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 A115730 #30 Jan 22 2025 05:59:40
%S A115730 0,0,1,2,6,16,42,110,289,756,1980,5184,13572,35532,93025,243542,
%T A115730 637602,1669264,4370190,11441306,29953729,78419880,205305912,
%U A115730 537497856,1407187656,3684065112,9645007681,25250957930,66107866110
%N A115730 a(n) = a(n-3) + A001654(n-1) with a(0)=0, a(1)=0 and a(2)=1.
%C A115730 The a(n+1) represent the Ca2 and Ze4 sums of the Golden Triangle A180662. Furthermore the a(3*n) represent the Ze1 (terms doubled) and Ca3 sums of the Golden triangle. See A180662 for more information about these and other triangle sums.
%H A115730 G. C. Greubel, <a href="/A115730/b115730.txt">Table of n, a(n) for n = 0..1000</a>
%H A115730 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,0,-2,-2,1).
%F A115730 a(n) = -floor(g(Fibonacci(n+1))) where g(x) = (1-x^2)^2/(-4*x^2).
%F A115730 G.f.: x^2/( (1-x)*(1+x)*(1+x+x^2)*(1-3*x+x^2) ). - _R. J. Mathar_, Jun 20 2015
%F A115730 a(n) - a(n-2) = A182890(n-1). - _R. J. Mathar_, Jun 20 2015
%F A115730 a(n) = (1/60)*((-1)^n*(6 - 5*ChebyshevU(n, 1/2) + 10*ChebyshevU(n-1, 1/2)) - (10 - 9*ChebyshevU(n, 3/2) + 6*ChebyshevU(n-1, 3/2))). - _G. C. Greubel_, Jan 20 2022
%F A115730 a(n) = floor((2*Fibonacci(2*n+1) + Fibonacci(2*n+2) + 2)/20). - _Michael Somos_, Sep 05 2023
%e A115730 G.f. = x^2 + 2*x^3 + 6*x^4 + 16*x^5 + 42*x^6 + 110*x^7 + 289*x^8 + ... - _Michael Somos_, Sep 05 2023
%p A115730 nmax:=31: with(combinat): for n from 0 to nmax do A001654(n):=fibonacci(n) * fibonacci(n+1) od: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):=a(n-3) + A001654(n-1) od: seq(a(n),n=0..nmax);
%t A115730 LinearRecurrence[{2,2,0,-2,-2,1}, {0,0,1,2,6,16}, 40] (* modified by _G. C. Greubel_, Jan 20 2022 *)
%t A115730 a[ n_] := Floor[(2*Fibonacci[2*n+1] + Fibonacci[2*n+2] + 2)/20]; (* _Michael Somos_, Sep 05 2023 *)
%o A115730 (Magma)
%o A115730 function A115730(n)
%o A115730 if n lt 3 then return Floor(n/2);
%o A115730 else return A115730(n-3) + Fibonacci(n-1)*Fibonacci(n);
%o A115730 end if; return A115730;
%o A115730 end function;
%o A115730 [A115730(n): n in [0..40]]; // _G. C. Greubel_, Jan 20 2022
%o A115730 (Sage)
%o A115730 U=chebyshev_U
%o A115730 def A115730(n): return (1/60)*((-1)^n*(6 - 5*U(n, 1/2) + 10*U(n-1, 1/2)) - (10 - 9*U(n, 3/2) + 6*U(n-1, 3/2)))
%o A115730 [A115730(n) for n in (0..40)] # _G. C. Greubel_, Jan 20 2022
%o A115730 (PARI) {a(n) = (2*fibonacci(2*n+1) + fibonacci(2*n+2) + 2)\20}; /* _Michael Somos_, Sep 05 2023 */
%Y A115730 Cf. A000045, A001654, A064831, A079962, A180662, A180664, A180665, A180666, A182890.
%K A115730 nonn,easy
%O A115730 0,4
%A A115730 _Roger L. Bagula_, Mar 13 2006
%E A115730 Corrected and information added by _Johannes W. Meijer_, Sep 22 2010
%E A115730 Edited by Editors-in-Chief. - _N. J. A. Sloane_, Jun 20 2015