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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.

%I A053088 #101 Jul 02 2025 16:01:59
%S A053088 1,0,3,2,9,12,31,54,117,224,459,906,1825,3636,7287,14558,29133,58248,
%T A053088 116515,233010,466041,932060,1864143,3728262,7456549,14913072,
%U A053088 29826171,59652314,119304657,238609284,477218599,954437166,1908874365
%N A053088 a(n) = 3*a(n-2) + 2*a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.
%C A053088 Growth of happy bug population in GCSE math course work assignment.
%C A053088 The generalized (3,2)-Padovan sequence p(3,2;n). See the W. Lang link under A000931. - _Wolfdieter Lang_, Jun 25 2010
%C A053088 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
%C A053088 From _Paul Curtz_, Nov 02 2021 (Start)
%C A053088 a(n-2) difference table (from 0, 0, a(n)):
%C A053088     0    0    1    0    3    2    9    12    31    54  ...
%C A053088     0    1   -1    3   -1    7    3    19    23    63  ...
%C A053088     1   -2    4   -4    8   -4   16     4    40    44  ...
%C A053088    -3    6   -8   12  -12   20  -12    36     4    84  ...
%C A053088     9  -14   20  -24   32  -32   48   -32    80     0  ...
%C A053088   -23   34  -44   56  -64   80  -80   112   -80   176  ...
%C A053088    57  -78  100 -120  144 -160  192  -192   256  -192  ...
%C A053088    ... .
%C A053088 The signature is valid for every row.
%C A053088 a(n-2) + a(n-1) = A001045(n).
%C A053088 a(n-2) + a(n+1) = A062510(n) = 3*A001045(n).
%C A053088 a(n-2) + a(n+3) = see A144472(n+1).
%C A053088 Second subdiagonal: 1, 6, 20, 56, 144, 352, ... = A014480(n).
%C A053088 First subdiagonal: -A036895(n) = -2*A001787(n).
%C A053088 Main diagonal: A001787(n) = -first and -third upper diagonals.
%C A053088 Second, fourth and fifth upper diagonals: A001792(n), A045891(n+2) and A172160(n+1). (End)
%H A053088 Stanislav Sykora, <a href="/A053088/b053088.txt">Table of n, a(n) for n = 0..1000</a>
%H A053088 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H A053088 Iwan Duursma, Xiao Li, and Hsin-Po Wang, <a href="https://arxiv.org/abs/2006.08911">Multilinear Algebra for Distributed Storage</a>, arXiv:2006.08911 [cs.IT], 2020.
%H A053088 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,2).
%F A053088 G.f.: 1 / (1-3*x^2-2*x^3).
%F A053088 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
%F A053088 a(n) = Sum_{k=0..floor(n/2)} A078008(n-2k). - _Paul Barry_, Nov 24 2003
%F A053088 a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)*3^k*(2/3)^(n-2k). - _Paul Barry_, Oct 16 2004
%F A053088 a(n) = Sum_{k=0..n} A078008(k)*(1 - (-1)^(n+k-1))/2. - _Paul Barry_, Apr 16 2005
%F A053088 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
%F A053088 From _Wolfdieter Lang_, Aug 26 2010: (Start)
%F A053088 a(n) = a(n-1) + 2*a(n-2) + (-1)^n for n > 1, a(0)=1, a(1)=0.
%F A053088 Due to the identity for the o.g.f. A(x): A(x) = x*(1+2*x)*A(x) + 1/(1+x).
%F A053088 (This recurrence was observed by _Gary Detlefs_ in a 08/25/10 e-mail to the author.) (End)
%F A053088 G.f.: Sum_{n>=0} binomial(3*n,n)*x^n / (1+x)^(3*n+3). - _Paul D. Hanna_, Mar 03 2012
%F A053088 E.g.f.: 1 + (1/9)*(exp(-x)*(3*x - 2) + 2*exp(2*x)). - _Stefano Spezia_, Sep 27 2019
%t A053088 CoefficientList[Series[1/(1 - 3 x^2 - 2 x^3), {x, 0, 32}], x] (* _Michael De Vlieger_, Sep 30 2019 *)
%o A053088 (PARI) c(n)=(2^(n+1)-(-1)^n*(3*n+2))/9; a(n)=c(n+1); \\ _Stanislav Sykora_, Nov 27 2013
%Y A053088 Cf. A232603, A232604.
%Y A053088 Cf. A001045, A001787, A001792, A014480, A036895, A062510, A045891, A172160.
%K A053088 nonn,easy
%O A053088 0,3
%A A053088 Pauline Gorman (pauline(AT)gorman65.freeserve.co.uk), Feb 26 2000
%E A053088 More terms from _James Sellers_, Feb 28 2000 and _Christian G. Bower_, Feb 29 2000