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A143625
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Decimal expansion of the constant E_3(0) := sum {n = 0.. inf} (-1)^floor(n/3)/n! = 1 + 1/1! + 1/2! - 1/3! - 1/4! - 1/5! + + + - - - ... = 2.28494 23824 ... .
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4
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2, 2, 8, 4, 9, 4, 2, 3, 8, 2, 4, 0, 9, 6, 3, 5, 2, 0, 8, 9, 9, 9, 0, 5, 0, 0, 1, 9, 2, 6, 3, 0, 8, 2, 7, 0, 2, 1, 6, 1, 5, 1, 3, 2, 6, 2, 9, 9, 4, 9, 5, 8, 9, 7, 8, 5, 9, 8, 2, 8, 8, 9, 8, 0, 0, 3, 7, 3, 7, 1, 0, 1, 5, 7, 5, 1, 9, 7, 3, 4, 5, 9, 4, 0, 3, 7, 4, 4, 9, 5, 1, 2, 5, 2, 4, 6, 3, 4, 4, 8, 8
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Define E_3(n) = sum {k = 0..inf} (-1)^floor(k/3)*k^n/k! = 0^n/0! + 1^n/1! + 2^n/2! - 3^n/3! - 4^n/4! - 5^n/5! + + + - - - ... for n = 0,1,2,... . It is easy to see that E_3(n+3) = 3*E_3(n+2) - 2*E_3(n+1) - sum {i = 0..n} 3^i*binomial(n,i) * E_3(n-i) for n >= 0. Thus E_3(n) is an integral linear combination of E_3(0), E_3(1) and E_3(2). See the examples below. The decimal expansions of E_3(1) and E_3(2) are given in A143626 and A143627. Compare with A143623 and A143624.
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REFERENCES
| Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.
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EXAMPLE
| E_3(n) as linear combination of E_3(i),
i = 0..2.
=======================================
..E_3(n)..|....E_3(0)...E_3(1)...E_3(2)
=======================================
..E_3(3)..|.....-1.......-2........3...
..E_3(4)..|.....-6.......-7........7...
..E_3(5)..|....-25......-23.......14...
..E_3(6)..|....-89......-80.......16...
..E_3(7)..|...-280.....-271......-77...
..E_3(8)..|...-700.....-750.....-922...
..E_3(9)..|...-380.....-647....-6660...
..E_3(10).|..13452....13039...-41264...
...
The columns are A143628, A143629 and A143630.
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CROSSREFS
| A143623, A143624, A143626, A143627, A143628, A143629 and A143630.
Sequence in context: A144816 A134812 A144847 * A003612 A103839 A135727
Adjacent sequences: A143622 A143623 A143624 * A143626 A143627 A143628
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KEYWORD
| cons,easy,nonn
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AUTHOR
| Peter Bala (pbala(AT)toucansurf.com), Aug 30 2008
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EXTENSIONS
| Offset corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2009
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