

A328351


Let S be any integer in the range 10 <= S <= 30. 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 four distinct nonzero digits d1, d2, d3 and d4 such that d1+d2+d3+d4=S.


8



0, 1, 45, 1821, 72925, 2917341, 116695005, 4667805661, 186712248285, 7468490018781, 298739601100765, 11949584045428701, 477983361822740445, 19119334472931987421, 764773378917368975325, 30590935156695116926941, 1223637406267806108733405, 48945496250712250075959261
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OFFSET

0,3


COMMENTS

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


LINKS

Table of n, a(n) for n=0..17.
PierreAlain Sallard, Integers sequences A328348 and A328350 to A328356
Index entries for linear recurrences with constant coefficients, signature (45,204,160).


FORMULA

a(n) = (30*40^n39*4^n+9)/1053.
From Stefano Spezia, Oct 15 2019: (Start)
G.f.: x/(1  45*x + 204*x^2  160*x^3).
E.g.f.: (1/351)*exp(x)*(3  13*exp(3*x) + 10*exp(39*x)).
a(n) = 45*a(n1)  204*a(n2) + 160*a(n3) for n > 2.
(End)
a(n) = 41*a(n1)  40*a(n2) + 4^(n1) for n > 1.  PierreAlain Sallard, Dec 22 2019


EXAMPLE

For n=2, the sum of all positive integers whose decimal notation is only made of the 4,5,6 or 7 digit with at most n=2 such digits, i.e., the sum 4+5+6+7+44+45+46+47+54+55+56+57+64+65+66+67+74+75+76+77, is equal to (4+5+6+7)*a(2)=990.
The formula is valid for any other quadruple, as soon as the four digits are different from each other. Another example: always with n=2 but let's say with the 1,2,3 and 4 digits, the sum 1+2+3+4+11+12+13+14+21+22+23+24+31+32+33+34+41+42+43+44 is equal to a(2)*(1+2+3+4) = 450.


MATHEMATICA

CoefficientList[Series[x/(1  45 x + 204 x^2  160 x^3), {x, 0, 17}], x] (* Michael De Vlieger, Dec 23 2019 *)


PROG

(Python) [(30*40**n39*4**n+9)//1053 for n in range(20)]


CROSSREFS

Cf. A328348, A328350, A328352, A328353, A328354, A328355, A328356.
Sequence in context: A049397 A215859 A143170 * A203828 A318221 A162885
Adjacent sequences: A328348 A328349 A328350 * A328352 A328353 A328354


KEYWORD

nonn,base


AUTHOR

PierreAlain Sallard, Oct 13 2019


STATUS

approved



