A213579 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.
1, 3, 2, 7, 5, 3, 14, 11, 7, 4, 26, 21, 15, 9, 5, 46, 38, 28, 19, 11, 6, 79, 66, 50, 35, 23, 13, 7, 133, 112, 86, 62, 42, 27, 15, 8, 221, 187, 145, 106, 74, 49, 31, 17, 9, 364, 309, 241, 178, 126, 86, 56, 35, 19, 10, 596, 507, 397, 295, 211, 146, 98, 63, 39, 21
Offset: 1
Examples
Northwest corner (the array is read by falling antidiagonals): 1....3....7....14...26...46 2....5....11...21...38...66 3....7....15...28...50...86 4....9....19...35...62...106 5....11...23...42...74...126 6....13...27...49...86...146
Links
- Clark Kimberling, Antidiagonas n = 1..60, flattened
Programs
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GAP
Flat(List([1..12], n-> List([1..n], k-> Fibonacci(k+3) + n*Fibonacci(k+2) -(n+k+2) ))); # G. C. Greubel, Jul 08 2019
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Magma
[[Fibonacci(k+3) + n*Fibonacci(k+2) -(n+k+2): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019
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Mathematica
(* First program *) b[n_]:= Fibonacci[n]; c[n_]:= 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}]] (* A213579 *) r[n_]:= Table[T[n, k], {k, 40}] d = Table[T[n, n], {n, 1, 40}] (* A213580 *) s[n_]:= Sum[T[i, n+1-i], {i, 1, n}] s1 = Table[s[n], {n, 1, 50}] (* A053808 *) (* Second program *) Table[Fibonacci[n-k+4] +k*Fibonacci[n-k+3] -(n+3), {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *) -
PARI
t(n,k) = fibonacci(n-k+4) + k*fibonacci(n-k+3) - (n+3); for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019
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Sage
[[fibonacci(k+3) + n*fibonacci(k+2) -(n+k+2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019
Formula
T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = n - (n-1)*x and g(x) = (1-x-x^2) *(1-x)^2.
T(n, k) = Fibonacci(k+3) + n*Fibonacci(k+2) - (n+k+2). - G. C. Greubel, Jul 08 2019
Comments