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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

Table of n, a(n) for n=1..15.

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: A013326 A202973 A074663 * A296984 A116584 A152477

Adjacent sequences:  A113119 A113120 A113121 * A113123 A113124 A113125

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Jan 04 2006

STATUS

approved

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Last modified November 18 07:55 EST 2018. Contains 317279 sequences. (Running on oeis4.)