cp's OEIS Frontend

A178574 a(n) = 2*n*(9*n-1).

%I A178574 #39 Feb 12 2025 10:46:32
%S A178574 16,68,156,280,440,636,868,1136,1440,1780,2156,2568,3016,3500,4020,
%T A178574 4576,5168,5796,6460,7160,7896,8668,9476,10320,11200,12116,13068,
%U A178574 14056,15080,16140,17236,18368,19536,20740,21980,23256,24568,25916,27300,28720,30176,31668,33196,34760,36360
%N A178574 a(n) = 2*n*(9*n-1).
%C A178574 Numbers with ordered partitions that have periods of length 6.
%C A178574 From each ordered partition of the numbers (15+j) with 0<j<6 one remove the first part z(1) and add 1 to the next z(1) parts to get a new partition until a period is reached.
%C A178574 The a(n) sequence begins with 16 and each member has 1 period; the b(n) sequence begins with 17 and each member has 2 periods; the c(n) sequence begins with 18 and each member has 3 periods; the d(n) sequence begins with 19 and each member has 2 periods; the e(n) sequence begins with 20 and each member has 1 period of length 6
%H A178574 Vincenzo Librandi, <a href="/A178574/b178574.txt">Table of n, a(n) for n = 1..1000</a>
%H A178574 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A178574 G.f. for a(n): (16 + 20*x)/(1-x)^3.
%F A178574      for b(n): (17 + 19*x)/(1-x)^3.
%F A178574      for c(n): (18 + 18*x)/(1-x)^3.
%F A178574      for d(n): (19 + 17*x)/(1-x)^3.
%F A178574      for e(n): (20 + 16*x)/(1-x)^3.
%F A178574 all sequences have the same recurrence:
%F A178574   s(n+3) = 3*s(n+2) - 3*s(n+1) + s(n);
%F A178574   with s(0) = 0, s(1) = 15 + j, s(2) = 66 + 2*j and 0<j<6.
%F A178574 a(n) = n*(18*n-2) = 4*A022266(n).
%F A178574 b(n) = n*(18*n-1) = a(n) + n.
%F A178574 c(n) = 18*n^2 = a(n) + 2*n.
%F A178574 d(n) = n*(18*n+1) = a(n) + 3*n.
%F A178574 e(n) = n*(18*n+2) = a(n) + 4*n.
%F A178574 The general formula for numbers with periods of length k: a(k,j,n) = n*(k^2*n - k + 2*j)/2 and 0<j<k.
%F A178574   For j=1 and j=(k-1) the numbers have 1 period.
%F A178574   For 1<j<(k-1) the numbers have A092964(k-4,j-1) periods.
%F A178574   G.f. (binomial(k,2)*(1+x) + j + (k-j)*x)/(1-x)^3.
%F A178574 E.g.f.: 2*exp(x)*x*(8 + 9*x). - _Elmo R. Oliveira_, Jan 28 2025
%e A178574 a(5) = 5*(18*5-2) = 440; b(5) = a(5) + 5 = 445; c(5) = a(5)*2*5 = 450; d(5) = a(5) + 3*5 = 455; e(5) = a(5) + 4*5 = 460.
%t A178574 Table[2n(9n-1), {n,1,50}] (* _Vincenzo Librandi_, Jul 10 2012 *)
%t A178574 LinearRecurrence[{3,-3,1},{16,68,156},50] (* _Harvey P. Dale_, Feb 15 2016 *)
%o A178574 (Magma) [2*n*(9*n-1): n in [1..50]]; // _Vincenzo Librandi_, Jul 10 2012
%o A178574 (PARI) a(n)=2*n*(9*n-1) \\ _Charles R Greathouse IV_, Jun 17 2017
%o A178574 (Sage) [2*n*(9*n-1) for n in (1..50)] # _G. C. Greubel_, Jan 30 2019
%Y A178574 Cf. A037306, A092964.
%K A178574 nonn,easy
%O A178574 1,1
%A A178574 _Paul Weisenhorn_, Dec 24 2010