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 A213504 Principal diagonal of the convolution array A213590. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Clark Kimberling, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (6,-10,-2,15,-2,-8,0,1). FORMULA 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 MATHEMATICA (* 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 *) PROG (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 CROSSREFS Cf. A213590, A213500. Sequence in context: A101077 A094952 A024526 * A089581 A132657 A161784 Adjacent sequences: A213501 A213502 A213503 * A213505 A213506 A213507 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jun 19 2012 STATUS approved

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Last modified September 9 07:24 EDT 2024. Contains 375762 sequences. (Running on oeis4.)