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%I M3482 #187 Jul 06 2023 20:53:52
%S 1,4,14,48,164,560,1912,6528,22288,76096,259808,887040,3028544,
%T 10340096,35303296,120532992,411525376,1405035520,4797091328,
%U 16378294272,55918994432,190919389184,651839567872
%N a(n) = 4*a(n-1) - 2*a(n-2) with a(0) = 1, a(1) = 4.
%C Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 4) is "size of raises in pot-limit poker, one blind, maximum raising."
%C It appears that this sequence is the BinomialMean transform of A002315 - see A075271. - _John W. Layman_, Oct 02 2002
%C Number of (s(0), s(1), ..., s(2n+3)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+3, s(0) = 1, s(2n+3) = 4. - _Herbert Kociemba_, Jun 11 2004
%C a(n) = number of distinct matrix products in (A+B+C+D)^n where commutators [A,B]=[C,D]=0 but neither A nor B commutes with C or D. - _Paul D. Hanna_ and _Joshua Zucker_, Feb 01 2006
%C The n-th term of the sequence is the entry (1,2) in the n-th power of the matrix M=[1,-1;-1,3]. - _Simone Severini_, Feb 15 2006
%C Hankel transform of this sequence is [1,-2,0,0,0,0,0,0,0,0,0,...]. - _Philippe Deléham_, Nov 21 2007
%C A204089 convolved with A000225, e.g., a(4) = 164 = (1*31 + 1*15 + 4*7 + 14*3 + 48*1) = (31 + 15 + 28 + 42 + 48). - _Gary W. Adamson_, Dec 23 2008
%C Equals INVERT transform of A000225: (1, 3, 7, 15, 31, ...). - _Gary W. Adamson_, May 03 2009
%C For n>=1, a(n-1) is the number of generalized compositions of n when there are 2^i-1 different types of the part i, (i=1,2,...). - _Milan Janjic_, Sep 24 2010
%C Binomial transform of A078057. - _R. J. Mathar_, Mar 28 2011
%C Pisano period lengths: 1, 1, 8, 1, 24, 8, 6, 1, 24, 24, 120, 8, 168, 6, 24, 1, 8, 24, 360, 24, ... . - _R. J. Mathar_, Aug 10 2012
%C a(n) is the diagonal of array A228405. - _Richard R. Forberg_, Sep 02 2013
%C From _Wolfdieter Lang_, Oct 01 2013: (Start)
%C a(n) appears together with A106731, both interspersed with zeros, in the representation of nonnegative powers of the algebraic number rho(8) = 2*cos(Pi/8) = A179260 of degree 4, which is the length ratio of the smallest diagonal and the side in the regular octagon.
%C The minimal polynomial for rho(8) is C(8,x) = x^4 - 4*x^2 + 2, hence rho(8)^n = A(n+1)*1 + A(n)*rho(8) + B(n+1)*rho(8)^2 + B(n)*rho(8)^3, n >= 0, with A(2*k) = 0, k >= 0, A(1) = 1, A(2*k+1) = A106731(k-1), k >= 1, and B(2*k) = 0, k >= 0, B(1) = 0, B(2*k+1) = a(k-1), k >= 1. See also the P. Steinbach reference given under A049310. (End)
%C The ratio a(n)/A006012(n) converges to 1+sqrt(2). - _Karl V. Keller, Jr._, May 16 2015
%C From _Tom Copeland_, Dec 04 2015: (Start)
%C An aerated version of this sequence is given by the o.g.f. = 1 / (1 - 4 x^2 + 2 x^4) = 1 / [x^4 a_4(1/x)] = 1 / determinant(I - x M) = exp[-log(1 -4 x + 2 x^4)], where M is the adjacency matrix for the simple Lie algebra B_4 given in A265185 with the characteristic polynomial a_4(x) = x^4 - 4 x^2 + 2 = 2 T_4(x/2) = A127672(4,x), where T denotes a Chebyshev polynomial of the first kind.
