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 A045891 First differences of A045623. 30
 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..n). - Milan Janjic, Nov 18 2007 Equals row sums of triangle A152194. - Gary W. Adamson, Nov 28 2008 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. - Johannes W. Meijer, Aug 15 2010 a(n) is the total number of runs of 1 in the compositions of n+1. For example, a(3) = A045623(3) - A045623(2) = 12 - 5 = 7 runs of only 1 in the compositions of 4, enumerated "()" as follows: 3,(1); (1),3; 2,(1,1);(1),2,(1); (1,1),2; (1,1,1,1). More generally, the total number of runs of only part k in the compositions of n+k is A045623(n) - A045623(n-k). - Gregory L. Simay, May 02 2017 This is essentially the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S - S^2 + S^3; see A291000.  - Clark Kimberling, Aug 24 2017 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5. F. Ellermann, Illustration of binomial transforms Milan Janjic, Two Enumerative Functions M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013. M. Janjic and 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 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 a(n) = Sum_{k=0..n} (k+1)*C(n-2,n-k). Peter Luschny, Apr 20 2015 From Amiram Eldar, Jan 13 2021: (Start) Sum_{n>=0} 1/a(n) = 128*log(2) - 1292/15. Sum_{n>=0} (-1)^n/a(n) = 782/15 - 128*log(3/2). (End) E.g.f.: (2 - x + exp(2*x)*(2 + x))/4. - Stefano Spezia, Mar 26 2022 EXAMPLE G.f. = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 36*x^5 + 80*x^6 + ... - Michael Somos, Mar 26 2022 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 *) LinearRecurrence[{4, -4}, {1, 1, 3, 7}, 40] (* Harvey P. Dale, May 03 2019 *) 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. A027656, A034007, A045623, A152194, A175655. Sequence in context: A106463 A238441 A173514 * A081037 A019489 A077852 Adjacent sequences:  A045888 A045889 A045890 * A045892 A045893 A045894 KEYWORD easy,nonn,nice AUTHOR STATUS approved

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Last modified August 14 17:00 EDT 2022. Contains 356122 sequences. (Running on oeis4.)