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Though the statement is similar to Ozsv\\'ath-Szab\\'o's integral surgery formula for Heegaard Floer homology, the proof is new and based on sutured instanton homology $SHI$ and the octahedral lemma in the derived category. As a corollary, we obtain an exact triangle between $I^\\sharp(Y_m(K))$, $I^\\sharp(Y_{m+k}(K))$ and $k$ copies of $I^\\sharp(Y)$ for any $m\\neq 0$ and large $k$. 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Though the statement is similar to Ozsv\\'ath-Szab\\'o's integral surgery formula for Heegaard Floer homology, the proof is new and based on sutured instanton homology $SHI$ and the octahedral lemma in the derived category. As a corollary, we obtain an exact triangle between $I^\\sharp(Y_m(K))$, $I^\\sharp(Y_{m+k}(K))$ and $k$ copies of $I^\\sharp(Y)$ for any $m\\neq 0$ and large $k$. 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