'And the days of' was used by the BBC's Hogmanay Party but I haven't seen that printed anywhere. Burns' version is 'for auld lang syne'. In Chambers' Songs of Scotland prior to Burns, many verses of an old song, pre 1700, called Old Long Syne is printed and he observes that Burns produced another song based on this, partly his own composition. However in Johnson's Scots Musical Museum Burns is recorded as saying that he found this 'fragment' which is reproduced there, no mention of his adding to it. He makes it quite clear where he added or altered other songs in The Museum. He used an old Lowland tune called Auld Lang Syne but later it was published by Thomson with the air we know today, 'I feed a Lass at Martinmas'.