A075412 -id:A075412 - cp's OEIS frontend

A178633 a(n) = 54*((10^n - 1)/9)^2.

54, 6534, 665334, 66653334, 6666533334, 666665333334, 66666653333334, 6666666533333334, 666666665333333334, 66666666653333333334, 6666666666533333333334, 666666666665333333333334, 66666666666653333333333334, 6666666666666533333333333334, 666666666666665333333333333334

Offset: 1

Reinhard Zumkeller, May 31 2010

			n = 1:                   54 = 9 * 6;
n = 2:                 6534 = 99 * 66;
n = 3:               665334 = 999 * 666;
n = 4:             66653334 = 9999 * 6666;
n = 5:           6666533334 = 99999 * 66666;
n = 6:         666665333334 = 999999 * 666666;
n = 7:       66666653333334 = 9999999 * 6666666;
n = 8:     6666666533333334 = 99999999 * 66666666;
n = 9:   666666665333333334 = 999999999 * 666666666.

Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.

Colin Barker, Table of n, a(n) for n = 1..499
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A059988, A075412, A075415, A178630, A178631, A178632, A178634, A178635.

Cf. A002277, A002280, A002283, A002477.

[54*((10^n-1)/9)^2: n in [1..50]]; // Vincenzo Librandi, Dec 28 2010

54 ((10^Range[20] - 1)/9)^2 (* Vincenzo Librandi, Dec 07 2015 *)

a(n)=54*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

Vec(54*x*(1+10*x)/((1-x)*(1-10*x)*(1-100*x)) + O(x^20)) \\ Colin Barker, Dec 07 2015

a(n) = 54*A002477(n) = A002283(n)*A002280(n).
a(n) = ((A002280(n-1)*10 + 5)*10^(n-1) + A002277(n-1))*10 + 4 = (2/3)*(10^n - 1)^2.
From Colin Barker, Dec 07 2015: (Start)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>3.
G.f.: 54*x*(1+10*x)/((1-x)*(1-10*x)*(1-100*x)). (End)
E.g.f.: 2*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/3. - Elmo R. Oliveira, Aug 01 2025

A178634 a(n) = 63*((10^n - 1)/9)^2.

Original entry on oeis.org

63, 7623, 776223, 77762223, 7777622223, 777776222223, 77777762222223, 7777777622222223, 777777776222222223, 77777777762222222223, 7777777777622222222223, 777777777776222222222223, 77777777777762222222222223, 7777777777777622222222222223, 777777777777776222222222222223

Offset: 1

Reinhard Zumkeller, May 31 2010

			n=1: ..................... 63 = 9 * 7;
n=2: ................... 7623 = 99 * 77;
n=3: ................. 776223 = 999 * 777;
n=4: ............... 77762223 = 9999 * 7777;
n=5: ............. 7777622223 = 99999 * 77777;
n=6: ........... 777776222223 = 999999 * 777777;
n=7: ......... 77777762222223 = 9999999 * 7777777;
n=8: ....... 7777777622222223 = 99999999 * 77777777;
n=9: ..... 777777776222222223 = 999999999 * 777777777.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 33 at p. 62.
Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.

G. C. Greubel, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A059988, A075412, A075415, A178630, A178631, A178632, A178633, A178635.

Cf. A002276, A002281, A002283, A002477.

List([1..20], n -> 63*((10^n - 1)/9)^2); # G. C. Greubel, Jan 28 2019

[63*((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Dec 28 2010

63((10^Range[15]-1)/9)^2 (* or *) Table[FromDigits[Join[PadRight[{},n,7],{6},PadRight[{},n,2],{3}]],{n,0,15}] (* Harvey P. Dale, Apr 23 2012 *)

a(n)=63*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

[63*((10^n - 1)/9)^2 for n in (1..20)] # G. C. Greubel, Jan 28 2019

a(n) = 63*A002477(n) = A002283(n)*A002281(n).
a(n) = ((A002281(n-1)*10 + 6)*10^(n-1) + A002276(n-1))*10 + 3.
G.f.: 63*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
E.g.f.: 7*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9. - Stefano Spezia, Jul 31 2024
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. - Elmo R. Oliveira, Aug 01 2025

A178635 a(n) = 72*((10^n - 1)/9)^2.

