cp's OEIS Frontend

A287324 a(n) = A008412(n-1) + A008412(n-2) for n>1, a(0)=0, a(1)=1.

%I A287324 #108 May 06 2025 11:26:10
%S A287324 0,1,9,40,120,280,552,968,1560,2360,3400,4712,6328,8280,10600,13320,
%T A287324 16472,20088,24200,28840,34040,39832,46248,53320,61080,69560,78792,
%U A287324 88808,99640,111320,123880,137352,151768,167160,183560,201000,219512,239128,259880,281800
%N A287324 a(n) = A008412(n-1) + A008412(n-2) for n>1, a(0)=0, a(1)=1.
%C A287324 Let's iteratively apply the summation of two consecutive terms to A000292. It generates A000330, then A005900, then A001845, then A008412, then this sequence. Every sequence in this series starts with 1 followed by the sum of 1 and the next term in the previous sequence; because of that, for A008412 and this sequence, the initial term(s) are exceptions from the general formula.
%C A287324 From _Leo James Borcherding_, May 23 2017: (Start)
%C A287324 a(n) = f(9,n), where f(k,n) is the set of all series derived from the anchored series.
%C A287324 k = (All whole numbers (including negative values))
%C A287324 n = (All whole numbers >= 1)
%C A287324 The anchored series is f(0,n).
%C A287324 See the attached file for an in-depth explanation of the family of tetrahedron sequences that f(9,n) (this sequence) is a part of.
%C A287324 A Visual Representation of the summation process is as follows:
%C A287324 a.) f(7,n) + f(7,n-1) = f(8,n)
%C A287324 b.) f(8,n) + f(8,n-1) = f(9,n)
%C A287324 a.)                      b.)
%C A287324 1   + 0   = 1            1   + 0   = 1
%C A287324 7   + 1   = 8            8   + 1   = 9
%C A287324 25  + 7   = 32           32  + 8   = 40
%C A287324 63  + 25  = 88           88  + 32  = 120
%C A287324 129 + 63  = 192          192 + 88  = 280
%C A287324 231 + 129 = 360          360 + 192 = 552
%C A287324 377 + 231 = 608          608 + 360 = 968
%C A287324 575 + 377 = 952          952 + 608 = 1560
%C A287324 ... iterate infinitely many times. (End)
%D A287324 William Dunham, Euler The Master of Us All, The Mathematical Association of America, 1999 p. 40.
%D A287324 Joseph and Frances Gies, Leonard of Pisa and the New Mathematics of the Middle Ages, Thomas Y. Crowell Company New York, 1969, p. 78.
%H A287324 Colin Barker, <a href="/A287324/b287324.txt">Table of n, a(n) for n = 0..1000</a>
%H A287324 Leo James Borcherding, <a href="/A287324/a287324_2.pdf">Tetrahedron Family of f(k,n)</a>
%H A287324 Leo James Borcherding, <a href="/A287324/a287324_4.pdf">The Unified Tetrahedral Family Defined by Pascal's Triangle</a>
%H A287324 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A287324 G.f.: x*(x + 1)^5 / (x - 1)^4.
%F A287324 a(n) = 8*(n - 1)*((n - 1)^2 + 2)/3 + 8*(n - 2)*((n - 2)^2 + 2)/3 = 8*(2*n - 3)*(n^2 - 3*n + 5)/3 for n>2, a(0)=0, a(1)=1, a(2)=9.
%F A287324 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>6. - _Colin Barker_, Jun 05 2017
%o A287324 (PARI) concat(0, Vec(x*(x+1)^5/(x-1)^4 + O(x^30))) \\ _Michel Marcus_, May 24 2017
%Y A287324 Cf. A000292, A000330, A001845, A002623, A005900, A006918, A008412.
%K A287324 nonn,easy
%O A287324 0,3
%A A287324 _Leo James Borcherding_, May 23 2017