A350177 a(n) is the number of nonnegative integers that can be represented by lighting only n segments on a 9-segment display, used by the Russian postal service.

0, 0, 0, 2, 3, 2, 5, 13, 17, 22, 47, 86, 127, 211, 387, 645, 1044, 1794, 3086, 5135, 8608, 14674, 24805, 41631, 70322, 119069, 200768, 338429, 571845, 965823, 1629253, 2749904, 4643876, 7838862, 13229487, 22333638, 37704236, 63642469, 107427241, 181351098, 306133271

Offset: 0

Author

Stefano Spezia, Dec 18 2021

Keywords

• nonn
• base
• easy

Comments

The nonnegative integers are displayed as in A350131.

Given the set S = {3, 4, 5, 6, 7}, the function f defined in S as f(3) = f(5) = 2, f(4) = 3, and f(6) = f(7) = 1, a(n) is equal to the difference between the number b(n) of S-restricted f-weighted integer compositions of n with that of n-6, i.e., b(n-6). The latter one provides the number of all those excluded cases where a nonnegative integer is displayed with leading zeros. b(n) is calculated as the sum of polynomial coefficients or extended binomial coefficients (see Equation 3 in Eger) where the index of summation is positive and it covers the numbers of possible digits that can be displayed by n segments (see first formula).

Examples

			a(6) = 5 since 0, 11, 17, 71 and 77 are displayed by 6 segments.
   _                 _    _        _  _
  | |    /| /|    /| /    /  /|    /  /
  |_|     |  |     | |    |   |    |  |
  (0)     (11)    (17)     (71)    (77)
		

Links

• Steffen Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients, Journal of Integer Sequences, Vol. 16, Article 13.1.3, (2013).
• Wikipedia, Postal code template.
• Wikipedia, Postal codes in Russia.
• Index entries for linear recurrences with constant coefficients, signature (0,0,2,3,2,1,1).
• Index entries for sequences related to calculator display
• Index entries for sequences related to compositions

Crossrefs

Cf. A002426, A004526, A331529, A331530, A343314, A343315, A350131.

Programs

Mathematica

P[x_]:=2x^3+3x^4+2x^5+x^6+x^7; b[n_]:=Coefficient[Sum[P[x]^k, {k, Max[1, Ceiling[n/7]], Floor[n/2]}], x, n]; a[n_]:=b[n]-b[n-6]; Array[a, 41, 0]

Formula

a(n) = b(n) - b(n-6), where b(n) = [x^n] Sum_{k=max(1,ceiling(n/7))..floor(n/2)} P(x)^k with P(x) = 2*x^3 + 3*x^4 + 2*x^5 + x^6 + x^7.

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

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