A029549 formula: (cosh((4*n+2)*log(1+sqrt(2)))-3)/16 . = binomial(A001652+1,2) = 2*binomial(A053141+1,2). (Correcting(?) M. Harris formula.) Mathematica: Intersection[#,2*#]&@Table[Binomial[n,2],{n,999999}] Macsyma: (makelist(binom(n,2),n,1,999999),intersection(%%,2*%%)) -------------------------------- A001652 formula: (sinh((2*n+1)*log(1+sqrt(2)))-1)/2 = (sqrt(1+8*A029549)-1)/2 binomial(A001652+1,2) = 2*binomial(A053141+1,2) = A029549 . See A053141. ------------------- A053141 formula: sqrt(2)*cosh((2*n+1)*log(1+sqrt(2))/4 - 1/2 = (sqrt(1+4*A029549)-1)/2 . Jason Holt observes: A pair drawn from a drawer with A053141(n)+1 red socks and A001652(n) - A053141(n) blue socks will likely as not be matching reds: A053141+1 A053141 1 --------- ------- = -, n>0. A001652+1 A001652 2 --rwg CHINA TREES CHINESE RAT CATHERINE'S CHESTERIAN HERITANCES