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 A123579 #21 Nov 06 2022 08:51:01
%S A123579 0,1,2,3,3,4,5,5,6,6,6,7,8,8,9,9,9,10,10,10,10,11,12,12,13,13,13,14,
%T A123579 14,14,14,15,15,15,15,15,16,17,17,18,18,18,19,19,19,19,20,20,20,20,20,
%U A123579 21,21,21,21,21,21,22,23,23,24,24,24,25,25,25,25,26,26,26,26,26,27,27,27
%N A123579 The Kruskal-Macaulay function M_3(n).
%C A123579 Write n (uniquely) as n = C(n_t,t) + C(n_{t-1},t-1) + ... + C(n_v,v) where n_t > n_{t-1} > ... > n_v >= v >= 1. Then M_t(n) = C(n_t-1,t-1) + C(n_{t-1}-1,t-2) + ... + C(n_v-1,v-1).
%C A123579 From _Samuel Harkness_, Sep 30 2022: (Start)
%C A123579 a(n) is the smallest number of balls needed on the base layer to stack n balls.
%C A123579 All nonrepeating terms other than a(0) occur at tetrahedral numbers + 1 (n = A000292 + 1).
%C A123579 The value of the nonrepeating terms other than a(0) are the Central Polygonal numbers (A000124). (End)
%D A123579 D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3.
%H A123579 Samuel Harkness, <a href="/A123579/b123579.txt">Table of n, a(n) for n = 0..10000</a>
%H A123579 B. M. Abrego, S. Fernandez-Merchant, and B. Llano, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Abrego/abrego2.html">An Inequality for Macaulay Functions</a>, J. Int. Seq. 14 (2011) # 11.7.4.
%p A123579 lowpol := proc(n,t) local x::integer ; x := floor( (n*factorial(t))^(1/t)) ; while binomial(x,t) <= n do x := x+1 ; od ; RETURN(x-1) ; end:
%p A123579 C := proc(n,t) local nresid,tresid,m,a ; nresid := n ; tresid := t ; a := [] ; while nresid > 0 do m := lowpol(nresid,tresid) ; a := [op(a),m] ; nresid := nresid - binomial(m,tresid) ; tresid := tresid-1 ; od ; RETURN(a) ; end:
%p A123579 M := proc(n,t) local a ; a := C(n,t) ; add( binomial(op(i,a)-1,t-i),i=1..nops(a)) ; end:
%p A123579 A123579 := proc(n) M(n,3) ; end:
%p A123579 for n from 0 to 120 do printf("%d, ",A123579(n)) ; od ; # _R. J. Mathar_, Mar 14 2007
%t A123579 c = 0; T = {0}; For[r = 1, r <= 7, r++, For[n = 1, n <= r, n++, c++; For[m = 1, m <= n, m++, AppendTo[T, c]]]]; Take[T, 75] (* _Samuel Harkness_, Sep 30 2022 *)
%Y A123579 For M_i(n), i=1, 2, 3, 4, 5 see A000127, A123578, A123579, A123580, A123731.
%Y A123579 Cf. A000124, A000292.
%K A123579 nonn,easy
%O A123579 0,3
%A A123579 _N. J. A. Sloane_, Nov 12 2006