A274977 - OEIS

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A274977

a(n) = a(n-1) + 3*a(n-2) with n>1, a(0)=1, a(1)=6.

1, 6, 9, 27, 54, 135, 297, 702, 1593, 3699, 8478, 19575, 45009, 103734, 238761, 549963, 1266246, 2916135, 6714873, 15463278, 35607897, 81997731, 188821422, 434814615, 1001278881, 2305722726, 5309559369, 12226727547, 28155405654, 64835588295, 149301805257, 343808570142 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	COMMENTS	`a(n)/a(n+1) converges to 1/A209927 as n approaches infinity.`
	LINKS	`Bruno Berselli, Table of n, a(n) for n = 0..1000` `Index entries for linear recurrences with constant coefficients, signature (1,3).`
	FORMULA	`G.f.: (1 + 5x)/(1 - x - 3x^2).` `a(n) = ((13 + 11sqrt(13))(1 + sqrt(13))^n + (13 - 11sqrt(13))(1 - sqrt(13))^n)/(262^n).` `3a(n) + a(n+1) = 9A105476(n+1).` `3a(n) - a(n+1) = 27A006130(n-3) with n>1, A006130(-1) = 0.` `a(n+1) - a(n) = 27A105476(n-3) with n>2.`
	EXAMPLE	`Table of similar sequences (not extendable on the left side) where this recurrence can be applied to the first two terms:` `----------------------------------------------------------------------` `() - - 1, -1, 2, -1, 5, 2, 17, 23, 74, 143, 365, ...` `A052533: - - 1, 0, 3, 3, 12, 21, 57, 120, 291, 651, 1524, ...` `(°) - 0, 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, ...` `A006138: - - 1, 2, 5, 11, 26, 59, 137, 314, 725, 1667, 3842, ...` `A105476: - - 1, 3, 6, 15, 33, 78, 177, 411, 942, 2175, 5001, ...` `(°) 0, 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, 6160, ...` `A105963: - - 1, 5, 8, 23, 47, 116, 257, 605, 1376, 3191, 7319, ...` `A274977: - - 1, 6, 9, 27, 54, 135, 297, 702, 1593, 3699, 8478, ...` `A075118: - 2, 1, 7, 10, 31, 61, 154, 337, 799, 1810, 4207, 9637, ...` `----------------------------------------------------------------------` `() see version A140165.` `(°) see A006130 and the signed versions A140167, A182228.`
	MATHEMATICA	`RecurrenceTable[{a[n] == a[n - 1] + 3 a[n - 2], a[0] == 1, a[1] == 6}, a, {n, 0, 40}]`
	PROG	`(PARI) v=vector(40); v[1]=1; v[2]=6; for(n=3, #v, v[n]=v[n-1]+3v[n-2]); v` `(Sage)` `from sage.combinat.sloane_functions import recur_gen2` `a=recur_gen2(1, 6, 1, 3)` `[a.next() for n in xrange(40)]` `(MAGMA) [n le 2 select 5n-4 else Self(n-1)+3*Self(n-2): n in [1..40]];`
	CROSSREFS	`Cf. A006130, A006138, A052533, A075118, A105476, A105963.` `Sequence in context: A243708 A024878 A007414 * A025493 A091519 A086491` `Adjacent sequences: A274974 A274975 A274976 * A274978 A274979 A274980`
	KEYWORD	`nonn,easy`
	AUTHOR	`Bruno Berselli, Sep 13 2016`
	STATUS	`approved`

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Last modified August 18 18:04 EDT 2017. Contains 290732 sequences.