%I A015518
%S A015518 0,1,2,7,20,61,182,547,1640,4921,14762,44287,132860,398581,1195742,
%T A015518 3587227,10761680,32285041,96855122,290565367,871696100,2615088301,
%U A015518 7845264902,23535794707,70607384120,211822152361,635466457082
%N A015518 a(n) = 2*a(n-1) + 3*a(n-2), with a(0)=0, a(1)=1.
%C A015518 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
%C A015518 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
%C A015518 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
%C A015518 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
%C A015518 (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
%C A015518 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
%C A015518 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
%C A015518 General form: k=3^n-k. Also: A001045, A078008, A097073, A115341 [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]
%C A015518 abs(A014983). [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 
               28 2009]
%D A015518 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see 
               p. 16.
%H A015518 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A015518 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A015518 G.f.: x/(1-2*x-3*x^2). a(n) = (3^n-(-1)^n)/4 = [3^n/4 + 1/2].
%F A015518 a(n) = 3^(n-1) - a(n-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Apr 01 2004
%F A015518 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
%F A015518 a(n) = sum{k=0..floor(n/2), C(n, 2k+1)*2^(2k) } - Paul Barry (pbarry(AT)wit.ie), 
               May 14 2003
%F A015518 a(n) = sum{k=1..n, binomial(n, k)(-1)^(n+k)*4^(k-1) }. - Paul Barry (pbarry(AT)wit.ie), 
               Apr 02 2003
%F A015518 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
%F A015518 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
%F A015518 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
%F A015518 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
%F A015518 E.g.f. : exp(x)sinh(2x)/2 - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004
%F A015518 a(n) = [3^n/4 + 1/2]. - M. F. Hasler, Mar 20 2008
%F A015518 a(2n+1) = A054880(n)+1 - M. F. Hasler, Mar 20 2008
%F A015518 2a(n) + (-1)^n = A046717(n) - M. F. Hasler, Mar 20 2008
%F A015518 ((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]
%t A015518 Table[(3^n-(-1)^n)/4,{n,0,30}] - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Nov 19 2006
%t A015518 CoefficientList[Series[1/(1-2x-3x^2), {x, 0, 25}], x] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Mar 22 2007
%t A015518 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]
%o A015518 (PARI) a(n)=round(3^n/4)
%o A015518 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
%o A015518 (Other) sage: [lucas_number1(n,2,-3) for n in xrange(0, 27)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%o A015518 (Other) sage: [abs(gaussian_binomial(n,1,-3)) for n in xrange(0,27)] 
               # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
%Y A015518 a(n) = A080926(n-1) + 1 = (1/3) A054878(n+1) = (1/3) |A084567(n+1)|.
%Y A015518 First differences of A033113 and A039300. Partial sums of A046717.
%Y A015518 The following sequences (and others) belong to the same family: A001333, 
               A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, 
               A002532, A083098, A083099, A083100, A015519.
%Y A015518 Cf. A046717.
%Y A015518 Cf. A007658, A111010.
%Y A015518 Cf. A001045, A078008, A097073, A115341 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Dec 11 2008]
%Y A015518 Sequence in context: A111017 A116408 A014983 this_sequence A083379 A000935 
               A035071
%Y A015518 Adjacent sequences: A015515 A015516 A015517 this_sequence A015519 A015520 
               A015521
%K A015518 nonn,walk,easy
%O A015518 0,3
%A A015518 Olivier Gerard (olivier.gerard(AT)gmail.com)
%E A015518 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004
%E A015518 Edited by Ralf Stephan, Aug 30 2004