A172173 Sums of NE-SW diagonals of triangle A172171.
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
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).
Programs
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Magma
[Lucas(n) +7*Fibonacci(n-1) -9*((n+1) mod 2): n in [0..50]]; // G. C. Greubel, Apr 25 2022
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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 *) -
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
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Sage
[fibonacci(n+1) +8*fibonacci(n-1) -9*((n+1)%2) for n in (0..50)] # G. C. Greubel, Apr 25 2022
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
Extensions
Offset corrected by Colin Barker, Feb 18 2013