A288492 Indices of terms of A288349 that are powers of 2.

1, 2, 3, 18, 95, 440, 1897, 7882, 32139, 129804, 521741, 2092046, 8378383, 33533968, 134176785, 536789010, 2147319827, 8589606932, 34359083029, 137437642774, 549753192471, 2199018012696, 8796082536473, 35184351117338, 140737446412315, 562949869535260

Offset

1,2

Comments

The sequence is derived from Chinese 2017 college entrance examination mathematics questions.

Links

Zhining Yang, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (8,-21,22,-8).

Formula

From Colin Barker, Jun 23 2017: (Start)

G.f.: x*(1 - 6*x + 8*x^2 + 14*x^3 - 22*x^4 + 8*x^5) / ((1 - x)^2*(1 - 2*x)*(1 - 4*x)).

a(n) = 2 - 5*2^(n-2) + 2^(2*n-3) + n for n>2.

a(n) = 8*a(n-1) - 21*a(n-2) + 22*a(n-3) - 8*a(n-4) for n>6.

(End)

Example

a(4) = 18 means the 18th element of the sum of the concatenate subsequences [2^0, 2^1, ..., 2^k] = 1+1+2+1+2+4+1+2+4+8+1+2+4+8+16+1+2+4 = 64, and 64 is power of 2.

Mathematica

Position[Accumulate@ Flatten@ Array[2^Range[0, #] &, 2000, 0], k_ /; IntegerQ@ Log2@ k][[All, 1]] (* per Name, or *)
Table[2 - 5*2^(n - 2) + 2^(2 n - 3) + n + Boole[n == 2], {n, 26}] (* or *)
LinearRecurrence[{8, -21, 22, -8}, {1, 2, 3, 18, 95, 440}, 26] (* or *)
Rest@ CoefficientList[Series[x (1 - 6 x + 8 x^2 + 14 x^3 - 22 x^4 + 8 x^5)/((1 - x)^2*(1 - 2 x) (1 - 4 x)), {x, 0, 26}], x] (* Michael De Vlieger, Jun 19 2017 *)

Prog

(PARI) for(k=0, 100, p=(2^k-3)*(2^k-2)/2+k; print1(p, ", "))
(PARI) ispower2(n) = (n==1) || (n==2) || (ispower(n, , &two) && (two==2));
lista(nn) = select(x->ispower2(x), vector(nn, n, t=floor(sqrt(2*n)+1/2); 2^t+2^(n-t*(t-1)/2)-t-2), 1); \\ Michel Marcus, Jun 20 2017
(PARI) Vec(x*(1 - 6*x + 8*x^2 + 14*x^3 - 22*x^4 + 8*x^5) / ((1 - x)^2*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jun 23 2017

Crossrefs

Cf. A059268, A288349.

Sequence in context: A048047 A114165 A166510 * A184719 A076016 A226961

Adjacent sequences: A288489 A288490 A288491 * A288493 A288494 A288495

Keyword

nonn,easy

Author

Zhining Yang, Jun 10 2017

Status

approved