A282504 - OEIS

%I #10 Mar 15 2017 04:01:51

%S 1,1,1,1,1,2,3,4,5,6,8,11,15,21,28,37,49,65,88,119,160,214,285,381,

%T 511,687,923,1237,1656,2217,2971,3985,5345,7166,9603,12867,17244,

%U 23115,30989,41543,55684,74634,100032,134081,179729,240919,322935,432858,580191,777680,1042407,1397262,1872911,2510457

%N Expansion of 1/(1 - Sum_{k>=0} x^(2*k*(k+1)+1)).

%C Number of compositions (ordered partitions) into centered square numbers (A001844).

%C Conjecture: every number > 1 is the sum of at most 6 centered square numbers.

%C Extended conjecture: every number > 1 is the sum of at most k+2 centered k-gonal numbers.

%H Indranil Ghosh, <a href="/A282504/b282504.txt">Table of n, a(n) for n = 0..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredSquareNumber.html">Centered Square Number</a>

%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F G.f.: 1/(1 - Sum_{k>=0} x^(2*k*(k+1)+1)).

%F a(n) ~ c / r^n, where r = 0.746043978237212782246711857485153004976647... is the root of the equation sqrt(r) * EllipticTheta(2, 0, r^2) = 2 and c = 0.453173429667590077751072798128748901015122665... . - _Vaclav Kotesovec_, Feb 17 2017

%e a(8) = 5 because we have [5, 1, 1, 1], [1, 5, 1, 1], [1, 1, 5, 1], [1, 1, 1, 5] and [1, 1, 1, 1, 1, 1, 1, 1].

%t nmax = 53; CoefficientList[Series[1/(1 - Sum[x^(2 k (k + 1) + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

%o (PARI) Vec(1/(1 - sum(k=0, 54, x^(2*k*(k + 1) + 1))) + O(x^54)) \\ _Indranil Ghosh_, Mar 15 2017

%Y Cf. A001844, A006456, A280951.

%K nonn

%O 0,6

%A _Ilya Gutkovskiy_, Feb 16 2017