OFFSET
1,2
COMMENTS
The numbers that are summed are the terms t of A247648 in the range 2^(n-1) <= t < 2^n.
There are Fibonacci(n) of these numbers (per Grimaldi's exercise, in which closed walks on the u-v graph there are a 1 bit at a visit to u and a 0 bit at a visit to v), and this allows recurrences etc. for a(n).
REFERENCES
R. Grimaldi, (2012). Fibonacci and Catalan Numbers: An Introduction, page 80, Example 12.1.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..1000
Iskender Ozturk, Walking paths.
Index entries for linear recurrences with constant coefficients, signature (3,3,-6,-4).
FORMULA
a(n) = a(n-1) + a(n-2) + F(n)*2^(n-1).
a(n) = 3*a(n-1) + 3*a(n-2) - 6*a(n-3) - 4*a(n-4).
a(n) = F(n)*(2^n-1) - Sum_{i=1..n-1} F(i)*F(n-i-1)*2^(n-i-1).
G.f.: x/((1 - x - x^2)*(1 - 2*x - 4*x^2)).
E.g.f.: 2*(exp(x)*(sqrt(5)*cosh(sqrt(5)*x) + 7*sinh(sqrt(5)*x)) - exp(x/2)*(sqrt(5)*cosh(sqrt(5)*x/2) + 4*sinh(sqrt(5)*x/2)))/(11*sqrt(5)). - Stefano Spezia, Jun 19 2024
EXAMPLE
For n=5, the terms of A247648 that are in the interval [16, 31] are 21, 23, 27, 29, and 31, so a(5) = 21+23+27+29+31 = 131.
MATHEMATICA
LinearRecurrence[{3, 3, -6, -4}, {1, 3, 12, 39}, 30] (* Paolo Xausa, Jun 19 2024 *)
PROG
(PARI) Vec(x/((1 - x - x^2)*(1 - 2*x - 4*x^2)) + O(x^40)) \\ Michel Marcus, Jun 16 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Iskender Ozturk, Melike Caliskan, Betül Küçükgök, Ecem Yanik, Irem Türker, Rüya Kılıçarslan, Jun 11 2024
STATUS
approved