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 A213576 #31 May 09 2025 19:52:42
%S A213576 1,3,1,7,4,2,14,10,7,3,26,21,17,11,5,46,40,35,27,18,8,79,72,66,56,44,
%T A213576 29,13,133,125,118,106,91,71,47,21,221,212,204,190,172,147,115,76,34,
%U A213576 364,354,345,329,308,278,238,186,123,55,596,585,575,557,533,498,450,385,301,199,89
%N A213576 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = F(n-1+h), where F=A000045 (Fibonacci numbers), n >= 1, h >= 1, and ** = convolution.
%C A213576 Principal diagonal: A213577.
%C A213576 Antidiagonal sums: A213578.
%C A213576 Row 1, (1,2,3,...)**(1,1,2,3,5,...): A001924;
%C A213576 Row 2, (1,2,3,...)**(1,2,3,5,8,...): A001891;
%C A213576 Row 3, (1,2,3,...)**(2,3,5,8,13,...): A033937;
%C A213576 Row 4, (1,2,3,...)**(3,5,8,13,21,...): A033960;
%C A213576 Row 5, (1,2,3,...)**(5,8,13,21,...): A037140;
%C A213576 Row 6, (1,2,3,...)**(8,13,21,34,...): A037157.
%C A213576 For a guide to related arrays, see A213500.
%C A213576 The falling antidiagonal rows can be computed by the sum Sum_{j=0..n-k} (n-k-j+1)*Fibonacci(k+j) which can also be seen as Fibonacci(n+4) - Lucas(k+2) - (n-k)*Fibonacci(k+1). - _G. C. Greubel_, Jul 05 2019
%H A213576 Clark Kimberling, <a href="/A213576/b213576.txt">Antidiagonals n = 1..60, flattened</a>
%F A213576 Rows: T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
%F A213576 Columns: T(n,k) = T(n-1,k) + T(n-2,k).
%F A213576 G.f. for row n: f(x)/g(x), where f(x) = F(n) - F(n-1)*x and g(x) = (1 - x - x^2)*(1 - x)^2.
%F A213576 T(n,k) = F(n+k+3) - k*F(n+1) - F(n+3). - _Ehren Metcalfe_, Jul 04 2019
%e A213576 Northwest corner (the array is read by falling antidiagonals):
%e A213576 1, 3, 7, 14, 26, 46, 79
%e A213576 1, 4, 10, 21, 40, 72, 125
%e A213576 2, 7, 17, 35, 66, 118, 204
%e A213576 3, 11, 27, 56, 106, 190, 329
%e A213576 5, 18, 44, 91, 172, 308, 533
%e A213576 8, 29, 71, 147, 278, 498, 862
%t A213576 (* First Program *)
%t A213576 b[n_]:= n; c[n_]:= Fibonacci[n];
%t A213576 t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213576 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213576 Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213576 *)
%t A213576 r[n_]:= Table[t[n, k], {k,1,40}] (* columns of antidiagonal triangle *)
%t A213576 d = Table[t[n, n], {n, 1, 40}] (* A213577 *)
%t A213576 s[n_]:= Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213576 s1 = Table[s[n], {n, 1, 50}] (* A213578 *)
%t A213576 (* Second Program *)
%t A213576 T[n_, k_]:= Fibonacci[n+4] - (n-k)*Fibonacci[k+1] - LucasL[k+2];
%t A213576 Table[T[n,k], {n,10}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 05 2019 *)
%o A213576 (PARI) T(n, k)= fibonacci(n+4) - (n-k+1)*fibonacci(k+1) - fibonacci(k+3);
%o A213576 for(n=1,10, for(k=1,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jul 05 2019
%o A213576 (Magma) [[Fibonacci(n+4) -(n-k)*Fibonacci(k+1) -Lucas(k+2): k in [1..n]]: n in [1..10]]; // _G. C. Greubel_, Jul 05 2019
%o A213576 (Sage) [[fibonacci(n+4) - (n-k+1)*fibonacci(k+1) - fibonacci(k+3) for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, Jul 05 2019
%o A213576 (GAP) Flat(List([1..10], n-> List([1..n], k-> Fibonacci(n+4) - (n-k+1) *Fibonacci(k+1) - Fibonacci(k+3)))); # _G. C. Greubel_, Jul 05 2019
%Y A213576 Cf. A213500.
%K A213576 nonn,tabl,easy
%O A213576 1,2
%A A213576 _Clark Kimberling_, Jun 18 2012