A176621 - OEIS

%I #35 Sep 08 2022 08:45:52

%S 2,3,4,11,40,157,656,2721,11346,47337,197398,823451,3434718,14326815,

%T 59760224,249271079,1039759044,4337038713,18090636780,75459593981,

%U 314756746798,1312912064069,5476413466674,22843193551799,95283436047290

%N a(n) = 2 + Sum_{k=0..n-1} A176513(4*k+1).

%C Old name was "a(n) is the minimum number that can be expressed as the sum of n terms of sequence A176513".

%C Lim_{n -> infinity} a(n+1)/a(n) = s^4 = 4.17119593178..., where s is the root of the characteristic equation s^5 = s^3 + s^2 + 1.

%H G. C. Greubel, <a href="/A176621/b176621.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,5,2,-11,3,-1).

%F From _Jianing Song_, Feb 04 2019: (Start)

%F a(n+5) = 2*a(n+4) + 7*a(n+3) + 9*a(n+2) - 2*a(n+1) + a(n) - 32.

%F a(n+6) = 3*a(n+5) + 5*a(n+4) + 2*a(n+3) - 11*a(n+2) + 3*a(n+1) - a(n). (End)

%t LinearRecurrence[{3, 5, 2, -11, 3, -1}, {2, 3, 4, 11, 40, 157}, 50]

%o (PARI) a(n) = my(v=vector(n+1), u=[2, 3, 4, 11, 40]); for(k=1, n+1, v[k]=if(k<=5, u[k], 2*v[k-1] + 7*v[k-2] + 9*v[k-3] - 2*v[k-4] + v[k-5] - 32)); v[n+1] \\ _Jianing Song_, Feb 04 2019

%o (Magma) I:=[2,3,4,11,40,157]; [n le 6 select I[n] else 3*Self(n-1) +5*Self(n-2) +2*Self(n-3) -11*Self(n-4) +3*Self(n-5) -Self(n-6): n in [1..51]]; // _G. C. Greubel_, Jul 01 2021

%o (Sage)

%o @CachedFunction

%o def b(n): return 1 if (n<6) else b(n-2) + b(n-3) + b(n-5) # b=A176513

%o def a(n): return 2 + sum(b(4*j+1) for j in (0..n-1))

%o [a(n) for n in (0..50)] # _G. C. Greubel_, Jul 01 2021

%Y Cf. A176513.

%K nonn,easy

%O 0,1

%A _Carmine Suriano_, Apr 22 2010

%E New name, a(0) = 2 prepended and a(1), a(2) corrected by _Jianing Song_, Feb 04 2019

%E Term a(11) corrected by _G. C. Greubel_, Jul 01 2021