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A213578
Antidiagonal sums of the convolution array A213576.
3
1, 4, 13, 34, 80, 174, 359, 712, 1371, 2580, 4768, 8684, 15629, 27852, 49225, 86390, 150704, 261530, 451795, 777360, 1332791, 2277864, 3882048, 6599064, 11191705, 18940564, 31992709, 53943562, 90807056, 152631750, 256190783
OFFSET
1,2
FORMULA
a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) - a(n-6).
G.f.: (1 + x^2)/(1 - 2*x + x^3)^2.
a(n) = n*F(n+4) - 2*(F(n+5) - n - 5), F = A000045. - Ehren Metcalfe, Jul 05 2019
MATHEMATICA
b[n_]:= n; 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}]] (* A213576 *)
r[n_] := Table[t[n, k], {k, 40}] (* columns of antidiagonal triangle *)
d = Table[t[n, n], {n, 1, 40}] (* A213577 *)
s[n_] := Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213578 *)
(* alternate program *)
LinearRecurrence[{4, -4, -2, 4, 0, -1}, {1, 4, 13, 34, 80, 174}, 40] (* Harvey P. Dale, Jul 04 2019 *)
PROG
(Magma) [n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5): n in [1..40]]; // Vincenzo Librandi, Jul 05 2019
(PARI) vector(40, n, n*fibonacci(n+4)-2*(fibonacci(n+5)-n-5)) \\ G. C. Greubel, Jul 05 2019
(Sage) [n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5) for n in (1..40)] # G. C. Greubel, Jul 05 2019
(GAP) List([1..40], n-> n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5)) # G. C. Greubel, Jul 05 2019
CROSSREFS
Sequence in context: A101946 A029860 A262200 * A212149 A357284 A208740
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 18 2012
STATUS
approved