A328352 Similar to A328350, but for 5 digits rather then 3.

0, 1, 56, 2831, 141706, 7086081, 354307956, 17715417331, 885770964206, 44288548698581, 2214427437370456, 110721371880729831, 5536068594097526706, 276803429705181511081, 13840171485260601432956, 692008574263037701042331, 34600428713151923199089206, 1730021435657596350689323581

Offset

0,3

Comments

This sequence is the building block for the calculation of the sums of positive integers whose decimal expansion only uses five distinct, nonzero digits: see the attached pdf document.

Links

Pierre-Alain Sallard, Table of n, a(n) for n = 0..50

Pierre-Alain Sallard, Integers sequences A328348 and A328350 to A328356

Index entries for linear recurrences with constant coefficients, signature (56, -305, 250).

Formula

a(n) = (40*50^n - 49*5^n + 9) / 1764.

a(n) = 51*a(n-1) - 50*a(n-2) + 5^(n-1) for n > 1.

G.f.: x / (1 - 56*x + 305*x^2 - 250*x^3).

a(n) = 56*a(n-1) - 305*a(n-2) + 250*a(n-3) for n > 2.

Example

For n=2, the sum of all positive integers whose decimal notation is only made of the 3,4,5,6 and 7 digit with at most n=2 such digits, i.e. the sum 3+4+5+6+7+33+34+35+36+37+43+44+45+46+47+53+54+55+56+57+63+64+65+66+67+73+74+75+76+77 is equal to a(2)*(3+4+5+6+7) = 56*25 = 1400.
The formula is valid for any other 5-tuple of digits, as soon as the 5 digits are different from each other. Always with n=2 but let's say with the 5,6,7,8 and 9 digits, the sum 5+6+7+8+9+55+56+57+58+59+65+66+67+68+69+75+76+77+78+79+85+86+87+88+89+95+96+97+98+99 is equal to a(2)*(5+6+7+8+9) = 56*35 = 1960.

Prog

(Python) [(40*50**n-49*5**n+9)//1764 for n in range(12)]

Crossrefs

Cf. A328348, A328350, A328351, A328353, A328354, A328355, A328356.

Sequence in context: A003696 A199709 A205227 * A224176 A223868 A223990

Adjacent sequences: A328349 A328350 A328351 * A328353 A328354 A328355

Keyword

nonn,base

Author

Pierre-Alain Sallard, Nov 26 2019

Status

approved