This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A095195 #33 Jan 13 2025 04:07:19
%S A095195 2,3,1,5,2,1,7,2,0,-1,11,4,2,2,3,13,2,-2,-4,-6,-9,17,4,2,4,8,14,23,19,
%T A095195 2,-2,-4,-8,-16,-30,-53,23,4,2,4,8,16,32,62,115,29,6,2,0,-4,-12,-28,
%U A095195 -60,-122,-237,31,2,-4,-6,-6,-2,10,38,98,220,457,37,6,4,8,14,20,22,12
%N A095195 T(n,0) = prime(n), T(n,k) = T(n,k-1)-T(n-1,k-1), 0<=k<n, triangle read by rows.
%C A095195 T(n,0)=A000040(n); T(n,1)=A001223(n-1) for n>1; T(n,2)=A036263(n-2) for n>2; T(n,n-1)=A007442(n) for n>1.
%C A095195 Row k of the array (not the triangle) is the k-th differences of the prime numbers. - _Gus Wiseman_, Jan 11 2025
%H A095195 Alois P. Heinz, <a href="/A095195/b095195.txt">Rows n = 1..141, flattened</a>
%e A095195 Triangle begins:
%e A095195 2;
%e A095195 3, 1;
%e A095195 5, 2, 1;
%e A095195 7, 2, 0, -1;
%e A095195 11, 4, 2, 2, 3;
%e A095195 13, 2, -2, -4, -6, -9;
%e A095195 Alternative: array form read by antidiagonals:
%e A095195 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,...
%e A095195 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6,...
%e A095195 1, 0, 2, -2, 2, -2, 2, 2, -4, 4, -2,...
%e A095195 -1, 2, -4, 4, -4, 4, 0, -6, 8, -6, 0,...
%e A095195 3, -6, 8, -8, 8, -4, -6, 14, -14, 6, 4,...
%e A095195 -9, 14, -16, 16, -12, -2, 20, -28, 20, -2, -8,...
%e A095195 23, -30, 32, -28, 10, 22, -48, 48, -22, -6, 10,..,
%e A095195 -53, 62, -60, 38, 12, -70, 96, -70, 16, 16, -12,...
%e A095195 115,-122, 98, -26, -82, 166,-166, 86, 0, -28, 28,...
%e A095195 -237, 220,-124, -56, 248,-332, 252, -86, -28, 56, -98,...
%e A095195 457,-344, 68, 304,-580, 584,-338, 58, 84,-154, 308,...
%p A095195 A095195A := proc(n,k) # array, k>=0, n>=0
%p A095195 option remember;
%p A095195 if n =0 then
%p A095195 ithprime(k+1) ;
%p A095195 else
%p A095195 procname(n-1,k+1)-procname(n-1,k) ;
%p A095195 end if;
%p A095195 end proc:
%p A095195 A095195 := proc(n,k) # triangle, 0<=k<n, n>=1
%p A095195 A095195A(k,n-k-1) ;
%p A095195 end proc: # _R. J. Mathar_, Sep 19 2013
%t A095195 T[n_, 0] := Prime[n]; T[n_, k_] /; 0 <= k < n := T[n, k] = T[n, k-1] - T[n-1, k-1]; Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Feb 01 2017 *)
%t A095195 nn=6;
%t A095195 t=Table[Differences[Prime[Range[nn]],k],{k,0,nn}];
%t A095195 Table[t[[j,i-j+1]],{i,nn},{j,i}] (* _Gus Wiseman_, Jan 11 2025 *)
%o A095195 (Haskell)
%o A095195 a095195 n k = a095195_tabl !! (n-1) !! (k-1)
%o A095195 a095195_row n = a095195_tabl !! (n-1)
%o A095195 a095195_tabl = f a000040_list [] where
%o A095195 f (p:ps) xs = ys : f ps ys where ys = scanl (-) p xs
%o A095195 -- _Reinhard Zumkeller_, Oct 10 2013
%Y A095195 Cf. A036262-A036271.
%Y A095195 Cf. A140119 (row sums).
%Y A095195 Below, the inclusive primes (A008578) are 1 followed by A000040. See also A075526.
%Y A095195 Rows of the array (columns of the triangle) begin: A000040, A001223, A036263.
%Y A095195 Column n = 1 of the array is A007442, inclusive A030016.
%Y A095195 The version for partition numbers is A175804, see A053445, A281425, A320590.
%Y A095195 First position of 0 is A376678, inclusive A376855.
%Y A095195 Absolute antidiagonal-sums are A376681, inclusive A376684.
%Y A095195 The inclusive version is A376682.
%Y A095195 For composite instead of prime we have A377033, see A377034-A377037.
%Y A095195 For squarefree instead of prime we have A377038, nonsquarefree A377046.
%Y A095195 Column n = 2 of the array is A379542.
%Y A095195 Cf. A002808, A064113, A065890, A073783, A258026, A293467, A333254, A376683, A377051.
%K A095195 sign,tabl,look
%O A095195 1,1
%A A095195 _Reinhard Zumkeller_, Jun 22 2004