A232731 -id:A232731 - cp's OEIS frontend

A232729 Number of n-digit numbers that yield an n-digit number after Reverse and Add.

4, 45, 405, 4095, 40500, 405450, 4050000, 40504500, 405000000, 4050045000, 40500000000, 405000450000, 4050000000000, 40500004500000, 405000000000000, 4050000045000000, 40500000000000000, 405000000450000000, 4050000000000000000, 40500000004500000000, 405000000000000000000

Offset: 1

Lars Blomberg, Nov 29 2013

			There are 4 1-digit numbers (1,2,3,4) that yield a 1-digit number (2,4,6,8), so a(1)=4.

Robert Israel, Table of n, a(n) for n = 1..990
Index entries for linear recurrences with constant coefficients, signature (10,10,-100).

Cf. A232730, A232731.

Vec(x*(4+5*x-85*x^2-5*x^3)/((1-10*x)*(1-10*x^2)) + O(x^50)) \\ Colin Barker, Apr 22 2016

a(n) + A232730(n) = 9*10^(n-1).
a(1) = 4, a(3) = 405, a(2k+1) = 100*a(2k-1), k > 1.
a(2) = 45, a(4) = 4095, a(2k) = 110*a(2k-2) - 1000*a(2k-4), k > 2.
From Colin Barker, Apr 21 2016: (Start)
a(n) = 10*a(n-1)+10*a(n-2)-100*a(n-3) for n>4.
G.f.: x*(4+5*x-85*x^2-5*x^3) / ((1-10*x)*(1-10*x^2)). (End)
E.g.f.: (81*cosh(10*x) + 90*cosh(sqrt(10)*x) + 81*sinh(10*x) - 10*x - 171)/200. - Stefano Spezia, Oct 27 2022

A232730 Number of n-digit numbers that yield an (n+1)-digit number after Reverse and Add.

Original entry on oeis.org

5, 45, 495, 4905, 49500, 494550, 4950000, 49495500, 495000000, 4949955000, 49500000000, 494999550000, 4950000000000, 49499995500000, 495000000000000, 4949999955000000, 49500000000000000, 494999999550000000, 4950000000000000000, 49499999995500000000

Offset: 1

Lars Blomberg, Nov 29 2013

A232729(n) + a(n) = 9*10^(n-1).

			There are 5 1-digit numbers (5,6,7,8,9) that yield a 2-digit number (10,12,14,16,18), so a(1)=5.

Robert Israel, Table of n, a(n) for n = 1..990
Index entries for linear recurrences with constant coefficients, signature (10,10,-100).

Cf. A232729, A232731.

a[1]:=5: t[0]:= 0: t[1]:= 5:
for n from 2 to 50 do
a[n]:= 45*10^(n-2) + 9*t[n-2];
t[n]:= a[n] + t[n-2];
od:
seq(a[n],n=1..50); # Robert Israel, Apr 21 2016

LinearRecurrence[{10,10,-100},{5,45,495,4905},20] (* Harvey P. Dale, Feb 29 2024 *)

Vec(5*x*(1+x)*(1-x)^2 / ((1-10*x)*(1-10*x^2)) + O(x^30)) \\ Colin Barker, Mar 20 2017

a(1) = 5, a(3) = 495, a(2*k+1) = 100*a(2*k-1), k > 1.
a(2) = 45, a(4) = 4905, a(2*k) = 110*a(2*k-2) - 1000*a(2*k-4), k > 2.
G.f. = 5*x*(1+x)*(1-x)^2 / ((1-10*x)*(1-10*x^2)). - M. F. Hasler, Nov 30 2013
From Colin Barker, Mar 20 2017: (Start)
a(n) = -45*(10^(n/2-2) - 11*10^(n-3)) for n>2 even.
a(n) = 99*2^(n-3)*5^(n-2) for n>2 odd.
a(n) = 10*a(n-1) + 10*a(n-2) - 100*a(n-3) for n>4. (End)
E.g.f.: (99*cosh(10*x) - 90*cosh(sqrt(10)*x) + 99*sinh(10*x) + 10*x - 9)/200. - Stefano Spezia, Oct 27 2022

G.f. corrected and empirical formulas proved by Robert Israel, Apr 21 2016

A190481 Number of distinct integers with n digits which are the image of integers by the function Reverse and Add!.

Original entry on oeis.org

4, 14, 93, 256, 1793, 4872, 34107, 92590, 648154, 1759313, 12315269, 33427272, 233991155, 635119194, 4445835138, 12067267861, 84470877438, 229278099157, 1604946701532, 4356283914175, 30493987422124, 82769394462323, 579385761306789, 1572618495070552

Offset: 1

Aldo González Lorenzo, May 25 2011

a(n) is the cardinality of the set of Image(Reverse and Add!) intersected with [10^(n-1), 10^n[. Here we suppose that the domain of the function Reverse and Add! is {1, 2, 3, ...}

There are 4, 50, 450, 4590, 45405,... (A232731) ways to obtain integers with n = 1,2,... digits as images under the function "Reverse and add!", but many result in the same image and are counted here only once. Example: 11+digrev(11) = 22 and 20+digrev(20)=22 contribute only once to the set of distinct images at n=2. - R. J. Mathar, Jun 17 2011

			Example: let RaA(x) be the function Reverse and Add!, then:
RaA(1)=2
RaA(2)=4
RaA(3)=6
RaA(4)=8
RaA(5)=10
RaA(6)=11, ...
So a(1) is the cardinal of {2,4,6,8}, which is 4:

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100

Cf. A056964, A067030.

A055642 := proc(n) max(1,1+ilog10(n)) ; end proc:
A056964 := proc(n) n+digrev(n) ; end proc:
A190481 := proc(n) local s,i,ra ; s := {} ; for i from 1 to 10^n do ra := A056964(i) ; if A055642( ra) = n then s := s union {ra}  ; end if; end do: nops(s) ; end proc:
for n from 1 do print(n,A190481(n)) ; end do: # R. J. Mathar, Jun 17 2011

Empirical g.f.: x*(4 + 18*x + 23*x^2 - 29*x^3 - 58*x^4 - 34*x^5 - 81*x^6 - 45*x^7 - 32*x^8 - 9*x^9) / ((1 + x)*(1 - 19*x^2)*(1 - 2*x + x^2 - x^3)*(1 + 2*x + x^2 + x^3)). - Colin Barker, Mar 20 2017

a(9)-a(10) from Lars Blomberg, Dec 01 2013

a(11)-a(24) from Hiroaki Yamanouchi, Sep 04 2014

cp's OEIS Frontend

A232729 Number of n-digit numbers that yield an n-digit number after Reverse and Add.

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A232730 Number of n-digit numbers that yield an (n+1)-digit number after Reverse and Add.

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A190481 Number of distinct integers with n digits which are the image of integers by the function Reverse and Add!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions