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

General form: k=3^n-k. Also: A001045, A078008, A097073, A115341 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]

abs(A014983). [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]

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

Index entries for sequences related to linear recurrences with constant coefficients

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

((1+sqrt4)^n-(1-sqrt4)^n)/4 in Fibonacci form or (3^n-(-1)^n)/4. Offset =1. a(3)=7. [From Al Hakanson (hawkuu(AT)gmail.com), Dec 31 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

k=0; lst={k}; Do[k=3^n-k; AppendTo[lst, k], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]

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

sage: from sage.combinat.sloane_functions import recur_gen2 sage: it = recur_gen2(0, 1, 2, 3) sage: [it.next() for i in range(30)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

(Other) sage: [lucas_number1(n, 2, -3) for n in xrange(0, 27)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

(Other) sage: [abs(gaussian_binomial(n, 1, -3)) for n in xrange(0, 27)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]

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.

Cf. A001045, A078008, A097073, A115341 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]

Sequence in context: A111017 A116408 A014983 this_sequence A083379 A000935 A035071

Adjacent sequences: A015515 A015516 A015517 this_sequence A015519 A015520 A015521

nonn,walk,easy

Olivier Gerard (olivier.gerard(AT)gmail.com)

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

Edited by Ralf Stephan, Aug 30 2004