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A213585 Principal diagonal of the convolution array A213584. 3
1, 7, 22, 54, 116, 232, 443, 821, 1490, 2664, 4710, 8256, 14373, 24883, 42878, 73594, 125880, 214664, 365087, 619425, 1048666, 1771852, 2988362, 5031744, 8459401, 14201887, 23811238, 39873726, 66695420, 111440104, 186016835 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
FORMULA
a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) + a(n-5).
G.f.: x*(1 + 3*x - 2*x^2 - 4*x^3 - 2*x^4)/(1 - 2*x + x^3)^2.
a(n) = Fibonacci(n+4) + n*Fibonacci(n+3) - (4*n + 3). - G. C. Greubel, Jul 08 2019
MATHEMATICA
(* First program *)
b[n_]:= Fibonacci[n+1]; 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}]] (* A213584 *)
r[n_]:= Table[T[n, k], {k, 40}] (* columns of antidiagonal triangle *)
d = Table[T[n, n], {n, 1, 40}] (* A213585 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213586 *)
(* Second program *)
Table[Fibonacci[n+4] + n*Fibonacci[n+3] -4*n-3, {n, 40}] (* G. C. Greubel, Jul 08 2019 *)
PROG
(PARI) vector(40, n, f=fibonacci; f(n+4) +n*f(n+3) -(4*n+3)) \\ G. C. Greubel, Jul 08 2019
(Magma) F:=Fibonacci; [F(n+4) +n*F(n+3) -(4*n+3): n in [1..40]]; // G. C. Greubel, Jul 08 2019
(Sage) f=fibonacci; [f(n+4) +n*f(n+3) -(4*n+3) for n in (1..40)] # G. C. Greubel, Jul 08 2019
(GAP) F:=Fibonacci;; List([1..40], n-> F(n+4) +n*F(n+3) -(4*n+3)) # G. C. Greubel, Jul 08 2019
CROSSREFS
Sequence in context: A101120 A151717 A188377 * A308579 A246831 A354430
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 18 2012
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)