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A004119
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3*2^n + 1. Alternatively, define the sequence S(a_0,a_1) by a_{n+2} is the least integer such that a_{n+2}/a_{n+1}>a_{n+1}/a_n for n >= 0. This is S(4,7).
(Formerly M3308)
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8
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1, 4, 7, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196609, 393217, 786433, 1572865, 3145729, 6291457, 12582913, 25165825, 50331649, 100663297, 201326593, 402653185, 805306369, 1610612737, 3221225473, 6442450945, 12884901889
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Also Pisot sequence L(4,7) (cf. A008776).
a(n) = number of terms of the arithmetic progression with first term 2^(2n-1) and last term 2^(2n+1). So common difference is 2^n. e.g. a(2)=7 corresponds to (8,12,16,20,24,28,32). - Augustine O. Munagi (amunagi(AT)yahoo.com), Feb 21 2007
Equals binomial transform of [1, 3, 0, 3, 0, 3, 0, 3,...] [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 27 2010]
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REFERENCES
| D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993;.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
A. O. Munagi and T. Shonhiwa, On the partitions of a number into arithmetic progressions, J. Integer Seq. 11 (2008), Article 08.5.4. [From Augustine O. Munagi (amunagi(AT)yahoo.com), Jan 08 2009]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| A. O. Munagi and T. Shonhiwa, On the partitions of a number into arithmetic progressions, J. Integer Sequences, 11 (2008), #08.5.4. [From Augustine O. Munagi (amunagi(AT)yahoo.com), Jan 08 2009]
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| a(n) = 3a(n-1) - 2a(n-2).
For n>3, a(3)=13, a(n)= a(n-1)+2*floor(a(n-1)/2). - Benoit Cloitre, Aug 14, 2002.
For n>=1, a(n) = A049775(n+1)/2^(n-2). For n>=2, a(n) = 2a(n-1)-1; see also A000051. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 20 2004
O.g.f.: -(-1-x+3*x^2)/((2*x-1)*(x-1)) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 23 2007
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MAPLE
| A004119:=-(-1-z+3*z**2)/(2*z-1)/(z-1); [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| s=4; lst={1, s}; Do[s=s+(s-1); AppendTo[lst, s], {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 30 2009]
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CROSSREFS
| A049988 [From Augustine O. Munagi (amunagi(AT)yahoo.com), Jan 08 2009]
Sequence in context: A118334 A205538 A181565 * A074864 A074865 A072683
Adjacent sequences: A004116 A004117 A004118 * A004120 A004121 A004122
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
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