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 A008725 #48 Nov 25 2023 14:29:34 %S A008725 1,2,3,4,5,6,7,9,11,13,15,17,19,21,24,27,30,33,36,39,42,46,50,54,58, %T A008725 62,66,70,75,80,85,90,95,100,105,111,117,123,129,135,141,147,154,161, %U A008725 168,175,182,189,196,204,212,220,228,236,244,252,261,270,279,288,297,306 %N A008725 Molien series for 3-dimensional group [2,n] = *22n. %C A008725 a(n) is the number of partitions of n into parts 1 and 7, where there are two kinds of part 1. - _Joerg Arndt_, Sep 27 2020 %C A008725 Define a general Somos-4 sequence by b(n) = (p1*b(n-1)*b(n-3) + p2*b(n-2)^2)/b(n-4) with b(0) = b0, b(1) = b1, b(2) = b2, b(3) = b3 and where p1 = (b1^3*b2 - b0^3*b3) / (b0*(b1^3 + b0^2*b2)), p2 = -b1*(b2^2 + b0*b3) / (b1^3 + b0^2*b2). Then b(n) = -b(-1-n) for all n in Z. The denominator of b(n) is a power of b0 times (b1^3 + b0^2*b2)^a(n-4). - _Michael Somos_, Nov 23 2023 %H A008725 Vincenzo Librandi, <a href="/A008725/b008725.txt">Table of n, a(n) for n = 0..1000</a> %H A008725 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=190">Encyclopedia of Combinatorial Structures 190</a> %H A008725 <a href="/index/Mo#Molien">Index entries for Molien series</a> %H A008725 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,1,-2,1). %F A008725 G.f.: 1/((1-x)^2*(1-x^7)). %F A008725 From _Mitch Harris_, Sep 08 2008: (Start) %F A008725 a(n) = Sum_{j=0..n+7} floor(j/7). %F A008725 a(n-7) = (1/2)*floor(n/7)*(2*n - 5 - 7*floor(n/7)). (End) %F A008725 a(n) = 2*a(n-1) - a(n-2) + a(n-7) - 2*a(n-8) + a(n-9). - _R. J. Mathar_, Apr 20 2010 %F A008725 a(n) = A011867(n+5). - _Pontus von Brömssen_, Sep 27 2020 %F A008725 a(n) = a(-9-n) = A174738(n+7) = floor((n+4)*(n+5)/14) for all n in Z. - _Michael Somos_, Nov 23 2023 %p A008725 1/((1-x)^2*(1-x^7)): seq(coeff(series(%, x, n+1), x, n), n=0..80); %t A008725 CoefficientList[Series[1/((1-x)^2*(1-x^7)), {x, 0, 80}], x] (* _Vincenzo Librandi_, Jun 11 2013 *) %t A008725 LinearRecurrence[{2,-1,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,9,11}, 80] (* _Harvey P. Dale_, Sep 27 2014 *) %t A008725 a[ n_] := Floor[(n+4)*(n+5)/14]; (* _Michael Somos_, Nov 23 2023 *) %o A008725 (PARI) my(x='x+O('x^80)); Vec(1/((1-x)^2*(1-x^7))) \\ _G. C. Greubel_, Sep 09 2019 %o A008725 (PARI) {a(n) = (n+4)*(n+5)\14}; /* _Michael Somos_, Nov 23 2023 */ %o A008725 (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x)^2*(1-x^7)) )); // _G. C. Greubel_, Sep 09 2019 %o A008725 (Sage) %o A008725 def A008725_list(prec): %o A008725 P.<x> = PowerSeriesRing(ZZ, prec) %o A008725 return P(1/((1-x)^2*(1-x^7))).list() %o A008725 A008725_list(80) # _G. C. Greubel_, Sep 09 2019 %o A008725 (GAP) a:=[1,2,3,4,5,6,7,9,11];; for n in [10..80] do a[n]:=2*a[n-1] -a[n-2]+a[n-7]-2*a[n-8]+a[n-9]; od; a; # _G. C. Greubel_, Sep 09 2019 %Y A008725 Cf. A011867, A174738. %K A008725 nonn,easy %O A008725 0,2 %A A008725 _N. J. A. Sloane_ %E A008725 More terms from _Vladimir Joseph Stephan Orlovsky_, Mar 14 2010