A352340 a(n) is the sum of digits of n in the maximal Pell representation of n (A352339).

0, 1, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 3, 4, 4, 5, 3, 4, 5, 3, 4, 4, 5, 6, 4, 5, 5, 6, 4, 5, 6, 4, 5, 5, 6, 7, 5, 6, 6, 7, 8, 4, 5, 5, 6, 4, 5, 6, 4, 5, 5, 6, 7, 5, 6, 6, 7, 5, 6, 7, 5, 6, 6, 7, 8, 6, 7, 7, 8, 9, 5, 6, 6, 7, 5, 6, 7, 5, 6, 6, 7, 8, 6, 7, 7, 8, 6

Offset: 0

Author

Amiram Eldar, Mar 12 2022

Keywords

• nonn
• base

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• A. F. Horadam, Maximal representations of positive integers by Pell numbers, The Fibonacci Quarterly, Vol. 32, No. 3 (1994), pp. 240-244.

Crossrefs

Cf. A000129, A007953, A352339.

Similar sequences: A053735, A007895, A112310, A265744, A278043, A352104.

Programs

Mathematica

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellp[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerDigits[Total[3^(s - 1)], 3]]; a[n_] := Module[{v = pellp[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] > 0 && v[[i + 1]] == 0 && v[[i + 2]] < 2, v[[i ;; i + 2]] += {-1, 2, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, Total[v[[i[[1, 1]] ;; -1]]]]]; Array[a, 100, 0]

Formula

a(n) = A007953(A352339(n)).

a(n) >= A265744(n).