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A053088
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a(n)=2a(n-3)+3a(n-2)
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8
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1, 0, 3, 2, 9, 12, 31, 54, 117, 224, 459, 906, 1825, 3636, 7287, 14558, 29133, 58248, 116515, 233010, 466041, 932060, 1864143, 3728262, 7456549, 14913072, 29826171, 59652314, 119304657, 238609284, 477218599, 954437166, 1908874365
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Growth of happy bug population in GCSE maths course work assignment.
The generalized (3,2)-Padovan sequence p(3,2;n). See the W. Lang link under A000931. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 25 2010]
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FORMULA
| G.f.: 1/(1-3x^2-2x^3).
With offset 1: a(1)=1; a(n)=2*a(n-1)-(-1)^n*n; a(n)=(1/9)*(2^(n+1)-(-1)^n*(3*n+2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 02 2002
a(n)=sum{k=0..floor(n/2), A078008(n-2k)} - Paul Barry (pbarry(AT)wit.ie), Nov 24 2003
a(n)=sum{k=0..floor(n/2), binomial(k, n-2k)3^k*(2/3)^(n-2k)}. - Paul Barry (pbarry(AT)wit.ie), Oct 16 2004
a(n)=sum{k=0..n, A078008(k)(1-(-1)^(n+k-1))/2; - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
a(n) = (2^(n+2) + ((-1)^n)*(3*n+5))/9 (see also the B. Cloitre comment above). From the o.g.f. 1/(1-3x^2-2x^3) = 1/((1-2*x)*(1+x)^2) = (3/(1+x)^2 + 2/(1+x) + 4/(1-2*x))/9. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 25 2010]
Contribution from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 26 2010: (Start)
a(n) = a(n-1) + 2*a(n-2) + (-1)^n, n>=2, a(0)=1, a(1)=0.
Due to the identity for the o.g.f. A(x): A(x) = x*(1+2*x)*A(x) + 1/(1+x).
(This recurrence was observed by Gary Detlefs in a 08/25/10 e-mail to the author.) (End)
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CROSSREFS
| Sequence in context: A099887 A038220 A053151 * A077898 A076584 A154343
Adjacent sequences: A053085 A053086 A053087 * A053089 A053090 A053091
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KEYWORD
| nonn,easy
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AUTHOR
| Pauline Gorman (pauline(AT)gorman65.freeserve.co.uk), Feb 26 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 28 2000 and Christian G. Bower (bowerc(AT)usa.net), Feb 29 2000.
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