A093137 - OEIS

%I #54 Sep 18 2024 11:24:09

%S 1,4,34,334,3334,33334,333334,3333334,33333334,333333334,3333333334,

%T 33333333334,333333333334,3333333333334,33333333333334,

%U 333333333333334,3333333333333334,33333333333333334,333333333333333334,3333333333333333334,33333333333333333334

%N Expansion of (1-7*x)/((1-x)*(1-10*x)).

%C Second binomial transform of 3*A001045(3n)/3+(-1)^n. Partial sums of A093138. A convex combination of 10^n and 1. In general the second binomial transform of k*Jacobsthal(3n)/3+(-1)^n is 1,1+k,1+11k,1+111k,... This is the case for k=3.

%C a(n) is the number of n-length sequences of decimal digits whose sum is divisible by 3. - _Geoffrey Critzer_, Jan 18 2014

%C This sequence appears in a family of curious cubic identities based on the Armstrong number 407 = A005188(13). See the formula section. For the analog identities based on 153 = A005188(10) see a comment on A246057 with the van der Poorten et al. reference and A281857. For those based on 370 = A005188(11) see A067275, A002277 and A281858. - _Wolfdieter Lang_, Feb 08 2017

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 3334 at p. 168.

%H Seiichi Manyama, <a href="/A093137/b093137.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-10).

%F a(n) = 3*10^n/9 + 6/9.

%F a(n) = 10*a(n-1)-6 with a(0)=1. - _Vincenzo Librandi_, Aug 02 2010

%F a(n)^3 + 0(n)^3 + A067275(n+1)^3 = concatenation(a(n), 0(n), A067275(n+1)) = A281859(n), where 0(n) denotes n 0's, n >= 1. - _Wolfdieter Lang_, Feb 08 2017

%F From _Elmo R. Oliveira_, Aug 17 2024: (Start)

%F E.g.f.: exp(x)*(exp(9*x) + 2)/3.

%F a(n) = 11*a(n-1) - 10*a(n-2) for n > 1. (End)

%e a(1)^2 = 16

%e a(2)^2 = 1156

%e a(3)^2 = 111556

%e a(4)^2 = 11115556

%e a(5)^2 = 1111155556

%e a(6)^2 = 111111555556

%e a(7)^2 = 11111115555556

%e a(8)^2 = 1111111155555556

%e a(9)^2 = 111111111555555556, etc... (see A102807). - _Philippe Deléham_, Oct 03 2011

%e Curious cubic identities: 407 = 4^3 + 0^3 + 7^3, 340067 = 34^3 + (00)^3 + 67^3, 334000677 = 334^3 + (000)^3 + 677^3, ... - _Wolfdieter Lang_, Feb 08 2017

%t nn=20; r=Solve[{s==4x s+3 x a+3x b+1,a==4x a+3x s+3x b,b==4x b+3x s+3x a},{s,a,b}]; CoefficientList[Series[s/.r,{x,0,nn}],x] (* _Geoffrey Critzer_, Jan 18 2014 *)

%t Table[3*10^n/9 + 6/9, {n, 0, 20}] (* or *) NestList[10 # - 6 &, 1, 20] (* _Michael De Vlieger_, Feb 08 2017 *)

%t LinearRecurrence[{11,-10},{1,4},20] (* _Harvey P. Dale_, Oct 07 2017 *)

%o (PARI) Vec((1-7*x)/((1-x)*(1-10*x)) + O (x^30)) \\ _Michel Marcus_, Feb 09 2017

%Y Cf. A005188, A067275, A246057, A281857, A281859.

%Y Cf. A001045, A002277, A093138, A281858.

%Y Cf. A102807 (squared).

%K nonn,easy

%O 0,2

%A _Paul Barry_, Mar 24 2004