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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 May 19 18:35 EDT 2022. Contains 353847 sequences. (Running on oeis4.)