%C A133314 relates a(n) to the reciprocal of the e.g.f. 1 - 4 x + 4 x^2/2!. (End)
%C a(n) is the number of vertices of the Minkowski sum of n simplices with vertices e_(2*i+1), e_(2*i+2), e_(2*i+3), e_(2*i+4) for i=0,...,n-1, where e_i is a standard basis vector. - _Alejandro H. Morales_, Oct 03 2022
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A007070/b007070.txt">Table of n, a(n) for n = 0..1000</a>
%H C. Bautista-Ramos and C. Guillen-Galvan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Bautista/bautista4.html">Fibonacci numbers of generalized Zykov sums</a>, J. Integer Seq., 15 (2012), Article 12.7.8
%H A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Rinaldi/rinaldi5.html">Permutations defining convex permutominoes</a>, J. Int. Seq. 10 (2007) # 07.9.7
%H A. Burstein, S. Kitaev and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0310379">Independent sets in certain classes of (almost) regular graphs</a>, arXiv:math/0310379 [math.CO], 2003.
%H Tomislav Doslic, <a href="http://dx.doi.org/10.1007/s10910-013-0167-2">Planar polycyclic graphs and their Tutte polynomials</a>, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. (See Cor. 3.7(e).)
%H Tomislav Doslic and I. Zubac, <a href="http://amc-journal.eu/index.php/amc/article/view/851">Counting maximal matchings in linear polymers</a>, Ars Mathematica Contemporanea 11 (2016) 255-276.
%H G. Dresden and Y. Li, <a href="https://arxiv.org/abs/2210.04322">Periodic Weighted Sums of Binomial Coefficients</a>, arXiv:2210.04322 [math.NT], 2022.
%H L. Escobar, P. Gallardo, J. González-Anaya, J. L. González, G. Montúfar, and A. H. Morales, <a href="https://arxiv.org/abs/2209.14978">Enumeration of max-pooling responses with generalized permutohedra</a>, arXiv:2209.14978 [math.CO], 2022. (See Ex. 5.1)
%H S. Falcon, <a href="http://dx.doi.org/10.9734/BJMCS/2014/11783">Iterated Binomial Transforms of the k-Fibonacci Sequence</a>, British Journal of Mathematics & Computer Science, 4 (22): 2014.
%H Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, <a href="https://arxiv.org/abs/2306.14192">alpha-beta-Factorization and the Binary Case of Simon's Congruence</a>, arXiv:2306.14192 [math.CO], 2023.
%H A. S. Fraenkel and C. Kimberling, <a href="http://dx.doi.org/10.1016/0012-365X(94)90259-3">Generalized Wythoff arrays, shuffles and interspersions</a>, Discr. Math. 126 (1-3) (1994) 137-149.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=440">Encyclopedia of Combinatorial Structures 440</a>
%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
%H J. Riordan, <a href="http://www.jstor.org/stable/2005477">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222.
%H J. Riordan, <a href="/A003480/a003480.pdf">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
%H <a href="/index/Poi#poker">Index entries for sequences related to poker</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2).
%F G.f.: 1/(1 - 4x + 2x^2).
