A350437 a(n) is the number of integers that can be represented in a 7-segment display by using only n segments (version A006942).

0, 0, 1, 2, 3, 7, 12, 18, 31, 52, 92, 158, 269, 460, 786, 1350, 2317, 3969, 6798, 11643, 19952, 34197, 58601, 100410, 172042, 294791, 505143, 865589, 1483206, 2541480, 4354847, 7462119, 12786520, 21909974, 37543133, 64330800, 110232005, 188884671, 323657539, 554593317

Offset: 0

Author

Stefano Spezia, Dec 31 2021

Keywords

• nonn
• base
• easy

Comments

The integers are displayed as in A006942, where a 7 is depicted by 3 segments. The negative integers are depicted by using 1 segment more for the minus sign.

Since the integer 0 is depicted by 6 segments, in order to avoid considering -0 in the case n = 7, a(7) is obtained by decreasing of a unit the result of the sum A331529(7) + A331529(6) = 12 + 7 = 19, i.e., a(7) = 19 - 1 = 18.

The same sequence is obtained when 7 and 9 are depicted respectively by 4 and 5 segments (A074458).

Examples

			a(7) = 18 since -111, -77, -41, -14, -9, -6, 8, 12, 13, 15, 21, 31, 47, 51, 74, 117, 171 and 711 are displayed by 7 segments.
segments.
                       __   __                                      __
   __   |  |  |     __   |    |    __ |__|  |    __   | |__|    __ |__|
        |  |  |          |    |          |  |         |    |        __|
        (-111)          (-77)         (-41)          (-14)        (-9)
       __      __        __         __         __      __        __
   __ |__     |__|    |  __|     |  __|     | |__      __|  |    __|  |
      |__|    |__|    | |__      |  __|     |  __|    |__   |    __|  |
     (-6)      (8)     (12)       (13)       (15)       (21)      (31)
        __      __        __               __       __        __
  |__|    |    |__   |      | |__|    |  |   |    |   |  |      |  |  |
     |    |     __|  |      |    |    |  |   |    |   |  |      |  |  |
     (47)        (51)        (74)       (117)       (171)        (711)
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,3,3,1).
• Index entries for sequences related to calculator display
• Index entries for sequences related to compositions

Crossrefs

Cf. A006942, A074458, A331529.

Programs

Mathematica

P[x_]:=x^2+x^3+x^4+3x^5+3x^6+x^7; c[n_]:=Coefficient[Sum[P[x]^k, {k, Max[1, Ceiling[n/7]], Floor[n/2]}], x, n]; b[n_]:=c[n]-c[n-6]; (* A331529 *)
a[n_]:=If[n!=7,b[n]+b[n-1],18]; Array[a,40,0]

Formula

a(7) = 18, otherwise a(n) = A331529(n) + A331529(n-1).

G.f.: x^2*(1 + 2*x + 2*x^2 + 4*x^3 + 6*x^4 + 3*x^5 - x^7 - x^8 - 3*x^9 - 3*x^10 - x^11)/(1 - x^2 - x^3 - x^4 - 3*x^5 - 3*x^6 - x^7).

a(n) = a(n-2) + a(n-3) + a(n-4) + 3*a(n-5) + 3*a(n-6) + a(n-7) for n > 13.