A045891 First differences of A045623.

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

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn
• nice

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

Examples

			G.f. = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 36*x^5 + 80*x^6 + ... - _Michael Somos_, Mar 26 2022
		

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.
• Frank Ellermann, Illustration of binomial transforms
• Milan Janjic, Two Enumerative Functions
• Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
• Thomas Selig and Haoyue Zhu, New combinatorial perspectives on MVP parking functions and their outcome map, arXiv:2309.11788 [math.CO], 2023. See p. 29.
• Index entries for linear recurrences with constant coefficients, signature (4,-4).

Crossrefs

Cf. A027656, A034007, A045623, A152194, A175655.

Programs

Magma

[1,1] cat [(n+4)*2^(n-3): n in [2..40]]; // G. C. Greubel, Sep 27 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 *)

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

SageMath

[1,1]+[(n+4)*2^(n-3) for n in range(2,40)] # G. C. Greubel, Sep 27 2022

Formula

a(n) = Sum_{k=0..n-2} (k+3)*binomial(n-2,k) for n >= 2. - N. J. A. Sloane, Jan 30 2008

a(n) = (n+4)*2^(n-3), n >= 2, with a(0) = a(1) = 1.

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

Equals 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

From Paul Barry, Nov 29 2004: (Start)

a(n) = ((n+4)*2^(n-1) + 3*C(0, n) - C(1, n))/4;

a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*(k+1). (End)

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