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A111297 First differences of A109975. 11
1, 2, 5, 11, 24, 52, 112, 240, 512, 1088, 2304, 4864, 10240, 21504, 45056, 94208, 196608, 409600, 851968, 1769472, 3670016, 7602176, 15728640, 32505856, 67108864, 138412032, 285212672, 587202560, 1207959552, 2483027968, 5100273664 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550, 2013

M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5

Index entries for linear recurrences with constant coefficients, signature (4,-4).

FORMULA

Equals binomial transform of [1, 1, 2, 1, 3, 1, 4, 1, 5,...] - Gary W. Adamson, Apr 25 2008

Contribution from Paul Barry, Mar 18 2009: (Start)

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

a(n)=sum{k=0..n, C(n,k)*sum{j=0..floor(k/2), C(j+1,k-j)}}=sum{k=0..n, C(n,k)*A158416(k)}. (End)

Sum_{k=0..n}(k+5)*binomial(n,k) gives 5, 11, 24, 52, 112, 240, ... [Philippe Deléham, Apr 20 2009]

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

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

EXAMPLE

11 = 2*5 + 1 ; 24 = 2*11 + 2 ; 52 = 2*24 + 4 ; 112 = 2* 52 + 8 ; 240 = 2* 112 + 16 ; 512 = 2*240 + 32 ; 1088 = 2*512 + 64 ; 2304 = 2*1088 + 128; ...

MATHEMATICA

CoefficientList[Series[(1-2x+x^2-x^3)/(1-2x)^2, {x, 0, 40}], x]  (* Vincenzo Librandi, Jun 27 2012 *)

PROG

(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

(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 *)

CROSSREFS

Sequence in context: A027934 A134389 A286945 * A077864 A052980 A190512

Adjacent sequences:  A111294 A111295 A111296 * A111298 A111299 A111300

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Jun 07 2007

STATUS

approved

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Last modified June 20 19:33 EDT 2019. Contains 324234 sequences. (Running on oeis4.)