Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #13 May 04 2022 08:28:07
%S 0,5,42,151,548,1185,2542,4403,7608,13621,20834,32535,47980,65609,
%T 88278,119947,162368,208869,269194,340007,416580,512305,622286,756003,
%U 925432,1114661,1314498,1537015,1771628,2031993,2393158,2786315
%N a(n) = Sum_{i=1..n} Sum_{j=1..i} (prime(i)^2 - prime(j)^2).
%H Robert Israel, <a href="/A062021/b062021.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 2*a(n-1) - a(n-2) + (n-1)*(prime(n)^2 - prime(n-1)^2) with a(1) = 0, a(2) = 5.
%e a(3) = (5^2 - 2^2) + (5^2 - 3^2) + (3^2 - 2^2) = 42.
%p N:= 100: # for a(1)..a(N)
%p P2:= [seq(ithprime(i)^2,i=1..N)]:
%p DP2:= P2[2..-1]-P2[1..-2]:
%p A[1]:= 0: A[2]:= 5:
%p for n from 3 to N do A[n]:= 2*A[n-1]+(n-1)*DP2[n-1]-A[n-2] od:
%p seq(A[i],i=1..N); # _Robert Israel_, Feb 02 2020
%t RecurrenceTable[{a[1]==0,a[2]==5,a[n]==2a[n-1]-a[n-2]+(n-1)(Prime[n]^2 - Prime[n-1]^2)}, a, {n,40}] (* _Harvey P. Dale_, May 16 2019 *)
%o (Magma) [(&+[(&+[NthPrime(i)^2 - NthPrime(j)^2: j in [1..i]]): i in [1..n]]): n in [1..40]]; // _G. C. Greubel_, May 04 2022
%o (SageMath)
%o @CachedFunction
%o def a(n):
%o if (n<3): return 5*(n-1)
%o else: return 2*a(n-1) - a(n-2) + (n-1)*(nth_prime(n)^2 - nth_prime(n-1)^2)
%o [a(n) for n in (1..40)] # _G. C. Greubel_, May 04 2022
%Y Cf. A000040, A062020, A062022.
%K nonn
%O 1,2
%A _Amarnath Murthy_, Jun 02 2001
%E More terms and formula from Larry Reeves (larryr(AT)acm.org), Jun 06 2001
%E Name edited by _G. C. Greubel_, May 04 2022