A331530 a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A010371).

0, 0, 1, 0, 3, 3, 8, 7, 16, 23, 48, 70, 125, 192, 345, 561, 972, 1578, 2683, 4436, 7537, 12536, 21114, 35163, 59123, 98837, 166006, 277650, 465619, 779296, 1306674, 2188248, 3667717, 6142653, 10293460, 17242678, 28892956, 48402553, 81099234, 135863965, 227636213

Offset

0,5

Comments

The nonnegative integers are displayed as in A010371, where a 7 is depicted by 4 segments.

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

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 (0,1,0,2,3,3,1).

Index entries for sequences related to calculator display

Index entries for sequences related to compositions

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

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

(End)

Example

a(6) = 8 since 0, 6, 9, 14, 17, 41, 71, 111 are displayed by 6 segments.
   __       __      __
  |  |     |__     |__|     |  |__|
  |__|     |__|     __|     |     |
  (0)      (6)      (9)       (14)
     __                   __
  | |  |     |__|  |     |  |  |    |  |  |
  |    |        |  |        |  |    |  |  |
   (17)        (41)        (71)      (111)

Mathematica

P[x_]:=x^2+2x^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]

Prog

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

Crossrefs

Cf. A002426, A004526, A010371, A331529, A343314, A343315.

Sequence in context: A021751 A302675 A305841 * A199624 A363220 A093366

Adjacent sequences: A331527 A331528 A331529 * A331531 A331532 A331533

Keyword

base,nonn,easy

Author

Stefano Spezia, Jan 19 2020

Status

approved