A075415 -id:A075415 - cp's OEIS frontend

A075414 Squares of A002279: a(n) = (5*(10^n - 1)/9)^2.

0, 25, 3025, 308025, 30858025, 3086358025, 308641358025, 30864191358025, 3086419691358025, 308641974691358025, 30864197524691358025, 3086419753024691358025, 308641975308024691358025, 30864197530858024691358025, 3086419753086358024691358025, 308641975308641358024691358025

Offset: 0

Michael Taylor (michael.taylor(AT)vf.vodafone.co.uk), Sep 14 2002

A transformation of the Wonderful Demlo numbers (A002477).

			a(2) = 55^2 = 3025.

Colin Barker, Table of n, a(n) for n = 0..100
Gérard Villemin, Variations sur les carrés.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417.

Cf. A002275, A002279, A002283, A002477.

concat(0, Vec(25*x*(1 + 10*x) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^20))) \\ Colin Barker, Jul 17 2019

a(n) = A002279(n)^2 = (5*A002275(n))^2 = 25*A002275(n)^2.
From Colin Barker, Jul 17 2019: (Start)
G.f.: 25*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2.
a(n) = 25*(10^n-1)^2/81. (End)
From Elmo R. Oliveira, Jul 29 2025: (Start)
E.g.f.: 25*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/81.
a(n) = 25*A002477(n). (End)

A075416 Squares of A002281.

Original entry on oeis.org

0, 49, 5929, 603729, 60481729, 6049261729, 604937061729, 60493815061729, 6049382595061729, 604938270395061729, 60493827148395061729, 6049382715928395061729, 604938271603728395061729, 60493827160481728395061729, 6049382716049261728395061729, 604938271604937061728395061729

Offset: 0

Michael Taylor (michael.taylor(AT)vf.vodafone.co.uk), Sep 14 2002

A transformation of the Wonderful Demlo numbers (A002477).

			a(2) = 77^2 = 5929.

Colin Barker, Table of n, a(n) for n = 0..100
Gérard Villemin, Variations sur les carrés.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417.

Cf. A002275, A002281, A002283, A002477.

concat(0, Vec(49*x*(1 + 10*x) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^20))) \\ Colin Barker, Jul 17 2019

a(n) = A002281(n)^2 = (7*A002275(n))^2 = 49*A002275(n)^2.
From Colin Barker, Jul 17 2019: (Start)
G.f.: 49*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2.
a(n) = 49*(10^n-1)^2/81. (End)
From Elmo R. Oliveira, Jul 29 2025: (Start)
E.g.f.: 49*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/81.
a(n) = 49*A002477(n). (End)

A075417 Squares of A002282: a(n) = (8*(10^n - 1)/9)^2.

Original entry on oeis.org

0, 64, 7744, 788544, 78996544, 7901076544, 790121876544, 79012329876544, 7901234409876544, 790123455209876544, 79012345663209876544, 7901234567743209876544, 790123456788543209876544, 79012345678996543209876544, 7901234567901076543209876544, 790123456790121876543209876544

Offset: 0

Michael Taylor (michael.taylor(AT)vf.vodafone.co.uk), Sep 14 2002

A transformation of the Wonderful Demlo numbers (A002477).

			a(2) = 88^2 = 7744.

Colin Barker, Table of n, a(n) for n = 0..100
Gérard Villemin, Variations sur les carrés.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A075411, A075412, A075413, A075414, A075415, A075416, A075417.

Cf. A002275, A002282, A002283, A002477.

concat(0, Vec(64*x*(1 + 10*x) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^20))) \\ Colin Barker, Jul 17 2019

a(n) = A002282(n)^2 = (8*A002275(n))^2 = 64*A002275(n)^2.
From Colin Barker, Jul 17 2019: (Start)
G.f.: 64*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2.
a(n) = 64*(10^n-1)^2/81. (End)
From Elmo R. Oliveira, Jul 30 2025: (Start)
E.g.f.: 64*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/81.
a(n) = 64*A002477(n). (End)

A177769 a(n) = 111*n.

Original entry on oeis.org

111, 222, 333, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443, 1554, 1665, 1776, 1887, 1998, 2109, 2220, 2331, 2442, 2553, 2664, 2775, 2886, 2997, 3108, 3219, 3330, 3441, 3552, 3663, 3774, 3885, 3996, 4107, 4218, 4329, 4440, 4551, 4662, 4773, 4884, 4995, 5106

Offset: 1

Paul Curtz, May 13 2010

The reference contains also sequences A102807, A109344, A075415, and A109492.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Émile Fourrey, Le nombre 37, Récreations arithmétiques, 1899 and later, Vuibert, Paris, page 11.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A075415, A102807, A109344, A109492.

