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A047926 a(n) = (3^(n+1) + 2*n + 1)/4. 14
1, 3, 8, 22, 63, 185, 550, 1644, 4925, 14767, 44292, 132866, 398587, 1195749, 3587234, 10761688, 32285049, 96855131, 290565376, 871696110, 2615088311, 7845264913, 23535794718, 70607384132, 211822152373, 635466457095, 1906399371260 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Density of regular language L{0}* over {0,1,2,3} (i.e., number of strings of length n in L), where L is described by regular expression with c=3: sum_{i=1..c}(prod_{j=1..i}(j(1+...+j)*) where sum stands for union and prod for concatenation. I.e., L=L((11*+11*2(1+2)*+11*2(1+2)*3(1+2+3)*)0*) - Nelma Moreira (nam(AT)ncc.up.pt), Oct 10 2004

Conjecture: Number of representations of 3^(2n) as a sum a^2 + b^2 + c^2 with 0 < a <= b <= c. That is, a(1)=3 because 3^2 = 1^2+2^2+2^2, a(2)=3 because 3^4 = 1^2+4^2+8^2 = 3^2+6^2+6^2 = 4^2+4^2+7^2. - Zak Seidov, Mar 01 2012

Construct an array with m(0,n) = 2*n^2 + 4*n + 1 = A056220(n+1), m(n,0) = 2*n^2 + 1 = A058331(n) and m(n,n) = 2*n*(n+1) + 1 = A001844(n). Row(n)begins with all the increasing odd numbers from A058331(n) to A001844(n) and column(n) begins with all the decreasing odd numbers from A056220(n+1) to A001844(n). The sum of the terms in row(n) plus those in column(n) minus m(n,n)=a(n). The first five rows are 1,7,17,31,49; 3,5,15,29,47; 9,11,13,27,45; 19,21,23,25,43; 33,35,37,39,41. - J. M. Bergot, Jul 16 2013

REFERENCES

M. Aigner, Combinatorial Search, Wiley, 1988, see Exercise 6.4.5.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto.

N. Moreira and R. Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.

Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

FORMULA

From Paul Barry, Sep 03 2003: (Start)

a(n) = Sum_{k=0..n} (3^k + 1)/2. Partial sums of A007051.

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

For c=3, a(c,n) = g(1,c)*n + Sum_{k=2..c} ((g(k,c)*k*(k^n - 1))/(k-1)) where g(1,1)=1, g(1,c) = g(1,c-1) + ((-1)^(c-1))/(c-1)!, c > 1 g(k,c) = g(k-1, c-1)/k, for c > 1 and 2 <= k <= c. - Nelma Moreira (nam(AT)ncc.up.pt), Oct 10 2004

a(n+1) = 3*a(n) - n. - Franklin T. Adams-Watters, Jul 05 2014

PROG

(Sage) [(gaussian_binomial(n, 1, 3)+n)/2 for n in xrange(1, 28)] # Zerinvary Lajos, May 29 2009

(MAGMA) [(3^(n+1)+2*n+1)/4: n in [0..40]]; // Vincenzo Librandi, May 02 2011

(PARI) a(n)=(3^(n+1)+2*n+1)/4 \\ Charles R Greathouse IV, Mar 02 2012

CROSSREFS

Sequence in context: A298260 A317997 A164934 * A192681 A014138 A099324

Adjacent sequences:  A047923 A047924 A047925 * A047927 A047928 A047929

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 16 08:40 EDT 2018. Contains 316259 sequences. (Running on oeis4.)