This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A103632 #38 Jan 05 2025 19:51:38 %S A103632 1,0,1,1,2,2,3,4,6,8,11,15,21,29,40,55,76,105,145,200,276,381,526,726, %T A103632 1002,1383,1909,2635,3637,5020,6929,9564,13201,18221,25150,34714, %U A103632 47915,66136,91286,126000,173915,240051,331337,457337,631252,871303,1202640 %N A103632 Expansion of (1 - x + x^2)/(1 - x - x^4). %C A103632 Diagonal sums of A103631. %C A103632 The Kn11 sums, see A180662, of triangle A065941 equal the terms of this sequence without a(0) and a(1). - _Johannes W. Meijer_, Aug 11 2011 %C A103632 For n >= 2, a(n) is the number of palindromic compositions of n-2 with parts in {2,1,3,5,7,9,...}. The generating function follows easily from Theorem 1,2 of the Hoggatt et al. reference. Example: a(9) = 8 because we have 7, 151, 11311, 232, 313, 12121, 21112, and 1111111. - _Emeric Deutsch_, Aug 17 2016 %C A103632 Essentially the same as A003411. - _Georg Fischer_, Oct 07 2018 %H A103632 Muniru A Asiru, <a href="/A103632/b103632.txt">Table of n, a(n) for n = 0..1000</a> %H A103632 V. E. Hoggatt, Jr., and Marjorie Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356. %H A103632 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1). %F A103632 G.f.: (1 - x + x^2)/(1 - x - x^4). %F A103632 a(n) = a(n-1) + a(n-4) with a(0)=1, a(1)=0, a(2)=1 and a(3)=1. %F A103632 a(n) = Sum_{k=0..floor(n/2)} binomial(floor((2*n-3*k-1)/2), n-2*k). %F A103632 a(n) = A003269(n+1) - A003269(n-4), n > 4. %p A103632 A103632 := proc(n): add( binomial(floor((2*n-3*k-1)/2), n-2*k), k=0..floor(n/2)) end: seq(A103632(n), n=0..46); # _Johannes W. Meijer_, Aug 11 2011 %t A103632 LinearRecurrence[{1,0,0,1}, {1,0,1,1}, 50] (* _G. C. Greubel_, Mar 10 2019 *) %o A103632 (GAP) a:=[1,0,1,1];; for n in [5..50] do a[n]:=a[n-1]+a[n-4]; od; a; # _Muniru A Asiru_, Oct 07 2018 %o A103632 (PARI) my(x='x+O('x^50)); Vec((1-x+x^2)/(1-x-x^4)) \\ _G. C. Greubel_, Mar 10 2019 %o A103632 (Magma) I:=[1,0,1,1]; [n le 4 select I[n] else Self(n-1) + Self(n-4): n in [1..50]]; // _G. C. Greubel_, Mar 10 2019 %o A103632 (Sage) ((1-x+x^2)/(1-x-x^4)).series(x, 50).coefficients(x, sparse=False) # _G. C. Greubel_, Mar 10 2019 %Y A103632 Cf. A275446. %K A103632 easy,nonn %O A103632 0,5 %A A103632 _Paul Barry_, Feb 11 2005 %E A103632 Formula corrected by _Johannes W. Meijer_, Aug 11 2011