Original entry on oeis.org

72, 8712, 887112, 88871112, 8888711112, 888887111112, 88888871111112, 8888888711111112, 888888887111111112, 88888888871111111112, 8888888888711111111112, 888888888887111111111112, 88888888888871111111111112, 8888888888888711111111111112, 888888888888887111111111111112

Offset: 1

Reinhard Zumkeller, May 31 2010

			n=1: ..................... 72 = 9 * 8;
n=2: ................... 8712 = 99 * 88;
n=3: ................. 887112 = 999 * 888;
n=4: ............... 88871112 = 9999 * 8888;
n=5: ............. 8888711112 = 99999 * 88888;
n=6: ........... 888887111112 = 999999 * 888888;
n=7: ......... 88888871111112 = 9999999 * 8888888;
n=8: ....... 8888888711111112 = 99999999 * 88888888;
n=9: ..... 888888887111111112 = 999999999 * 888888888.

Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A059988, A075412, A075415, A178630, A178631, A178632, A178633, A178634.

Cf. A002275, A002282, A002283, A002477.

[72*((10^n-1)/9)^2: n in [1..50]]; // Vincenzo Librandi, Dec 28 2010

72 (FromDigits/@Table[PadRight[{}, n, 1], {n, 40}])^2 (* Vincenzo Librandi, Mar 21 2014 *)

a(n)=72*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

a(n) = 72*A002477(n) = A002283(n)*A002282(n).
a(n) = ((A002282(n-1)*10 + 7)*10^(n-1) + A002275(n-1))*10 + 2.
G.f.: 72*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017
From Elmo R. Oliveira, Aug 01 2025: (Start)
E.g.f.: 8*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. (End)

A075411 Squares of A002276.

Original entry on oeis.org

0, 4, 484, 49284, 4937284, 493817284, 49382617284, 4938270617284, 493827150617284, 49382715950617284, 4938271603950617284, 493827160483950617284, 49382716049283950617284, 4938271604937283950617284, 493827160493817283950617284, 49382716049382617283950617284

Offset: 0

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

A transformation of the Wonderful Demlo numbers (A002477).

			a(2) = 22^2 = 484.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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, A002276, A002283, A002477.

I:=[0,4,484]; [n le 3 select I[n] else 111*Self(n-1)-1110*Self(n-2)+1000*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Apr 25 2017

LinearRecurrence[{111, -1110, 1000}, {0, 4, 484}, 30] (* Vincenzo Librandi, Apr 25 2017 *)

a(n) = A002276(n)^2 = (2*A002275(n))^2 = 4*A002275(n)^2.
From Chai Wah Wu, Apr 24 2017: (Start)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
G.f.: 4*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). (End)
a(n) = 4*(10^n - 1)^2/81. - Colin Barker, Apr 25 2017
From Elmo R. Oliveira, Jul 28 2025: (Start)
E.g.f.: 4*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/81.
a(n) = 4*A002477(n). (End)

A075413 Squares of A002278.

Original entry on oeis.org

0, 16, 1936, 197136, 19749136, 1975269136, 197530469136, 19753082469136, 1975308602469136, 197530863802469136, 19753086415802469136, 1975308641935802469136, 197530864197135802469136, 19753086419749135802469136, 1975308641975269135802469136, 197530864197530469135802469136

Offset: 0

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

A transformation of the Wonderful Demlo numbers (A002477).

			a(2) = 44^2 = 1936.

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, A002278, A002283, A002477.

concat(0, Vec(16*x*(1 + 10*x) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^20))) \\ Colin Barker, Jul 17 2019

a(n) = A002278(n)^2 = (4*A002275(n))^2 = 16*A002275(n)^2.
From Colin Barker, Jul 17 2019: (Start)
G.f.: 16*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) = 16*(10^n-1)^2/81. (End)
From Elmo R. Oliveira, Jul 29 2025: (Start)
E.g.f.: 16*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/81.
a(n) = 16*A002477(n). (End)

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

Original entry on oeis.org

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)

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.

cp's OEIS Frontend

A178633 a(n) = 54*((10^n - 1)/9)^2.

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A178634 a(n) = 63*((10^n - 1)/9)^2.

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A178635 a(n) = 72*((10^n - 1)/9)^2.

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A075411 Squares of A002276.

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A075413 Squares of A002278.

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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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