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A083318
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a(0) = 1; for n>0, a(n) = 2^n+1.
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10
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1, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Inverse binomial transform of A005056.
Also, A000533 interpreted as binary numbers, written in base 10. Numbers whose representation in base 2 is has n+1 digits and the digit "1" is the initial and final digit and if n>1 then the internal digits are "0" (see example). - Omar E. Pol (info(AT)polprimos.com), Feb 24 2008
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (3,-2).
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FORMULA
| a(n) = 2^n+1^n-0^n.
G.f.: (1-2*x^2)/((1-x)*(1-2x)).
E.g.f.: exp(2*x)+exp(x)-exp(0).
a(n) = sum{k=0..n, 0^(k*(n-k))*2^(n-k)} - Paul Barry, Feb 09 2005
a(n) = Min{m: A008687(m) = n+1}. - Reinhard Zumkeller, Jul 25 2006
Row sums of triangle A132749; = binomial transform of [1, 2, 0, 2, 0, 2, 0, 2,...]. - Gary W. Adamson, Aug 28 2007
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EXAMPLE
| ------------------------------
n .... a(n) .. a(n) in base 2
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0 ..... 1 ..... 1
1 ..... 3 ..... 11
2 ..... 5 ..... 101
3 ..... 9 ..... 1001
4 .... 17 ..... 10001
5 .... 33 ..... 100001
6 .... 65 ..... 1000001
7 ... 129 ..... 10000001
8 ... 257 ..... 100000001
9 ... 513 ..... 1000000001
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PROG
| (MAGMA) [2^n+1-0^n : n in [0..40]]; // Vincenzo Librandi, Sep 01 2011
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CROSSREFS
| Except for the leading term, the same as A000051. Cf. A083319.
Cf. A132749.
Cf. A000533.
Sequence in context: A205537 A135728 * A127904 A048578 A087312 A099170
Adjacent sequences: A083315 A083316 A083317 * A083319 A083320 A083321
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KEYWORD
| nonn,easy
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Apr 25 2003
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 28 2007
Examples provided by Omar E. Pol (info(AT)polprimos.com), Feb 24 2008.
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