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Ramanujan's Sum Identity

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Ramanujan's sum identity is an identity for the sequences defined by the generating functions

(1+53x+9x^2)/(1-82x-82x^2+x^3)=sum_(n=0)^(infty)a_nx^n
(1)
(2-26x-12x^2)/(1-82x-82x^2+x^3)=sum_(n=0)^(infty)b_nx^n
(2)
(2+8x-10x^2)/(1-82x-82x^2+x^3)=sum_(n=0)^(infty)c_nx^n
(3)

(OEIS A051028, A051029, and A051030). It states that

a_n^3+b_n^3=c_n^3+(-1)^n
(4)

for nonnegative integers n. Hirschhorn (1995) gave the explicit formulas

a_n=1/(85)[(64+8sqrt(85))alpha^n+(64-8sqrt(85))beta^n-43(-1)^n]
(5)
b_n=1/(85)[(77+7sqrt(85))alpha^n+(77-7sqrt(85))beta^n+16(-1)^n]
(6)
c_n=1/(85)[(93+9sqrt(85))alpha^n+(93-9sqrt(85))beta^n-16(-1)^n],
(7)

where alpha=1/2(83+9sqrt(85)) and beta=1/2(83-9sqrt(85)). Hirschhorn (1996) showed that checking the first seven cases n=0 to 6 is sufficient to prove the result.

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References

Hirschhorn, M. D. "An Amazing Identity of Ramanujan." Math. Mag. 68, 199-201, 1995. https://doi.org/10.1080/0025570X.1995.11996312.Hirschhorn, M. D. "A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan." Math. Mag. 69, 267-269, 1996. https://doi.org/10.1080/0025570X.1996.11996450.Sloane, N. J. A. Sequences A051028, A051029, and A051030 in "The On-Line Encyclopedia of Integer Sequences."

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Ramanujan's Sum Identity

Cite this as:

Weisstein, Eric W. "Ramanujan's Sum Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RamanujansSumIdentity.html

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