A331529 a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A006942).

0, 0, 1, 1, 2, 5, 7, 12, 19, 33, 59, 99, 170, 290, 496, 854, 1463, 2506, 4292, 7351, 12601, 21596, 37005, 63405, 108637, 186154, 318989, 546600, 936606, 1604874, 2749973, 4712146, 8074374, 13835600, 23707533, 40623267, 69608738, 119275933, 204381606, 350211711, 600094277

Offset: 0

Author

Stefano Spezia, Jan 19 2020

Keywords

• base
• nonn
• easy

Comments

The nonnegative integers are displayed as in A006942, where a 7 is depicted by 3 segments.

Given the set S = {2, 3, 4, 5, 6, 7}, the function f defined in S as f(5) = f(6) = 3 and f(s) = 1 elsewhere, 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).

The same sequence is obtained when 7 and 9 are depicted respectively by 4 and 5 segments (A074458). - Stefano Spezia, Apr 11 2021

Examples

			a(5) = 5 since 2, 3, 5, 17 and 71 are displayed by 5 segments.
   __      __       __         __      __
   __|     __|     |__       |   |       |  |
  |__      __|      __|      |   |       |  |
   (2)     (3)      (5)       (17)       (71)
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Steffen Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients, Journal of Integer Sequences, Vol. 16, Article 13.1.3, (2013).
• 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. A002426, A004526, A006942, A216261, A331530, A343314, A343315.

Programs

Mathematica

P[x_]:=x^2+x^3+x^4+3x^5+3x^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]

PARI

concat([0,0], Vec(x^2*(1 - x)*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)*(1 + x^2 + 2*x^3 + x^4) / (1 - x^2 - x^3 - x^4 - 3*x^5 - 3*x^6 - x^7) + O(x^41))) \\ Colin Barker, Jan 20 2020

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) = x^2 + x^3 + x^4 + 3*x^5 + 3*x^6 + x^7.

From Colin Barker, Jan 20 2020: (Start)

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

(End)