, 4, 12, 52, 204, 820, 3276,...) where we just add "1" to coeff of x (but the entry has a slightly different formula. Anyway, the same formatted formula would be (1 - 3x) / (1 - 3x - 4x^2). But we want the version with alternate signs, so that formula can be tweaked to equate to: (1, 4, 4, 4,....) / (1, 4, 0, 0, 0).; proof should follow. Next, case N=5 = A109500: (1, 0, 5, 20, 105, 520, 2605, 13020,...); and the analogous formula should be: (1 - 4x) / (1 - 4x - 5x^2); with the alternating sign version = (1, 5, 5, 5,...) / (1, 5, 0, 0, 0,...). proof should follow. ... ...but I'm not into proofs so you can have this project, if desired. ...Gary Hotmail: Trusted email with powerful SPAM protection. Sign up now.
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New article: http://www.scipress.org/ /Then click on to No.2 Date: Thu, 7 Jan 2010 20:23:26 -0800 From: qntmpkt@yahoo.com Subject: Toothpick / Tetrahedral "closed walks" - proof - Gary proof of the generalization should be within each reach, where we state that: ... Theorem: (toothpick sequence )/ ("tends to" sequence of triangle rows) = corresponding "closed walks of length n along edges of (tetrahedron, etc) based at vertex. First, by way of example, let's cover the left side of the equation. Given our triangle A160552: 1 1, 3 1, 3, 5, 7 1, 3, 5, 7, 5, 11 17, 15; ... where rows tend to A151548: (1, 3, 5, 7, 5, 11 17, 15, 5, 11, 17, 15, 5, 11, 17, 19,...). Already in your comments section, we have triangle (A160552) as a string * (1, 2, 2, 2, ....) = toothpick, sequence A139250. But (A160552) * (1, 2, 0, 0, 0, ...) = A151548., the "tends to" sequence. So we put A160552 * (1, 2, 2, 2,...) / A160552*(1, 2, 0, 0, 0,...); = (toothpick / "tends to"); then the A160552 cancels., leaving (1, 2, 2, 2,...) / (1, 2, 0, 0, 0,...). ... Now we cover the right side of the equation by listing some examples. Case N=1 turns out to be (1, 0, 1, 0, 1, 0, 1,...) trivial. So next we address A151575: (1, 0, 2, -2, 6, -10, 22, -42, 86,...). But first we access the unsigned version: (1, 0, 2, 2, 6, 10, 22, 42, 86,....); noting that the g.f. is (1-x) / (1 - x - 2x^2). It can be shown that the signed version = (1, 2, 2, 2,....) / (1, 2, 0, 0, 0,...).; so the previous formula should be tweaked to reflect the alternating signs. But (1, 2, 2, 2,...) / (1, 2, 0, 0,....) is what we had before on the left side of the formula. Proof is complete for this case N=2. ... Next, case N=3: Right side of the formula is A054878: (1, 0, 3, 6, 21, 60 183, 546, 1641,...) but with alternating signs. A054878 g.f. is (1 - 2x)/ (1-2x-3x^2) where from the previous g.f. we just add "1" to coeff of x's. it can be shown that the alternating sign version of A054878 = (1, 3, 3, 3,....) / (1, 3, 0, 0, 0); where again this = (toothpick seq N=3) / ("tends to" sequence); where the generating triangle for N=3 = 1 1, 4 1, 4, 7, 13; 1, 4, 7, 13, 7, 19, 34, 40; ... again left side of equation = right side. .... Next, case N=4, right side of equation = A109499: (1, 0, 4, 12, 52, 204, 820, 3276,...) where we just add "1" to coeff of x (but the entry has a slightly different formula. Anyway, the same formatted formula would be (1 - 3x) / (1 - 3x - 4x^2). But we want the version with alternate signs, so that formula can be tweaked to equate to: (1, 4, 4, 4,....) / (1, 4, 0, 0, 0).; proof should follow. Next, case N=5 = A109500: (1, 0, 5, 20, 105, 520, 2605, 13020,...); and the analogous formula should be: (1 - 4x) / (1 - 4x - 5x^2); with the alternating sign version = (1, 5, 5, 5,...) / (1, 5, 0, 0, 0,...). proof should follow. ... ...but I'm not into proofs so you can have this project, if desired. ...Gary Hotmail: Trusted email with powerful SPAM protection. Sign up now.
| New window Print all Sponsored Links Trading with Triangles Free issue! Stocks & Commodities magazine. How-to guide for traders. Geometry for grades K-6 Practice area, volume, perimeter, shape names, terms, so much more! Math Manipulatives The multi-sensory math programme that is transforming math teaching "Raise Your Credit Score" "I Raised My Bad Credit Score To Over 725 Using These Free Tips!" About these links |
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