login
Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} Legendre(i,prime(n)).
2

%I #14 Oct 06 2024 09:15:52

%S 1,1,0,1,0,-1,0,1,2,1,2,1,0,1,0,1,2,3,2,1,0,1,0,1,0,1,2,1,0,-1,-2,-1,

%T 0,-1,0,1,2,1,2,1,0,-1,0,1,0,-1,-2,-1,-2,-1,0,1,0,-1,0,1,2,3,2,3,2,3,

%U 2,1,0,-1,0,1,0,1,2,3,4,3,4,3,4,5,4,3,4,5,4,3,4,3,4,3,2,1,0

%N Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} Legendre(i,prime(n)).

%C Strictly speaking, the symbol in the definition is the Legendre-Jacobi-Kronecker symbol, since the Legendre symbol is defined only for odd primes.

%D József Beck, Inevitable randomness in discrete mathematics, University Lecture Series, 49. American Mathematical Society, Providence, RI, 2009. xii+250 pp. ISBN: 978-0-8218-4756-5; MR2543141 (2010m:60026). See page 23.

%H G. C. Greubel, <a href="/A226519/b226519.txt">Rows n = 1..50 of the irregular triangle, flattened</a>

%e Triangle begins:

%e 1;

%e 1, 0;

%e 1, 0, -1, 0;

%e 1, 2, 1, 2, 1, 0;

%e 1, 0, 1, 2, 3, 2, 1, 0, 1, 0;

%e 1, 0, 1, 2, 1, 0, -1, -2, -1, 0, -1, 0;

%e 1, 2, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, -2, -1, 0;

%e ...

%p with(numtheory);

%p T:=(n,k)->add(legendre(i,ithprime(n)),i=1..k);

%p f:=n->[seq(T(n,k),k=1..ithprime(n)-1)];

%p [seq(f(n),n=1..15)];

%t Table[P = Prime[n]; Table[JacobiSymbol[k,P], {k,P-1}]//Accumulate, {n,15}]// Flatten (* _G. C. Greubel_, Oct 05 2024 *)

%o (Magma)

%o A226519:= func< n,k | n eq 1 select k else (&+[JacobiSymbol(j, NthPrime(n)): j in [0..k]]) >;

%o [A226519(n,k) : k in [1..NthPrime(n)-1], n in [1..15]]; // _G. C. Greubel_, Oct 05 2024

%o (SageMath)

%o def A226519(n,k): return k if n==1 else sum(jacobi_symbol(j, nth_prime(n)) for j in range(k+1))

%o flatten([[A226519(n,k) for k in range(1,nth_prime(n))] for n in range(1,16)]) # _G. C. Greubel_, Oct 05 2024

%Y A variant of A226518, which is the main entry for this triangle.

%Y Cf. A165582, A226518.

%K sign,tabf

%O 1,9

%A _N. J. A. Sloane_, Jun 19 2013