%F Preceded by 0, this is the binomial transform of the Pell numbers A000129. Its e.g.f. is then exp(2x)*sinh(sqrt(2)x)/sqrt(2). - _Paul Barry_, May 09 2003
%F a(n) = ((2+sqrt(2))^(n+1) - (2-sqrt(2))^(n+1))/sqrt(8). - Al Hakanson (hawkuu(AT)gmail.com), Dec 27 2008, corrected Mar 28 2011
%F a(n) = (2 - sqrt(2))^n*(1/2 - sqrt(2)/2) + (2 + sqrt(2))^n*(1/2 + sqrt(2)/2). - _Paul Barry_, May 09 2003
%F a(n) = ceiling((2 + sqrt(2))*a(n-1)). - _Benoit Cloitre_, Aug 15 2003
%F a(n) = U(n, sqrt(2))*sqrt(2)^n. - _Paul Barry_, Nov 19 2003
%F a(n) = (1/4)*Sum_{r=1..7} sin(r*Pi/8)*sin(r*Pi/2)*(2*cos(r*Pi/8))^(2n+3)). - _Herbert Kociemba_, Jun 11 2004
%F a(n) = center term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 1 1 / 1 2 1 / 1 1 1]. M^n * [1 1 1] = [A007052(n) a(n) A007052(n)]. E.g., a(3) = 48 since M^3 * [1 1 1] = [34 48 34], where 34 = A007052(3). - _Gary W. Adamson_, Dec 18 2004
%F This is the binomial mean transform of A002307. See Spivey and Steil (2006). - Michael Z. Spivey (mspivey(AT)ups.edu), Feb 26 2006
%F a(2n) = Sum_{r=0..n} 2^(2n-1-r)*(4*binomial(2n-1,2r) + 3*binomial(2n-1,2r+1)) a(2n-1) = Sum_{r=0..n} 2^(2n-2-r)*(4*binomial(2n-2,2r) + 3*binomial(2n-2,2r+1)). - _Jeffrey Liese_, Oct 12 2006
%F a(n) = 3*a(n - 1) + a(n - 2) + a(n - 3) + ... + a(0) + 1. - _Gary W. Adamson_, Feb 18 2011
%F G.f.: 1/(1 - 4*x + 2*x^2) = 1/( x*(1 + U(0)) ) - 1/x where U(k)= 1 - 2^k/(1 - x/(x - 2^k/U(k+1) )); (continued fraction 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Dec 05 2012
%F G.f.: A(x) = G(0)/(1-2*x) where G(k) = 1 + 2*x/(1 - 2*x - x*(1-2*x)/(x + (1-2*x)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 04 2013
%F G.f.: G(0)/(2*x) - 1/x, where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - (1-x)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013
%F a(n-1) = Sum_{k=0..n} binomial(2*n, n+k)*(k|8) where (k|8) is the Kronecker symbol. - _Greg Dresden_, Oct 11 2022
%e a(3) = 48 = 3 * 4 + 4 + 1 + 1 = 3*a(2) + a(1) + a(0) + 1.
%e Example for the octagon rho(8) powers: rho(8)^4 = 2 + sqrt(2) = -2*1 + 4*rho(8)^2 = A(5)*1 + A(4)*rho(8) + B(5)*rho(8)^2 + B(4)*rho(8)^3, with a(5) = A106731(1) = -2, B(5) = a(1) = 4, A(4) = 0, B(4) = 0. - _Wolfdieter Lang_, Oct 01 2013
%p A007070 :=proc(n) option remember; if n=0 then 1 elif n=1 then 4 else 4*procname(n-1)-2*procname(n-2); fi; end:
%p seq(A007070(n), n=0..30); # _Wesley Ivan Hurt_, Dec 06 2015
%t LinearRecurrence[{4,-2}, {1,4}, 30] (* _Harvey P. Dale_, Sep 16 2014 *)
%o (PARI) a(n)=polcoeff(1/(1-4*x+2*x^2)+x*O(x^n),n)
%o (PARI) a(n)=if(n<1,1,ceil((2+sqrt(2))*a(n-1)))
%o (Sage) [lucas_number1(n,4,2) for n in range(1, 24)]# _Zerinvary Lajos_, Apr 22 2009
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-8); S:=[ ((4+r)^(1+n)-(4-r)^(1+n))/((2^(1+n))*r): n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Vincenzo Librandi_, Mar 27 2011
%o (Magma) [n le 2 select 3*n-2 else 4*Self(n-1)-2*Self(n-2): n in [1..23]]; // _Bruno Berselli_, Mar 28 2011
%o (Haskell)
%o a007070 n = a007070_list !! n
%o a007070_list = 1 : 4 : (map (* 2) $ zipWith (-)
%o (tail $ map (* 2) a007070_list) a007070_list)
%o -- _Reinhard Zumkeller_, Jan 16 2012
%Y Row sums of A059474. - _David W. Wilson_, Aug 14 2006
%Y Cf. A007052, A006012 (same recurrence).
%Y Equals 2 * A003480, n>0.
%Y Cf. A007052.
%Y Row sums of A140071.
%Y Cf. A127672, A265185, A133314.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _Mira Bernstein_, _Simon Plouffe_
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