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Consider the triangle shown below in which the n-th row contains the n smallest numbers greater than those in the previous row such that the arithmetic mean is an integer. Sequence contains the leading diagonal.
4

%I #27 Feb 09 2024 08:40:57

%S 1,4,7,13,18,27,34,46,55,70,81,99,112,133,148,172,189,216,235,265,286,

%T 319,342,378,403,442,469,511,540,585,616,664,697,748,783,837,874,931,

%U 970,1030,1071,1134,1177,1243,1288,1357,1404,1476,1525,1600,1651,1729

%N Consider the triangle shown below in which the n-th row contains the n smallest numbers greater than those in the previous row such that the arithmetic mean is an integer. Sequence contains the leading diagonal.

%H Colin Barker, <a href="/A111710/b111710.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F a(1)=1, a(2n) = a(2n-1)+3n, a(2n+1)=a(2n)+2n+1. - _Franklin T. Adams-Watters_, May 01 2006

%F G.f.: -x*(1+3*x+x^2) / ( (1+x)^2*(x-1)^3 ). a(n+1)-a(n) = A080512(n+1). - _R. J. Mathar_, May 02 2013

%F From _Colin Barker_, Jan 26 2016: (Start)

%F a(n) = (10*n^2+2*(-1)^n*n+10*n+(-1)^n-1)/16.

%F a(n) = (5*n^2+6*n)/8 for n even.

%F a(n) = (5*n^2+4*n-1)/8 for n odd. (End)

%e The fourth row is 8,9,10 and 13,(8+9+10 +13)/4 = 10.

%e Triangle begins:

%e 1

%e 2 4

%e 5 6 7

%e 8 9 10 13

%e 14 15 16 17 18

%e 19 20 21 22 23 27

%e 28 29 30 31 32 33 34

%t LinearRecurrence[{1, 2, -2, -1, 1}, {1, 4, 7, 13, 18}, 100] (* _Paolo Xausa_, Feb 09 2024 *)

%o (PARI) Vec(x*(1+3*x+x^2)/((1-x)^3*(1+x)^2) + O(x^100)) \\ _Colin Barker_, Jan 26 2016

%Y Cf. A111711, A111712.

%Y Cf. A085787. - _R. J. Mathar_, Aug 15 2008

%K easy,nonn

%O 1,2

%A _Amarnath Murthy_, Aug 24 2005

%E More terms from _Franklin T. Adams-Watters_, May 01 2006