# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a002843 Showing 1-1 of 1 %I A002843 M1072 N0405 #61 Jan 15 2024 12:02:23 %S A002843 1,1,2,4,7,13,24,43,78,141,253,456,820,1472,2645,4749,8523,15299, %T A002843 27456,49267,88407,158630,284622,510683,916271,1643963,2949570, %U A002843 5292027,9494758,17035112,30563634,54835835,98383803,176515310,316694823,568197628,1019430782 %N A002843 Number of partitions of n into parts 1/2, 3/4, 7/8, 15/16, etc. %C A002843 Row sums of A049286 and A047913. [_Vladeta Jovovic_, Dec 02 2009] %C A002843 Also number of compositions (a_1,a_2,...) of n with each a_i <= 2*a_(i-1). [_Vladeta Jovovic_, Dec 02 2009] %D A002843 Minc, H.; A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid. Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224. %D A002843 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A002843 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A002843 Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 201 terms from Vincenzo Librandi) %H A002843 David Benson, Pavel Etingof, On cohomology in symmetric tensor categories in prime characteristic, arXiv:2008.13149 [math.RT], 2020. %H A002843 R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] %H A002843 H. Minc, A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid, Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224. %H A002843 Simon Plouffe, Approximations de sÃ©ries gÃ©nÃ©ratrices et quelques conjectures, Dissertation, UniversitÃ© du QuÃ©bec Ã MontrÃ©al, 1992; arXiv:0911.4975 [math.NT], 2009. %H A002843 Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 %F A002843 The g.f. (z**2+z+1)*(z-1)**2/(1-2*z-z**3+3*z**4) conjectured by _Simon Plouffe_ in his 1992 dissertation is wrong. %e A002843 A straightforward partition problem: 1 = 1/2 + 1/2 and there is no other partition of 1, so a(1)=1. %e A002843 a(3)=4 since 3 = 6(1/2) = 4(3/4) = 2(3/4) + 3(1/2) = 2(7/8) + 3/4 + 1/2. %e A002843 a(4)=7 since 4 = 8(1/2) = 5(1/2) + 2(3/4) = 2(1/2) + 4(3/4) = 3(1/2) + 3/4 + 2(7/8) = 3(3/4) + 2(7/8) = 1/2 + 4(7/8) = 2(15/16) + 7/8 + 3/4 + 1/2. %e A002843 From _Joerg Arndt_, Dec 28 2012: (Start) %e A002843 There are a(6)=24 compositions of 6 where part(k) <= 2 * part(k-1): %e A002843 [ 1] [ 1 1 1 1 1 1 ] %e A002843 [ 2] [ 1 1 1 1 2 ] %e A002843 [ 3] [ 1 1 1 2 1 ] %e A002843 [ 4] [ 1 1 2 1 1 ] %e A002843 [ 5] [ 1 1 2 2 ] %e A002843 [ 6] [ 1 2 1 1 1 ] %e A002843 [ 7] [ 1 2 1 2 ] %e A002843 [ 8] [ 1 2 2 1 ] %e A002843 [ 9] [ 1 2 3 ] %e A002843 [10] [ 2 1 1 1 1 ] %e A002843 [11] [ 2 1 1 2 ] %e A002843 [12] [ 2 1 2 1 ] %e A002843 [13] [ 2 2 1 1 ] %e A002843 [14] [ 2 2 2 ] %e A002843 [15] [ 2 3 1 ] %e A002843 [16] [ 2 4 ] %e A002843 [17] [ 3 1 1 1 ] %e A002843 [18] [ 3 1 2 ] %e A002843 [19] [ 3 2 1 ] %e A002843 [20] [ 3 3 ] %e A002843 [21] [ 4 1 1 ] %e A002843 [22] [ 4 2 ] %e A002843 [23] [ 5 1 ] %e A002843 [24] [ 6 ] %e A002843 (End) %p A002843 b:= proc(n, i) option remember; `if`(n=0, 1, %p A002843 add(b(n-j, min(n-j, 2*j)), j=1..i)) %p A002843 end: %p A002843 a:= n-> b(n$2): %p A002843 seq(a(n), n=0..40); # _Alois P. Heinz_, Jun 24 2017 %t A002843 v[c_, d_] := v[c, d] = If[d < 0 || c < 0, 0, If[d == c, 1, Sum[v[i, d - c], {i, 1, 2*c}]]]; Join[{1}, Plus @@@ Table[v[d, c], {c, 1, 34}, {d, 1, c}]] (* _Jean-FranÃ§ois Alcover_, Dec 10 2012, after _Vladeta Jovovic_ *) %Y A002843 Cf. A047913, A049286. %K A002843 nonn,nice %O A002843 0,3 %A A002843 _N. J. A. Sloane_ %E A002843 More terms from _John W. Layman_, Nov 24 2001 %E A002843 Examples and offset corrected by Larry Reeves (larryr(AT)acm.org), Jan 06 2005 %E A002843 Further terms from _Vladeta Jovovic_, Mar 13 2006 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE