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 A298172 Decimal expansion of limiting power-ratio for A296776; see Comments. 3
 1, 1, 6, 5, 8, 3, 7, 7, 4, 8, 5, 0, 5, 6, 4, 6, 6, 6, 8, 1, 7, 0, 8, 6, 8, 2, 6, 0, 6, 2, 9, 3, 9, 4, 9, 4, 7, 3, 9, 2, 0, 5, 5, 6, 7, 8, 1, 6, 7, 0, 6, 2, 8, 1, 8, 0, 6, 9, 4, 5, 7, 6, 0, 9, 0, 5, 4, 8, 1, 9, 3, 4, 6, 0, 0, 2, 0, 5, 9, 7, 2, 8, 1, 3, 5, 9 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 2,3 COMMENTS Suppose that A = (a(n)), for n >= 0, is a sequence, and g is a real number such that a(n)/a(n-1) -> g. The limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A296776, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296462 for related sequences. LINKS EXAMPLE limiting power-ratio = 11.65837748505646668170868260629394947392... MATHEMATICA a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 2 n; j = 1; While[j < 16, k = a[j] - j - 1; While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; u = Table[a[n], {n, 0, k}];  (* A296776 *) z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}]; StringJoin[StringTake[ToString[h[[z]]], 41], "..."] Take[RealDigits[Last[h], 10][[1]], 120]   (* A298172 *) CROSSREFS Cf. A001622, A296462, A296776. Sequence in context: A147313 A242759 A021607 * A225113 A329247 A133618 Adjacent sequences:  A298169 A298170 A298171 * A298173 A298174 A298175 KEYWORD nonn,easy,cons AUTHOR Clark Kimberling, Feb 09 2018 STATUS approved

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Last modified June 23 13:00 EDT 2021. Contains 345401 sequences. (Running on oeis4.)