A393532 The difference between the maximum and minimum voucher costs for n distinct positive integer vouchers v_1, v_2, ..., v_n, where the cost is defined as v_1 + v_1*v_2 + v_2*v_3 + ... + v_(n-1)*v_n.

0, 1, 4, 10, 22, 39, 64, 96, 138, 189, 252, 326, 414, 515, 632, 764, 914, 1081, 1268, 1474, 1702, 1951, 2224, 2520, 2842, 3189, 3564, 3966, 4398, 4859, 5352, 5876, 6434, 7025, 7652, 8314, 9014, 9751, 10528, 11344, 12202, 13101, 14044, 15030, 16062, 17139, 18264

Offset

1,3

Comments

The question originated from the first puzzle 'Low Budget' in the book Mathematical Puzzles and Curiosities.

References

I. David, T. Khovanova, and Y. Shpilman, Mathematical Puzzles and Curiosities, World Scientific, 2026, p. 2.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Formula

a(n) = n*(n^2+3*n-19)/6 + (51-3*(-1)^n)/12, for n > 1.

a(n) = A110610(n) - A393145(n).

G.f.: x^2*(1 + x + 2*x^3 - 2*x^4)/((1 - x)^4*(1 + x)). - Stefano Spezia, Feb 20 2026

From Enrique Navarrete, Feb 26 2026: (Start)

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) for n > 6.

E.g.f.: (1/12)*(2*x^3 + 12*x^2 - 30*x + 51)*exp(x) - (1/4)*exp(-x) - 2*x - 4. (End)

Example

For n=4, the minimum cost is achieved by buying 4 vouchers in the following order: 3, 2, 1, 4. The cost is 3+3*2+2*1+1*4=15. The maximum cost is achieved when we buy 4 vouchers in the following order: 2, 4, 3, 1. The cost is 2+2*4+4*3+3*1=25. Thus, a(4) = 25-15 = 10.

Mathematica

A393532[n_] := If[n == 1, 0, (2*n*(n*(n + 3) - 19) - 3*(-1)^n + 51)/12]; Array[A393532, 50] (* or *)
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 4, 10, 22, 39}, 50] (* Paolo Xausa, Feb 25 2026 *)

Prog

(PARI) a(n)= binomial(n+2, 3) - 7*n\2 + 4 - 2*(1==n); \\ Ruud H.G. van Tol, Mar 04 2026

Crossrefs

Cf. A026035, A101986, A110610, A110611, A393145.

Sequence in context: A339609 A008248 A301243 * A177736 A061777 A298030

Adjacent sequences: A393529 A393530 A393531 * A393533 A393534 A393535

Keyword

nonn,easy

Author

Tanya Khovanova and PRIMES STEP junior group, Feb 18 2026

Status

approved