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A045891 First differences of A045623. 24
1, 1, 3, 7, 16, 36, 80, 176, 384, 832, 1792, 3840, 8192, 17408, 36864, 77824, 163840, 344064, 720896, 1507328, 3145728, 6553600, 13631488, 28311552, 58720256, 121634816, 251658240, 520093696, 1073741824, 2214592512, 4563402752 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Let M_n be the n X n matrix m_(i,j)=3+abs(i-j) then det(M_n)=(-1)^(n+1)*a(n+1) - Benoit Cloitre, May 28 2002

If X_1,X_2,...,X_n are 2-blocks of a (2n+3)-set X then, for n>=1, a(n+2) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007

Equals row sums of triangle A152194. [Gary W. Adamson, Nov 28 2008]

Contribution from Johannes W. Meijer, Aug 15 2010: (Start)

An elephant sequence, see A175655. For the central square 16 A[5] vectors, with decimal values between 19 and 400, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A045623.

(End)

REFERENCES

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

LINKS

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

F. Ellermann, Illustration of binomial transforms

Milan Janjic, Two Enumerative Functions

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

Index to sequences with linear recurrences with constant coefficients, signature (4,-4).

FORMULA

Sum_{ k = 0..n } (k+3)*binomial(n,k) gives the sequence with a different offset: 3, 7, 16, 36, 80, 176, 384, 832, 1792, 3840, 8192, ... - N. J. A. Sloane, Jan 30 2008

a(n) = (n+4)*2^(n-3), n >= 2; a(0)=1=a(1); G.f.: (1-x)^3/(1-2*x)^2.

Binomial transform of A027656.

Starting 1, 3, 7, 16.. this is ((n+5)2^n-0^n)/4, the binomial transform of (1, 2, 2, 3, 3, ...). - Paul Barry, May 20 2003

a(n)=(n+4)*2^(n-3)+3C(0, n)/4-C(1, n)/4; a(n)=sum{k=0..floor(n/2), C(n, 2k)(k+1)}. - Paul Barry, Nov 29 2004

a(n)=A045623(n-1)+2^n-2)=A034007(n+1)-2^(n-2) for n>=2 . [Philippe Deléham, Apr 20 2009]

G.f.: 1 + Q(0)*x/(1-x)^2, 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 25 2013

MATHEMATICA

Join[{1, 1, a=3, b=7}, Table[c=4*b-4*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)

Table[ If[n < 2, 1, 2^(n-3)*(n+4)], {n, 0, 30}] (* Jean-François Alcover, Sep 12 2012 *)

PROG

(PARI) v=[1, 1, 3, 7]; for(i=1, 99, v=concat(v, 4*(v[#v]-v[#v-1]))); v \\ Charles R Greathouse IV, Jun 01 2011

CROSSREFS

Cf. A152194.

Sequence in context: A238441 A173514 * A081037 A019489 A077852 A218983

Adjacent sequences:  A045888 A045889 A045890 * A045892 A045893 A045894

KEYWORD

easy,nonn,nice

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified July 28 20:19 EDT 2014. Contains 245007 sequences.