A328348 Let S be any integer in the range 3 <= S <= 17. Sequence has the property that a(n)*S is the sum of all positive integers whose decimal expansion has <= n digits and contains at most two distinct nonzero digits p and q such that p+q=S.

0, 1, 23, 467, 9355, 187131, 3742683, 74853787, 1497075995, 29941520411, 598830409243, 11976608186907, 239532163742235, 4790643274852891, 95812865497074203, 1916257309941516827, 38325146198830402075, 766502923976608172571, 15330058479532163713563, 306601169590643274795547

Offset

0,3

Comments

This sequence is the building block for the calculation of the sums of positive integers whose decimal notation only uses two distinct, nonzero digits. See the attached pdf document.

Links

Table of n, a(n) for n=0..19.

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

Index entries for linear recurrences with constant coefficients, signature (23,-62,40).

Formula

a(n) = (10*20^n - 19*2^n + 9)/171.

From Stefano Spezia, Oct 15 2019: (Start)

G.f.: x/(1 - 23*x + 62*x^2 - 40*x^3).

E.g.f.: (1/171)*exp(x)*(9 - 19*exp(x) + 10*exp(19*x)).

a(n) = 23*a(n-1) - 62*a(n-2) + 40*a(n-3) for n > 2.

(End)

a(n) = 21*a(n-1) - 20*a(n-2) + 2^(n-1) for n > 1. - Pierre-Alain Sallard, Dec 10 2019

Example

For n=3, the sum of all positive integers whose decimal expansion contains only the digits 5 and 8 (then S=5+8=13) with at most n=3 such digits, i.e., the sum 5 + 8 + 55 + 58 + 85 + 88 + 555 + 558 + 585 + 588 + 855 + 858 + 885 + 888, is equal to a(3)*13=6071.
The formula is valid for any other choice of two distinct digits. Other examples: always with n=3 but let's say with the 2 and 3 digits (then S=2+3=5), the sum 2+3+22+23+32+33+222+223+232+233+322+323+332+333 is equal to a(3)*5=2335.
Or with the 6 and 7 digits (and which case S=6+7 is the same as with the 5 and 8 digits), the sum 6+7+66+67+...+776+777 is equal to a(3)*13=6071 (same sum as with the 5 and 8 digits).

Prog

(Python) [(10*20**n-19*2**n+9)/(9*19) for n in range(20)]

Crossrefs

Cf. A328350, A328351, A328352, A328353, A328354, A328355, A328356.

Sequence in context: A053067 A036906 A134733 * A095256 A273008 A015678

Adjacent sequences: A328345 A328346 A328347 * A328349 A328350 A328351

Keyword

nonn,base

Author

Pierre-Alain Sallard, Oct 13 2019

Status

approved