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.
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, 29, 13, 133, 125, 118, 106, 91, 71, 47, 21, 221, 212, 204, 190, 172, 147, 115, 76, 34, 364, 354, 345, 329, 308, 278, 238, 186, 123, 55, 596, 585, 575, 557, 533, 498, 450, 385, 301, 199, 89
Offset: 1
Examples
Northwest corner (the array is read by falling antidiagonals): 1, 3, 7, 14, 26, 46, 79 1, 4, 10, 21, 40, 72, 125 2, 7, 17, 35, 66, 118, 204 3, 11, 27, 56, 106, 190, 329 5, 18, 44, 91, 172, 308, 533 8, 29, 71, 147, 278, 498, 862
Links
- Clark Kimberling, Antidiagonals n = 1..60, flattened
Crossrefs
Cf. A213500.
Programs
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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
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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
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Mathematica
(* First Program *) b[n_]:= n; c[n_]:= Fibonacci[n]; t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}] TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]] Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213576 *) r[n_]:= Table[t[n, k], {k,1,40}] (* columns of antidiagonal triangle *) d = Table[t[n, n], {n, 1, 40}] (* A213577 *) s[n_]:= Sum[t[i, n + 1 - i], {i, 1, n}] s1 = Table[s[n], {n, 1, 50}] (* A213578 *) (* Second Program *) T[n_, k_]:= Fibonacci[n+4] - (n-k)*Fibonacci[k+1] - LucasL[k+2]; Table[T[n,k], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Jul 05 2019 *) -
PARI
T(n, k)= fibonacci(n+4) - (n-k+1)*fibonacci(k+1) - fibonacci(k+3); for(n=1,10, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 05 2019
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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
Formula
Rows: T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
Columns: T(n,k) = T(n-1,k) + T(n-2,k).
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.
T(n,k) = F(n+k+3) - k*F(n+1) - F(n+3). - Ehren Metcalfe, Jul 04 2019
Comments