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 A322675 a(n) = n * (4*n + 3)^2. 3
 0, 49, 242, 675, 1444, 2645, 4374, 6727, 9800, 13689, 18490, 24299, 31212, 39325, 48734, 59535, 71824, 85697, 101250, 118579, 137780, 158949, 182182, 207575, 235224, 265225, 297674, 332667, 370300, 410669, 453870, 499999, 549152, 601425, 656914, 715715, 777924, 843637 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^3. sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^3. Sum_{n>=1} 1/a(n) = 8/27 + 2*c/3 + Pi/18 - Pi^2/12 - log(2)/3 = 0.027956857336446942649782759291008857522041405948099294509008..., where c is the Catalan constant A006752. - Vaclav Kotesovec, Dec 23 2018 From Colin Barker, Dec 23 2018: (Start) G.f.: x*(49 + 46*x + x^2) / (1 - x)^4. a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End) EXAMPLE (sqrt(2) - sqrt(1))^3 = 5*sqrt(2) - 7 = sqrt(50) - sqrt(49). So a(1) = 49. PROG (PARI) {a(n) = n*(4*n+3)^2} (PARI) concat(0, Vec(x*(49 + 46*x + x^2) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Dec 23 2018 CROSSREFS Column 3 of A322699. sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^k: A033996(n) (k=2), this sequence (k=3), A322677 (k=4), A322745 (k=5). Sequence in context: A266799 A211741 A211761 * A260198 A243904 A017246 Adjacent sequences:  A322672 A322673 A322674 * A322676 A322677 A322678 KEYWORD nonn,easy AUTHOR Seiichi Manyama, Dec 23 2018 STATUS approved

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Last modified April 7 12:07 EDT 2020. Contains 333305 sequences. (Running on oeis4.)