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%I #59 Sep 08 2022 08:44:49
%S 1,1,1,2,3,3,4,5,6,7,8,9,11,12,13,15,17,18,20,22,24,26,28,30,33,35,37,
%T 40,43,45,48,51,54,57,60,63,67,70,73,77,81,84,88,92,96,100,104,108,
%U 113,117,121,126,131,135,140,145,150,155,160,165,171,176,181,187,193,198
%N Expansion of 1/((1-x)*(1-x^3)*(1-x^4)).
%C Apply the Riordan array (1/(1-x^4),x) to floor((n+3)/3). - _Paul Barry_, Jan 20 2006
%C Number of partitions of n into parts 1, 3, and 4. - _David Neil McGrath_, Aug 30 2014
%C Also, a(n-4) is equal to the number of partitions mu of n of length 3 such that mu_1-mu_2 is even and mu_2-mu_3 is odd or vice versa (see below example). - _John M. Campbell_, Jan 29 2016
%C With four 0's prepended and offset 0, a(n) is the number of partitions of n into four parts whose 2nd and 3rd largest parts are equal. - _Wesley Ivan Hurt_, Jan 05 2021
%H Vincenzo Librandi, <a href="/A025767/b025767.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,0,-1,0,-1,1).
%F G.f.: 1/((1-x)*(1-x^3)*(1-x^4)).
%F a(n) = floor(n^2/24+n/3+1).
%F a(n) = Sum_{k=0..floor(n/4)} floor((n-4*k+3)/3)}. - _Paul Barry_, Jan 20 2006
%F Euler transform of length 4 sequence [1, 0, 1, 1]. - _Michael Somos_, Nov 09 2007
%F a(n) = a(-8 - n) for all n in Z. - _Michael Somos_, Nov 09 2007
%F a(n) = n^2/24 + n/3 + 83/144 + (-1)^n/16 + A061347(n+1)/9 + A056594(n)/4. - _R. J. Mathar_, Mar 31 2011
%F a(n) = a(n-1)+a(n-3)-a(n-5)-a(n-7)+a(n-8). - _David Neil McGrath_, Aug 30 2014
%F a(n) = Sum_{k=1..floor((n+4)/4)} Sum_{j=k..floor((n+4-k)/3)} Sum_{i=j..floor((n+4-j-k)/2)} [j = i], where [ ] is the Iverson bracket. - _Wesley Ivan Hurt_, Jan 17 2021
%F a(n)-a(n-1) = A008679(n). - _R. J. Mathar_, Jun 23 2021
%F a(n)-a(n-4) = A008620(n). - _R. J. Mathar_, Jun 23 2021
%e The a(4)=3 partitions of 4 into parts 1, 3, and 4 are (4), (3,1), and (1,1,1,1). - _David Neil McGrath_, Aug 30 2014
%e From _John M. Campbell_, Jan 29 2016: (Start)
%e Letting n=12, there are a(n-4)=a(8)=6 partitions mu of n=12 of length 3 such that mu_1-mu_2 is even and mu_2-mu_3 is odd or vice versa:
%e (10,1,1) |- n
%e (8,3,1) |- n
%e (7,3,2) |- n
%e (6,5,1) |- n
%e (6,3,3) |- n
%e (5,5,2) |- n
%e (End)
%p A056594 := proc(n) op(1+(n mod 4),[1,0,-1,0]) ; end proc:
%p A061347 := proc(n) op(1+(n mod 3),[-2,1,1]) ; end proc:
%p A025767 := proc(n) n^2/24+n/3+83/144+(-1)^n/16 +A061347(n+1)/9 +A056594(n)/4 ; end proc: # _R. J. Mathar_, Mar 31 2011
%t Table[Floor[n^2/24 + n/3 + 1], {n, 0, 60}] (* _Vincenzo Librandi_, Aug 31 2014 *)
%o (PARI) a(n)=if(n<0,0,(n^2+8*n)\24+1)
%o (PARI) {a(n) = round( ((n + 4)^2 - 1) / 24 )}; /* _Michael Somos_, Nov 09 2007 */
%o (PARI) Vec(1/((1-x)*(1-x^3)*(1-x^4)) + O(x^80)) \\ _Michel Marcus_, Jan 29 2016
%o (Magma) [Floor(n^2/24 + n/3 + 1): n in [0..70]]; // _Vincenzo Librandi_, Aug 31 2014
%Y A008621(n) = A002265(n+4) = a(n) - a(n-3).
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_
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