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A113122 Sum of the first n Fibonacci numbers, in ascending order, as bases, with the same, in descending order, as exponents. 18
1, 2, 4, 7, 14, 32, 107, 724, 18616, 4117597, 28878084584, 53183366452504936, 794001316484619940422835765, 25210343943654420841949267608211227900299990, 14311021641196256564899251685012421154803682074917148917844556724305980 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Primes begin a(2) = 2, a(4) = 7, a(7) = 107; what is the next prime? This transform can be reflexively applied to any integer sequence which does not give an indeterminate 0^0 term.
LINKS
FORMULA
a(n) = Sum_{i=1..n} F(i)^F(n-i+1).
EXAMPLE
a(1) = F(1)^F(1) = 1^1 = 1.
a(2) = F(1)^F(2) + F(2)^F(1) = 1^1 + 1^1 = 2.
a(3) = F(1)^F(3) + F(2)^F(2) + F(3)^F(1) = 1^2 + 1^1 + 2^1 = 4.
a(4) = F(1)^F(4) + F(2)^F(3) + F(3)^F(2) + F(4)^F(1) = 1^3 + 1^2 + 2^1 + 3^1 = 7.
a(5) = 1^5 + 1^3 + 2^2 + 3^1 + 5^1 = 14.
a(6) = 1^8 + 1^5 + 2^3 + 3^2 + 5^1 + 8^1 = 32.
a(7) = 1^13 + 1^8 + 2^5 + 3^3 + 5^2 + 8^1 + 13^1 = 107.
a(8) = 1^21 + 1^13 + 2^8 + 3^5 + 5^3 + 8^2 + 13^1 + 21^1 = 724.
a(9) = 1^34 + 1^21 + 2^13 + 3^8 + 5^5 + 8^3 + 13^2 + 21^1 + 34^1 = 18616.
a(10) = 1^55 + 1^34 + 2^21 + 3^13 + 5^8 + 8^5 + 13^3 + 21^2 + 34^1 + 55^1 = 4117597.
a(11) = 1^89 + 1^55 + 2^34 + 3^21 + 5^13 + 8^8 + 13^5 + 21^3 + 34^2 + 55^1 + 89^1 = 28878084584.
a(12) = 1^144 + 1^89 + 2^55 + 3^34 + 5^21 + 8^13 + 13^8 + 21^5 + 34^3 + 55^2 + 89^1 + 144^1 = 53183366452504936.
a(13) = 1^233 + 1^144 + 2^89 + 3^55 + 5^34 + 8^21 + 13^13 + 21^8 + 34^5 + 55^3 + 89^2 + 144^1 + 233^1 = 794001316484619940422835765.
a(14) = 1^377 + 1^233 + 2^144 + 3^89 + 5^55 + 8^34 + 13^21 + 21^13 + 34^8 + 55^5 + 89^3 + 144^2 + 233^1 + 377^1 = 25210343943654420841949267608211227900299990.
MAPLE
F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= n-> add(F(i)^F(n-i+1), i=1..n):
seq(a(n), n=1..16); # Alois P. Heinz, Aug 09 2018
MATHEMATICA
Table[Sum[(Fibonacci[k])^((Fibonacci[n - k + 1]), {k, 1, n}], {n, 1, 10}] (* G. C. Greubel, May 18 2017 *)
PROG
(PARI) for(n=1, 10, print1(sum(k=1, n, (fibonacci(k))^(fibonacci(n-k+1))), ", ")) \\ G. C. Greubel, May 18 2017
CROSSREFS
Cf. A000045.
Sequence in context: A074663 A325303 A356781 * A296984 A116584 A152477
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Jan 04 2006
STATUS
approved

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Last modified March 28 16:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)