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A172173
Sums of NE-SW diagonals of triangle A172171.
1
0, 1, 1, 11, 12, 32, 44, 85, 129, 223, 352, 584, 936, 1529, 2465, 4003, 6468, 10480, 16948, 27437, 44385, 71831, 116216, 188056, 304272, 492337, 796609, 1288955, 2085564, 3374528, 5460092, 8834629, 14294721, 23129359, 37424080, 60553448, 97977528, 158530985
OFFSET
0,4
FORMULA
For n=even: a(n) = a(n-1) + a(n-2); for n=odd: a(n) = a(n-1) + a(n-2) + 9 ; with a(0) = 0 and a(1) = 1.
From Colin Barker, Feb 18 2013: (Start)
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>3.
G.f.: x*(1+8*x^2) / ((1-x)*(1+x)*(1-x-x^2)).
(End)
a(n) = (2^(-1-n)*(-45*((-2)^n+2^n) + (45-7*sqrt(5))*(1+sqrt(5))^n + (1-sqrt(5))^n*(45+7*sqrt(5)))) / 5. - Colin Barker, Jul 13 2017
a(n) = Fibonacci(n+1) + 8*Fibonacci(n-1) - 9*((1+(-1)^n)/2). - G. C. Greubel, Apr 25 2022
MATHEMATICA
CoefficientList[Series[x*(1+8*x^2)/((1-x^2)*(1-x-x^2)), {x, 0, 50}], x] (* G. C. Greubel, Jul 13 2017 *)
PROG
(PARI) concat(0, Vec(x*(1+8*x^2)/((1-x)*(1+x)*(1-x-x^2)) + O(x^50))) \\ Colin Barker, Jul 13 2017
(Magma) [Lucas(n) +7*Fibonacci(n-1) -9*((n+1) mod 2): n in [0..50]]; // G. C. Greubel, Apr 25 2022
(Sage) [fibonacci(n+1) +8*fibonacci(n-1) -9*((n+1)%2) for n in (0..50)] # G. C. Greubel, Apr 25 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mark Dols, Jan 28 2010
EXTENSIONS
Offset corrected by Colin Barker, Feb 18 2013
STATUS
approved