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First differences of A109975.
12

%I #57 Sep 27 2024 13:44:23

%S 1,2,5,11,24,52,112,240,512,1088,2304,4864,10240,21504,45056,94208,

%T 196608,409600,851968,1769472,3670016,7602176,15728640,32505856,

%U 67108864,138412032,285212672,587202560,1207959552,2483027968,5100273664

%N First differences of A109975.

%H Vincenzo Librandi, <a href="/A111297/b111297.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013.

%H Milan Janjic and Boris Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).

%F Equals binomial transform of [1, 1, 2, 1, 3, 1, 4, 1, 5, ...] - _Gary W. Adamson_, Apr 25 2008

%F From _Paul Barry_, Mar 18 2009: (Start)

%F G.f.: (1-2*x+x^2-x^3)/(1-2*x)^2.

%F a(n) = Sum_{k=0..n} C(n,k)*Sum_{j=0..floor(k/2)} C(j+1,k-j).

%F a(n) = Sum_{k=0..n} C(n,k)*A158416(k). (End)

%F a(n) = Sum_{k=0..n-2} (k+5)*binomial(n-2,k) for n >= 2. - _Philippe Deléham_, Apr 20 2009

%F a(n) = 2*a(n-1) + 2^(n-3) for n > 2, a(0) = 1, a(1) = 2, a(2) = 5. - _Philippe Deléham_, Mar 02 2012

%F G.f.: Q(0), where Q(k) = 1 + (k+1)*x/(1 - x - x*(1-x)/(x + (k+1)*(1-x)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 24 2013

%F From _Amiram Eldar_, Jan 13 2021: (Start)

%F a(n) = (n+8) * 2^(n-3), for n >= 2.

%F Sum_{n>=0} 1/a(n) = 2048*log(2) - 893149/630.

%F Sum_{n>=0} (-1)^n/a(n) = 523549/630 - 2048*log(3/2). (End)

%F E.g.f.: (1/4)*((4+x)*exp(2*x) - x). - _G. C. Greubel_, Sep 27 2022

%e 11 = 2 * 5 + 1;

%e 24 = 2 * 11 + 2;

%e 52 = 2 * 24 + 4;

%e 112 = 2 * 52 + 8;

%e 240 = 2 * 112 + 16;

%e 512 = 2 * 240 + 32;

%e 1088 = 2 * 512 + 64;

%e 2304 = 2 * 1088 + 128; ...

%p 1,2, seq((n+8)*2^(n-3), n = 2..30); # _G. C. Greubel_, Sep 27 2022

%t CoefficientList[Series[(1-2x+x^2-x^3)/(1-2x)^2, {x,0,40}], x] (* _Vincenzo Librandi_, Jun 27 2012 *)

%t LinearRecurrence[{4,-4},{1,2,5,11},40] (* _Harvey P. Dale_, Sep 27 2024 *)

%o (PARI) a=[1,2,5,11]; for(i=1,99,a=concat(a,4*a[#a]-4*a[#a-1])); a \\ _Charles R Greathouse IV_, Jun 01 2011

%o (Magma) I:=[1, 2, 5, 11]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Jun 27 2012

%o (SageMath) [(n+8)*2^(n-3) - int(n==1)/4 for n in range(40)] # _G. C. Greubel_, Sep 27 2022

%Y Cf. A109975, A158416.

%K nonn,easy

%O 0,2

%A _Paul Curtz_, Jun 07 2007