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 A048495 #45 Feb 14 2025 17:31:23
%S A048495 1,2,6,18,50,130,322,770,1794,4098,9218,20482,45058,98306,212994,
%T A048495 458754,983042,2097154,4456450,9437186,19922946,41943042,88080386,
%U A048495 184549378,385875970,805306370,1677721602,3489660930,7247757314
%N A048495 a(n) = (n-1)*2^n + 2.
%C A048495 Binomial transform of 1 followed by the odd numbers (2n-1+2*0^n, or abs(A060747)). Binomial transform is A084643. - _Paul Barry_, Jun 09 2003
%C A048495 Total number of bits of all binary numbers less than 2^n (see example).
%C A048495 Total number of zero bits of all binary numbers less than 2^(n+1). - _Olivier Gérard_, Feb 25 2014.
%C A048495 Number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>2, 4>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second element, and the fourth element is larger than the third element. - _Sergey Kitaev_, Dec 08 2020
%H A048495 Vincenzo Librandi, <a href="/A048495/b048495.txt">Table of n, a(n) for n = 0..1000</a>
%H A048495 Alice L. L. Gao, Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.
%H A048495 Alice L. L. Gao, Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
%H A048495 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,4).
%F A048495 a(n) - 1 = Sum_{i=0..n-1} (n-i) * 2^(n-i-1) = n*2^(n-1) + (n-1)*2^(n-2) + (n-2)*2^(n-3) + ... + 1*(2^0). - Matthew Erbst (matt(AT)erbst.org), Apr 19 2006
%F A048495 a(n) = 2 * A002064(n-1), n >= 1. - _Omar E. Pol_, Sep 30 2012
%F A048495 a(n) = a(n-1) + (2^n - 2^(n-1)) * n = a(n-1) + n*2^(n-1). - _Olivier Gérard_, Feb 25 2014
%F A048495 G.f.: -(4*x^2-3*x+1) / ((x-1)*(2*x-1)^2). - _Colin Barker_, Jun 29 2014
%F A048495 E.g.f.: exp(x)*(2 + exp(x)*(2*x - 1)). - _Stefano Spezia_, Feb 14 2025
%e A048495 a(1)=2 : 0 1
%e A048495 a(2)=6 : 0 1 10 11
%e A048495 a(3)=18 : 0 1 10 11 100 101 110 111
%e A048495 a(4)=50 : 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111
%e A048495 ...
%p A048495 A048495:=n->(n-1)*2^n + 2; seq(A048495(n), n=0..30); # _Wesley Ivan Hurt_, Jun 29 2014
%t A048495 f[n_]:=(n-1)2^n+2;Array[f,29,0] (* _Robert G. Wilson v_, Jun 29 2014 *)
%t A048495 LinearRecurrence[{5,-8,4},{1,2,6},30] (* _Harvey P. Dale_, Jan 23 2015 *)
%o A048495 (Magma) [(n-1)*2^n + 2: n in [0..30]]; // _Vincenzo Librandi_, Sep 25 2011
%o A048495 (PARI) a(n) = (n-1)*2^n + 2; \\ _Joerg Arndt_, Feb 25 2014
%o A048495 (PARI) Vec(-(4*x^2-3*x+1)/((x-1)*(2*x-1)^2) + O(x^100)) \\ _Colin Barker_, Jun 29 2014
%Y A048495 a(n) = T(1, n), array T given by A048494.
%Y A048495 Cf. A002064, A060747, A084643.
%K A048495 nonn,easy
%O A048495 0,2
%A A048495 _Clark Kimberling_
%E A048495 Better description from _John W. Layman_, May 04 1999