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.
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).
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)
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
(Magma) [1, 1] cat [(n+4)*2^(n-3): n in [2..40]]; // G. C. Greubel, Sep 27 2022
(SageMath) [1, 1]+[(n+4)*2^(n-3) for n in range(2, 40)] # G. C. Greubel, Sep 27 2022
CROSSREFS
KEYWORD
easy,nonn,nice
AUTHOR
STATUS
approved