A181718 -id:A181718 - cp's OEIS frontend

A181719 a(n) = A133473(n+1)^2.

25, 1225, 112225, 11122225, 1111222225, 111112222225, 11111122222225, 1111111222222225, 111111112222222225, 11111111122222222225, 1111111111222222222225, 111111111112222222222225, 11111111111122222222222225, 1111111111111222222222222225, 111111111111112222222222222225

Offset: 1

Paul Curtz, Nov 17 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A109344, A133473, A181718.

[(100^n+10*10^n+25)/9: n in [1..20]]; // Vincenzo Librandi, Jun 02 2011

(5+10^Range[30])^2/9 (* G. C. Greubel, Mar 25 2024 *)
LinearRecurrence[{111,-1110,1000},{25,1225,112225},20] (* Harvey P. Dale, Feb 22 2025 *)

a(n)=(100^n+10*10^n+25)/9 \\ Charles R Greathouse IV, Jun 01 2011

Vec(5*x*(1 - 62*x + 160*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^17)) \\ Colin Barker, Aug 21 2019

[(5+10^n)^2//9 for n in range(1,31)] # G. C. Greubel, Mar 25 2024

a(n) = 100 * A181718(n-1) + 25.
a(n) = 25 * A109344(n-1), for n > 1.
From Colin Barker, Aug 21 2019: (Start)
G.f.: x*(1 - 62*x + 160*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = (5 + 10^n)^2 / 9.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>3. (End)
E.g.f.: (1/9)*exp(x)*(25 + 10*exp(9*x) + exp(99*x)). - Stefano Spezia, Aug 21 2019 after Colin Barker

Formulas edited by Eric M. Schmidt, Oct 29 2012

A289980 Oblong (or pronic) numbers of the form x_n.y_n where z_n indicates the digit z repeated n times.

Original entry on oeis.org

12, 20, 30, 42, 56, 72, 90, 1122, 4422, 9900, 111222, 444222, 999000, 11112222, 44442222, 99990000, 1111122222, 4444422222, 9999900000, 111111222222, 444444222222, 999999000000, 11111112222222, 44444442222222, 99999990000000, 1111111122222222

Offset: 1

Bernard Schott, Jul 17 2017

There are only three infinite subsequences, that is, x_n.y_n is pronic for all n, and they occur with (x,y) = (1,2), (4,2), or (9,0). (See Crux Mathematicorum link.)

			6 * 7 = 42, 66 * 67 = 4422, 666 * 667 = 444222, 6666 * 6667 = 44442222.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Charles W. Trigg, Problem 938, Crux Mathematicorum, page 199, Vol. 11, Jun. 1985.

Cf. A002378, A181718, A276532.

seq(coeff(series(2*x*(15000*x^14+22000*x^13+21000*x^12-1650*x^11-14420*x^10-17265*x^9-14865*x^8-8553*x^7-4374*x^6+1629*x^5+1082*x^4+645*x^3-15*x^2-10*x-6)/((x-1)*(100*x^3-1)*(10*x^3-1)*(x^2+x+1)),x,n+1), x, n), n = 1 .. 30); # Muniru A Asiru, Dec 02 2018

Table[Select[Map[FromDigits[Join @@ Map[ConstantArray[#, n] &, #]] &, Drop[Tuples[Range[0, 9], {2}], 10]], IntegerQ@ Sqrt[4 # + 1] &], {n, 7}] // Flatten (* Michael De Vlieger, Jul 17 2017, after Robert G. Wilson v at A002378 *)
CoefficientList[Series[2 (15000 x^14 + 22000 x^13 + 21000 x^12 - 1650 x^11 - 14420 x^10 - 17265 x^9 - 14865 x^8 - 8553 x^7 - 4374 x^6 + 1629 x^5 + 1082 x^4 + 645 x^3 - 15 x^2 - 10 x - 6) / ((x - 1) (100 x^3 - 1) (10 x^3 - 1) (x^2 + x + 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 02 2018 *)

3_n * (3_n + 1) = 1_n.2_n as 33 * 34 = 1122.
6_n * (6_n + 1)  = 4_n.2_n as 666 * 667 = 444222
9_n * (9_n + 1) = 9_n.0_n as 9999 * 10000 = 99990000.
G.f.: 2*x*(15000*x^14 + 22000*x^13 + 21000*x^12 - 1650*x^11 - 14420*x^10 - 17265*x^9 - 14865*x^8 - 8553*x^7 - 4374*x^6 + 1629*x^5 + 1082*x^4 + 645*x^3 - 15*x^2 - 10*x -6) / ((x-1) * (100*x^3-1) * (10*x^3-1) * (x^2+x+1)). - Alois P. Heinz, Jul 17 2017

cp's OEIS Frontend

A181719 a(n) = A133473(n+1)^2.

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