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A061165 Polynomial extrapolation of 2, 3, 5, 7, 11. 2

%I #16 Jan 04 2016 11:36:25

%S 2,3,5,7,11,22,48,100,192,341,567,893,1345,1952,2746,3762,5038,6615,

%T 8537,10851,13607,16858,20660,25072,30156,35977,42603,50105,58557,

%U 68036,78622,90398,103450,117867,133741,151167,170243,191070,213752

%N Polynomial extrapolation of 2, 3, 5, 7, 11.

%H Harry J. Smith, <a href="/A061165/b061165.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).

%F a(n) = (3n^4-34n^3+141n^2-206n+144)/24.

%F G.f.: x*(2-7*x+10*x^2-8*x^3+6*x^4)/(1-x)^5. [Colin Barker, Mar 28 2012]

%F a(1)=2, a(2)=3, a(3)=5, a(4)=7, a(5)=11, a(n)=5*a(n-1)-10*a(n-2)+ 10*a(n-3)- 5*a(n-4)+a(n-5). - _Harvey P. Dale_, Oct 05 2012

%e a(6)=22 since first differences of (2,3,5,7,11) are (1,2,2,4), second differences (1,0,2), third differences (-1,2) and fourth differences (3), so a(6)=11+4+2+2+3=22.

%t Table[(3n^4-34n^3+141n^2-206n+144)/24,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{2,3,5,7,11},40] (* _Harvey P. Dale_, Oct 05 2012 *)

%o (PARI) for (n=1, 1000, write("b061165.txt", n, " ", (3*n^4 - 34*n^3 + 141*n^2 - 206*n + 144)/24)) \\ _Harry J. Smith_, Jul 18 2009

%Y Cf. A061166.

%K nonn,easy

%O 1,1

%A _Henry Bottomley_, Apr 18 2001

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