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A055795 a(n) = binomial(n,4) + binomial(n,2).

%I A055795 #81 Jul 02 2025 16:02:00
%S A055795 0,1,3,7,15,30,56,98,162,255,385,561,793,1092,1470,1940,2516,3213,
%T A055795 4047,5035,6195,7546,9108,10902,12950,15275,17901,20853,24157,27840,
%U A055795 31930,36456,41448,46937,52955,59535,66711,74518,82992,92170,102090,112791,124313,136697
%N A055795 a(n) = binomial(n,4) + binomial(n,2).
%C A055795 Answer to the question: if you have a tall building and 4 plates and you need to find the highest story from which a plate thrown does not break, what is the number of stories you can handle given n tries?
%C A055795 If Y is a 2-subset of an n-set X then, for n >= 4, a(n-3) is the number of 4-subsets of X which do not have exactly one element in common with Y. - _Milan Janjic_, Dec 28 2007
%C A055795 Antidiagonal sums of A139600. - _Johannes W. Meijer_, Apr 29 2011
%C A055795 Also the number of maximal cliques in the n-tetrahedral graph for n > 5. - _Eric W. Weisstein_, Jun 12 2017
%C A055795 Mark each point on an 8^(n-2) grid with the number of points that are visible from the point; for n > 3, a(n) is the number of distinct values in the grid. - _Torlach Rush_, Mar 25 2021
%C A055795 Antidiagonal sums of both A057145 and also A134394 yield this sequence without the initial term 0. - _Michael Somos_, Nov 23 2021
%H A055795 James Spahlinger, <a href="/A055795/b055795.txt">Table of n, a(n) for n = 1..1000</a>
%H A055795 Michael Boardman, <a href="http://www.jstor.org/stable/3219201">The Egg-Drop Numbers</a>, Mathematics Magazine, 77 (2004), 368-372.
%H A055795 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A055795 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H A055795 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>
%H A055795 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>
%H A055795 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TetrahedralGraph.html">Tetrahedral Graph</a>
%H A055795 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A055795 a(n) = A000127(n)-1. Differences give A000127.
%F A055795 a(1) = 1; a(n) = a(n-1) + 1 + A004006(n-1).
%F A055795 a(n+1) = C(n, 1) + C(n, 2) + C(n, 3) + C(n, 4). - _James Sellers_, Mar 16 2002
%F A055795 Row sums of triangle A134394. Also, binomial transform of [1, 2, 2, 2, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Oct 23 2007
%F A055795 O.g.f.: -x^2(1-2x+2x^2)/(x-1)^5. a(n) = A000332(n) + A000217(n-1). - _R. J. Mathar_, Apr 13 2008
%F A055795 a(n) = n*(n^3 - 6*n^2 + 23*n - 18)/24. - _Gary Detlefs_, Dec 08 2011
%F A055795 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(1)=0, a(2)=1, a(3)=3, a(4)=7, a(5)=15. - _Harvey P. Dale_, Dec 07 2015
%p A055795 A055795:=n->binomial(n,4)+binomial(n,2); # _Zerinvary Lajos_, Jul 24 2006
%t A055795 Table[Binomial[n, 4] + Binomial[n, 2], {n, 50}] (* _Vladimir Joseph Stephan Orlovsky_, May 24 2009 *)
%t A055795 Table[n (n^3 - 6 n^2 + 23 n - 18)/24, {n, 100}] (* _Wesley Ivan Hurt_, Sep 29 2013 *)
%t A055795 LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 3, 7, 15}, 50] (* _Harvey P. Dale_, Dec 07 2015 *)
%t A055795 Total[Binomial[Range[20], #] & /@ {2, 4}] (* _Eric W. Weisstein_, Dec 01 2017 *)
%t A055795 CoefficientList[Series[x (-1 + 2 x - 2 x^2)/(-1 + x)^5, {x, 0, 20}], x] (* _Eric W. Weisstein_, Dec 01 2017~ *)
%o A055795 (Magma) [n*(n^3-6*n^2+23*n-18)/24: n in [1..100]]; // _Wesley Ivan Hurt_, Sep 29 2013
%o A055795 (Maxima) A055795(n):=n*(n^3-6*n^2+23*n-18)/24$ makelist(A055795(n), n, 1, 100); /* _Wesley Ivan Hurt_, Sep 29 2013 */
%o A055795 (PARI) a(n)= n*(n^3-6*n^2+23*n-18)/24 \\ _Wesley Ivan Hurt_, Sep 29 2013
%Y A055795 T(2n+1, n), array T as in A055794. Cf. A004006, A000127.
%Y A055795 Cf. A134394, A051601, A057145.
%K A055795 nonn,easy
%O A055795 1,3
%A A055795 _Clark Kimberling_, May 28 2000
%E A055795 Better description from _Leonid Broukhis_, Oct 24 2000
%E A055795 Edited by _Zerinvary Lajos_, Jul 24 2006
%E A055795 Offset corrected and Sellers formula adjusted by _Gary Detlefs_, Nov 28 2011