0,3

Number of walks of length n between any two distinct vertices of the complete graph K_4. - Paul Barry and Emeric Deutsch, Apr 01 2004

For n>=1, a(n) is the number of integers k, 1<=k<=3^(n-1), such that their ternary representation ends in even number of zeros (see A007417). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 31 2004

Form the digraph with matrix A=[0,1,1,1;1,0,1,1;1,1,0,1;1,0,1,1]. A015518(n) corresponds to the (1,3) term of A^n. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 4 times the bottom to get the new top. The limit of the sequence of fractions is 2. - Cino Hilliard (hillcino368(AT)gmail.com), Sep 25 2005

(A046717(n))^2 + (2*a(n))^2 = A046717(2n). E.g. A046717(3) = 13, 2*a(3) = 14, A046717(6) = 365. 13^2 + 14^2 = 365. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 17 2006

For n>=2, number of ordered partitions of n-1 into parts of sizes 1 and 2 where there are two types of 1 (singletons) and three types of 2 (twins). For example, the number of possible configurations of families of n-1 male (M) and female (F) offspring considering only single births and twins, where the birth order of M/F/pair-of-twins is considered and there are three types of twins; namely, both F, both M, or one F and one M - where birth order within a pair of twins itself is disregarded. In particular, for a(3)=7, two children could be either: (1) F, then M; (2) M, then F; (3) F,F; (4) M,M; (5) F,F twins; (6) M,M twins; or (7) M,F twins (emphasizing that birth order is irrelevant here when both/all children are the same gender and when two children are within the same pair of twins). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 18 2004

a(n) is prime for n = {2, 3, 5, 7, 13, 23, ...}, where only a(2) = 2 corresponds to a prime of the form (3^n - 1)/4. All prime a(n), except a(2) = 2, are the primes of the form (3^n + 1)/4. Numbers n such that (3^n + 1)/4 is prime are listed in A007658(n) = {3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, ...}. Note that all prime a(n) have prime indices. Prime a(n) are listed in A111010(n) = {2, 7, 61, 547, 398581, 23535794707, 82064241848634269407, ...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 19 2006

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

Index entries for sequences related to Chebyshev polynomials.

G.f.: x/(1-2*x-3*x^2). a(n) = (3^n-(-1)^n)/4 = [3^n/4 + 1/2].

a(n) = 3^(n-1) - a(n-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004

E.g.f. (exp(3x)-exp(-x))/4. Second inverse binomial transform of (5^n-1)/4, A003463. Inverse binomial transform for powers of 4, A000302 (when preceded by 0). - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003

a(n) = sum{k=0..floor(n/2), C(n, 2k+1)*2^(2k) } - Paul Barry (pbarry(AT)wit.ie), May 14 2003

a(n) = sum{k=1..n, binomial(n, k)(-1)^(n+k)*4^(k-1) }. - Paul Barry (pbarry(AT)wit.ie), Apr 02 2003

a(n+1) = sum{k=0..floor(n/2), binomial(n-k, k)2^(n-2k)3^k} - Paul Barry (pbarry(AT)wit.ie), Jul 13 2004

a(n) = U(n-1, i/sqrt(3))(-i*sqrt(3))^(n-1), i^2=-1. - Paul Barry (pbarry(AT)wit.ie), Nov 17 2003

G.f.: x(1+x)^2/(1-6x^2-8x^3-3x^4) = x(1+x)^2/characteristic polynomial(x^4*adj(K_4)(1/x)). - Paul Barry (pbarry(AT)wit.ie), Feb 03 2004

a(n) = sum_{k=0..3^(n-1)} A014578(k) = -(-1)^n*A014983(n) = A051068(3^(n-1)), for n>0. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 31 2004

E.g.f. : exp(x)sinh(2x)/2 - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004

a(n) = [3^n/4 + 1/2]. - M. F. Hasler, Mar 20 2008

a(2n+1) = A054880(n)+1 - M. F. Hasler, Mar 20 2008

2a(n) + (-1)^n = A046717(n) - M. F. Hasler, Mar 20 2008

Table[(3^n-(-1)^n)/4, {n, 0, 30}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 19 2006

CoefficientList[Series[1/(1-2x-3x^2), {x, 0, 25}], x] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 22 2007

(PARI) a(n)=round(3^n/4)

a(n) = A080926(n-1) + 1 = (1/3) A054878(n+1) = (1/3) |A084567(n+1)|.

First differences of A033113 and A039300. Partial sums of A046717.

The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.

Cf. A046717.

Cf. A007658, A111010.

Adjacent sequences: A015515 A015516 A015517 this_sequence A015519 A015520 A015521

Sequence in context: A111017 A116408 A014983 this_sequence A083379 A000935 A035071

nonn,walk,easy

Olivier Gerard (ogerard(AT)ext.jussieu.fr)

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004

Edited by Ralf Stephan, Aug 30 2004