This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A034494 #44 Aug 14 2025 15:47:53
%S A034494 1,4,25,172,1201,8404,58825,411772,2882401,20176804,141237625,
%T A034494 988663372,6920643601,48444505204,339111536425,2373780754972,
%U A034494 16616465284801,116315256993604,814206798955225,5699447592686572,39896133148806001,279272932041642004
%N A034494 a(n) = (7^n+1)/2.
%C A034494 Binomial transform of A081341. Inverse binomial transform of A081342. - _R. J. Mathar_, Oct 23 2008
%C A034494 Number of compositions of even natural numbers into n parts <=6. - _Adi Dani_, May 28 2011
%C A034494 From _Charlie Marion_, Jun 24 2011: (Start)
%C A034494 a(n)+(a(n)+1)+...+(a(n+1)-7^n-1)=(a(n+1)-7^n)+...+(a(n+1)-1). Let S(2n) and S(2n+1) be the sets of addends on the left- and right-hand sides, respectively, of the preceding equations. Then, since the intersection of any 2 different S(i) is null and the union of all of them is the positive integers, {S(i)} forms a partition of the positive integers. See also A034659.
%C A034494 In general, for k>0, let b(n)=((4k+3)^n+1)/2. Then b(n)+(b(n)+1)+ ... +(b(n+1)-(4k+3)^n-1)=k*((b(n+1)-(4k+3)^n)+ ... +(b(n+1)-1)). Then, for each k, the set of addends on the two sides of these equations also forms a partition of the positive integers. Also, with b(0)=1, b(n)=(4k+3)*b(n-1)-(2k+1).
%C A034494 For k>0, let c(0)=1 and, for n>0, c(n)=(2*(2k+1))^n/2. Then the sequence b(0),b(1),... is the binomial transform of the sequence c(0),c(1),....
%C A034494 For k>0, let d(2n)=(2k+1)^(2n) and d(2n+1)=0. Then the sequence b(0),b(1),... is the (2k+2)nd binomial transform of the sequence d(0),d(1),.... (End)
%H A034494 Vincenzo Librandi, <a href="/A034494/b034494.txt">Table of n, a(n) for n = 0..200</a>
%H A034494 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-7).
%F A034494 E.g.f.: exp(4*x)*cosh(3*x). - _Paul Barry_, Apr 20 2003
%F A034494 a(n) = 7*a(n-1) - 3, a(0) = 1.
%F A034494 G.f.: (1-4*x)/((1-x)*(1-7*x)). - _Philippe Deléham_, Jul 11 2005
%F A034494 a(n) = 8*a(n-1)-7*a(n-2), a(0)=1, a(1)=4. - _Philippe Deléham_, Nov 15 2008
%F A034494 a(n) = ((4+sqrt(9))^n+(4-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
%e A034494 From _Adi Dani_, May 28 2011: (Start)
%e A034494 a(2)=25: there are 25 compositions of even numbers into 2 parts <=6:
%e A034494 (0,0)
%e A034494 (0,2),(2,0),(1,1)
%e A034494 (0,4),(4,0),(1,3),(3,1),(2,2)
%e A034494 (0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3)
%e A034494 (2,6),(6,2),(3,5),(5,3),(4,4)
%e A034494 (4,6),(6,4),(5,5)
%e A034494 (6,6)
%e A034494 (End)
%p A034494 A034494:=n->(7^n+1)/2: seq(A034494(n), n=0..30); # _Wesley Ivan Hurt_, Apr 09 2017
%t A034494 (7^Range[0, 25] + 1)/2 (* or *)
%t A034494 LinearRecurrence[{8, -7}, {1, 4}, 26] (* _Paolo Xausa_, Aug 14 2025 *)
%o A034494 (Magma) [(7^n+1)/2: n in [0..30]]; // _Vincenzo Librandi_, Jun 16 2011
%o A034494 (PARI) a(n)=(7^n+1)/2 \\ _Charles R Greathouse IV_, Jul 02 2013
%o A034494 (PARI) Vec((1-4*x)/((1-x)*(1-7*x)) + O(x^100)) \\ _Altug Alkan_, Nov 01 2015
%K A034494 nonn,easy
%O A034494 0,2
%A A034494 _N. J. A. Sloane_