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A062021 a(n) = Sum_{i=1..n} Sum_{j=1..i} (prime(i)^2 - prime(j)^2).

%I A062021 #13 May 04 2022 08:28:07
%S A062021 0,5,42,151,548,1185,2542,4403,7608,13621,20834,32535,47980,65609,
%T A062021 88278,119947,162368,208869,269194,340007,416580,512305,622286,756003,
%U A062021 925432,1114661,1314498,1537015,1771628,2031993,2393158,2786315
%N A062021 a(n) = Sum_{i=1..n} Sum_{j=1..i} (prime(i)^2 - prime(j)^2).
%H A062021 Robert Israel, <a href="/A062021/b062021.txt">Table of n, a(n) for n = 1..10000</a>
%F A062021 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 A062021 a(3) = (5^2 - 2^2) + (5^2 - 3^2) + (3^2 - 2^2) = 42.
%p A062021 N:= 100: # for a(1)..a(N)
%p A062021 P2:= [seq(ithprime(i)^2,i=1..N)]:
%p A062021 DP2:= P2[2..-1]-P2[1..-2]:
%p A062021 A[1]:= 0: A[2]:= 5:
%p A062021 for n from 3 to N do A[n]:= 2*A[n-1]+(n-1)*DP2[n-1]-A[n-2] od:
%p A062021 seq(A[i],i=1..N); # _Robert Israel_, Feb 02 2020
%t A062021 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 A062021 (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 A062021 (SageMath)
%o A062021 @CachedFunction
%o A062021 def a(n):
%o A062021     if (n<3): return 5*(n-1)
%o A062021     else: return 2*a(n-1) - a(n-2) + (n-1)*(nth_prime(n)^2 - nth_prime(n-1)^2)
%o A062021 [a(n) for n in (1..40)] # _G. C. Greubel_, May 04 2022
%Y A062021 Cf. A000040, A062020, A062022.
%K A062021 nonn
%O A062021 1,2
%A A062021 _Amarnath Murthy_, Jun 02 2001
%E A062021 More terms and formula from Larry Reeves (larryr(AT)acm.org), Jun 06 2001
%E A062021 Name edited by _G. C. Greubel_, May 04 2022