|
%I #33 Sep 08 2022 08:46:04
%S 1,249,8537,186073,3427545,58664153,970097881,15776875737,
%T 254486643929,4088295982297,65545039643865,1049779971687641,
%U 16804957869966553,268947166998693081,4303697458594972889,68863501862374868185
%N a(n) = (392*16^n -1620*8^n +1890*4^n -767)/105.
%C Starting at n=1, a cube has an edge=2^(n+1)-3. The beginning cube has a value of 1 and is surrounded by 2^n layers of cubes each valued at 2^n. The sum of all cubes with values of 2^n is a(n).
%C Indices of primes in this sequence: 3, 10, 12, 21, 37, 70, 102, 201, 961, 1854, ....
%H G. C. Greubel, <a href="/A225209/b225209.txt">Table of n, a(n) for n = 1..825</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (29,-252,736,-512).
%F a(n) = 29*a(n-1) - 252*a(n-2) + 736*a(n-3) - 512*a(n-4).
%F a(n) = a(n-1) + 7*2^(4*n-1) - 27*2^(3*n-1) + 27*2^(2*n-1), for n>0.
%F G.f. x*(1 +220*x +1568*x^2 +512*x^3)/( (1-x)*(1-4*x)*(1-8*x)*(1-16*x) ). - _R. J. Mathar_, May 09 2013
%F a(n) = a(n-1) +2^(n-1)*(A036563(n+1)^3 -A036563(n)^3). - _R. J. Mathar_, May 18 2013
%e The first cubes has value 1=a(1). The second cube has 2 layers of cubes each valued at 2 surrounding the cube of value 1 to give (5^3-1)*2+1=249=a(2). Next surround by 2^2 layers of cubes each valued at 2^2: (13^3-5^3)*4+249=8537=a(3). Finally, surround by 2^3 layers of cubes each of value 2^3 to get (29^3-13^3)*8 + 8537 = 186073 = a(4).
%p seq( (392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105, n=1..20); # _G. C. Greubel_, Dec 31 2019
%t LinearRecurrence[{29,-252,736,-512},{1,249,8537,186073},20] (* _Harvey P. Dale_, Apr 22 2018 *)
%o (PARI) vector(20, n, (392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105) \\ _G. C. Greubel_, Dec 31 2019
%o (Magma) [(392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105: n in [1..20]]; // _G. C. Greubel_, Dec 31 2019
%o (Sage) [(392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105 for n in (1..20)] # _G. C. Greubel_, Dec 31 2019
%o (GAP) List([1..20], n-> (392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105); # _G. C. Greubel_, Dec 31 2019
%K nonn,easy
%O 1,2
%A _J. M. Bergot_, May 01 2013
|
