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 A172316 #27 Mar 28 2025 11:29:40
%S A172316 1,2,4,8,16,32,64,127,252,500,992,1968,3904,7744,15361,30470,60440,
%T A172316 119888,237808,471712,935680,1855999,3681528,7302616,14485344,
%U A172316 28732880,56994048,113052416,224248833,444816138,882329660
%N A172316 7th column of the array A172119.
%H A172316 O. Dunkel, <a href="http://www.jstor.org/stable/2298801">Solutions of a probability difference equation</a>, Amer. Math. Monthly, 32 (1925), 354-370; see p. 356 with r = 6.
%H A172316 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,0,0,0,-1)
%F A172316 G.f.: 1/(1 - 2*z + z^7).
%F A172316 Recurrence formula: a(n+7) = 2*a(n+6) - a(n).
%F A172316 a(n) = Sum_{j=0..floor(n/(k+1))} ((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j)) with k=6.
%e A172316 a(3) = binomial(3,3)*2^3 = 8.
%e A172316 a(7) = binomial(7,7)*2^7 - binomial(1,0)*2^0 = 127.
%p A172316 for k from 0 to 20 do for n from 0 to 30 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od:k: seq(b(n),n=0..30):od; k:=6:taylor(1/(1-2*z+z^(k+1)),z=0,30);
%Y A172316 Cf. A000071, A001949, A008937, A107066, A172119.
%Y A172316 Partial sums of A001592.
%K A172316 easy,nonn
%O A172316 0,2
%A A172316 _Richard Choulet_, Jan 31 2010