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A213504
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Principal diagonal of the convolution array A213590.
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4
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1, 6, 35, 138, 488, 1564, 4733, 13734, 38711, 106846, 290496, 781264, 2084753, 5531846, 14619811, 38527834, 101328712, 266119228, 698218525, 1830665830, 4797572551, 12568780126, 32920653120, 86214096768, 225758326273
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = 6*a(n-1) - 10*a(n-2) - 2*a(n-3) + 15*a(n-4) - 2*a(n-5)- 8*a(n-6) + a(n-8).
G.f.: x*(1 + 9*x^2 - 10*x^3 + 7*x^4 - 2*x^5)/((1 - 3*x + x^2)*(1 - x - x^2)^3). [corrected by Georg Fischer, May 11 2019]
a(n) = Fibonacci(2*n+6) - Fibonacci(n+6) - 2*n*Fibonacci(n+3) - n^2*Fibonacci(n+1). - G. C. Greubel, Jul 06 2019
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MATHEMATICA
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(* First program *)
b[n_]:= n^2; 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}]] (* A213590 *)
r[n_]:= Table[T[n, k], {k, 40}] (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213504 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213557 *)
(* Second program *)
With[{F = Fibonacci}, Table[F[2*n+6] -F[n+6] -2*n*F[n+3] -n^2*F[n+1], {n, 40}]] (* G. C. Greubel, Jul 06 2019 *)
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PROG
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(PARI) vector(40, n, my(f=fibonacci); f(2*n+6) - f(n+6) - 2*n*f(n+3) - n^2*f(n+1)) \\ G. C. Greubel, Jul 06 2019
(Magma) F:=Fibonacci; [F(2*n+6) -F(n+6) -2*n*F(n+3) -n^2*F(n+1): n in [1..40]]; // G. C. Greubel, Jul 06 2019
(Sage) f=fibonacci; [f(2*n+6) -f(n+6) -2*n*f(n+3) -n^2*f(n+1) for n in (1..40)] # G. C. Greubel, Jul 06 2019
(GAP) F:=Fibonacci;; List([1..40], n-> F(2*n+6) -F(n+6) -2*n*F(n+3) -n^2*F(n+1)) # G. C. Greubel, Jul 06 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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