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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). 6
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 (list; graph; refs; listen; history; text; internal format)
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
Steffen Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients, Journal of Integer Sequences, Vol. 16, Article 13.1.3, (2013).
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
Sequence in context: A021751 A302675 A305841 * A199624 A363220 A093366
KEYWORD
base,nonn,easy
AUTHOR
Stefano Spezia, Jan 19 2020
STATUS
approved

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Last modified July 23 06:44 EDT 2024. Contains 374544 sequences. (Running on oeis4.)