[111*n: n in [1..50]]; // Vincenzo Librandi, Aug 04 2011

G.f.: 111*x/(x-1)^2.
a(n) = 2*a(n-1) - a(n-2).
a(n) = a(n-1) + 111.
E.g.f.: 111*x*exp(x). - Stefano Spezia, Sep 15 2023

A271528 a(n) = 2*(10^n - 1)^2/27.

Original entry on oeis.org

0, 6, 726, 73926, 7405926, 740725926, 74073925926, 7407405925926, 740740725925926, 74074073925925926, 7407407405925925926, 740740740725925925926, 74074074073925925925926, 7407407407405925925925926, 740740740740725925925925926, 74074074074073925925925925926

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

All terms are multiple of 6.
Converges in a 10-adic sense to ...925925925926.
A transformation of the Wonderful Demlo numbers (A002477).
More generally, the ordinary generating function for the transformation of the Wonderful Demlo numbers, is k*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).

			n=1:                  6 = 2 * 3;
n=2:                726 = 22 * 33;
n=3:              73926 = 222 * 333;
n=4:            7405926 = 2222 * 3333;
n=5:          740725926 = 22222 * 33333;
n=6:        74073925926 = 222222 * 333333;
n=7:      7407405925926 = 2222222 * 3333333;
n=8:    740740725925926 = 22222222 * 33333333;
n=9:  74074073925925926 = 222222222 * 333333333, etc.

Ilya Gutkovskiy, Transformation of the Wonderful Demlo numbers
Eric Weisstein's World of Mathematics, Demlo Number
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275, A002276, A002277, A002477.

Cf. similar sequences of the form k*((10^n - 1)/9)^2: A075411 (k=4), this sequence (k=6), A075412 (k=9), A075413 (k=16), A178630 (k=18), A075414 (k=25), A178631 (k=27), A075415 (k=36), A178632 (k=45), A075416 (k=49), A178633 (k=54), A178634 (k=63), A075417 (k=64), A178635 (k=72), A059988 (k=81).

Table[2 ((10^n - 1)^2/27), {n, 0, 15}]
LinearRecurrence[{111, -1110, 1000}, {0, 6, 726}, 16]

x='x+O('x^99); concat(0, Vec(6*x*(1+10*x)/(1-111*x+1110*x^2-1000*x^3))) \\ Altug Alkan, Apr 09 2016

for n in range(0,10**1):print((int)((2*(10**n-1)**2)/27))
# Soumil Mandal, Apr 10 2016

O.g.f.: 6*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).
E.g.f.: 2 (exp(x) - 2*exp(10*x) + exp(100*x))/27.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
a(n) = 6*A002477(n) = 6*A002275(n)^2 = A002276(n)*A002277(n) = sqrt(A075411(n)*A075412(n)).
Sum_{n>=1} 1/a(n) = 0.1680577405662077350849154881928636039793563...
Lim_{n -> infinity} a(n + 1)/a(n) = 100.

A102832 Number of n-digit squares which contain the string "666" but not "6666".

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 21, 78, 302, 1139, 4156, 14791, 52529, 183565, 635691, 2183533, 7314869, 25303217

Offset: 1

James R. Buddenhagen, Feb 27 2005

Cf. A052041, A075415, A102807 A032746.

fQ[n_] := StringPosition[ ToString[n^2], "666"] != {} && StringPosition[ ToString[n^2], "6666"] == {}; t = Table[0, {20}]; Do[ If[ fQ[n], t[[ Ceiling[ Log[10, n^2]] + 1]]++ ], {n, 3162277661}]; t (* Robert G. Wilson v, Mar 10 2005 *)
t6[digs_]:=Module[{a=Floor[Sqrt[10^(digs-1)]],b=Floor[ Sqrt[ 10^digs]]},Count[ Range[a,b]^2,?(SequenceCount[IntegerDigits[#],{6,6,6}]> 0&&SequenceCount[IntegerDigits[#],{6,6,6,6}]==0&)]]; Table[ t6[n],{n,20}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Mar 26 2021 *)

a(13)-a(19) from Robert G. Wilson v, Mar 03 2005

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A075414 Squares of A002279: a(n) = (5*(10^n - 1)/9)^2.

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A075416 Squares of A002281.

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A075417 Squares of A002282: a(n) = (8*(10^n - 1)/9)^2.

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A177769 a(n) = 111*n.

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Magma

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A271528 a(n) = 2*(10^n - 1)^2/27.

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A102832 Number of n-digit squares which contain the string "666" but not "6666".

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