A343315 a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A277116).

0, 0, 1, 1, 2, 6, 6, 14, 19, 36, 67, 103, 194, 315, 560, 971, 1651, 2895, 4933, 8581, 14798, 25515, 44165, 76067, 131563, 227034, 392032, 677152, 1168742, 2018769, 3485255, 6018422, 10392472, 17943750, 30985861, 53501944, 92385050, 159523542, 275451221, 475633952

Offset

0,5

Comments

The nonnegative integers are displayed as in A277116.

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

Links

Table of n, a(n) for n=0..39.

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 (0,1,1,1,4,2,1).

Index entries for sequences related to calculator display

Index entries for sequences related to compositions

Formula

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

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

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 + 4*x^5 + 2*x^6 + x^7.

Example

a(5) = 6 since 2, 3, 5, 9, 17 and 71 are displayed by 5 segments.
   __        __        __        __          __       __
   __|       __|      |__       |__|      |    |        |    |
  |__        __|       __|         |      |    |        |    |
   (2)       (3)       (5)       (9)       (17)          (71)

Mathematica

P[x_]:=x^2+x^3+x^4+4x^5+2x^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, 40, 0]

Crossrefs

Cf. A002426, A004526, A216261, A277116, A331529, A331530, A343314.

Sequence in context: A051398 A073131 A206490 * A321302 A294735 A367072

Adjacent sequences: A343312 A343313 A343314 * A343316 A343317 A343318

Keyword

nonn,base,easy

Author

Stefano Spezia, Apr 11 2021

Status

approved