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 A053088 a(n) = 3*a(n-2) + 2*a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3. 11
 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; text; internal format)
 OFFSET 0,3 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. - Wolfdieter Lang, Jun 25 2010 With offset 1: a(n) = -2^n*Sum_{k=0..n} k^p*q^k for p=1, q=-1/2. See also A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2). - Stanislav Sykora, Nov 27 2013 LINKS Stanislav Sykora, Table of n, a(n) for n = 0..1000 Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5. Index entries for linear recurrences with constant coefficients, signature (0, 3, 2). FORMULA G.f.: 1 / (1-3*x^2-2*x^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, Nov 02 2002 a(n) = Sum_{k=0..floor(n/2)} A078008(n-2k). - Paul Barry, Nov 24 2003 a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)*3^k*(2/3)^(n-2k). - Paul Barry, Oct 16 2004 a(n) = Sum_{k=0..n} A078008(k)*(1 - (-1)^(n+k-1))/2. - Paul Barry, 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/(1-3*x^2-2*x^3) = 1/((1-2*x)*(1+x)^2) = (3/(1+x)^2 + 2/(1+x) + 4/(1-2*x))/9. - Wolfdieter Lang, Jun 25 2010 From Wolfdieter Lang, Aug 26 2010: (Start) a(n) = a(n-1) + 2*a(n-2) + (-1)^n for n > 1, 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) G.f.: Sum_{n>=0} binomial(3*n,n)*x^n / (1+x)^(3*n+3). - Paul D. Hanna, Mar 03 2012 E.g.f.: 1 + (1/9)*(exp(-x)*(3*x - 2) + 2*exp(2*x)). - Stefano Spezia, Sep 27 2019 MATHEMATICA CoefficientList[Series[1/(1 - 3 x^2 - 2 x^3), {x, 0, 32}], x] (* Michael De Vlieger, Sep 30 2019 *) PROG (PARI) c(n)=(2^(n+1)-(-1)^n*(3*n+2))/9; a(n)=c(n+1); \\ Stanislav Sykora, Nov 27 2013 CROSSREFS Cf. A232603, A232604. Sequence in context: A038220 A303901 A053151 * A077898 A303631 A076584 Adjacent sequences:  A053085 A053086 A053087 * A053089 A053090 A053091 KEYWORD nonn,easy AUTHOR Pauline Gorman (pauline(AT)gorman65.freeserve.co.uk), Feb 26 2000 EXTENSIONS More terms from James A. Sellers, Feb 28 2000 and Christian G. Bower, Feb 29 2000 STATUS approved

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Last modified June 7 05:20 EDT 2020. Contains 334837 sequences. (Running on oeis4.)