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A130423 Main diagonal of array A[k,n] = n-th sum of 3 consecutive k-gonal numbers, k>2.

%I A130423 #24 Feb 16 2025 08:33:06
%S A130423 4,14,39,88,170,294,469,704,1008,1390,1859,2424,3094,3878,4785,5824,
%T A130423 7004,8334,9823,11480,13314,15334,17549,19968,22600,25454,28539,31864,
%U A130423 35438,39270,43369,47744,52404,57358,62615,68184,74074,80294,86853
%N A130423 Main diagonal of array A[k,n] = n-th sum of 3 consecutive k-gonal numbers, k>2.
%C A130423 The first row of the array is the sum of 3 consecutive triangular numbers = A000217(n) + A000217(n+1) + A000217(n+2) = Centered triangular numbers: 3*n*(n-1)/2 + 1, for n>1. The second row of the array is the sum of 3 consecutive squares = Number of points on surface of square pyramid: 3*n^2 + 2 (n>1). The first column of the array is k+1 = 4, 5, 6, 7, 8, 9, ... The second column of the array is A016825 = 4*n + 2 (for n>2). The third column of the array is A017377 = 10*n + 9 (for n>0).
%H A130423 Vincenzo Librandi, <a href="/A130423/b130423.txt">Table of n, a(n) for n = 1..1000</a>
%H A130423 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>
%H A130423 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A130423 a(n) = A[n+2,n] = P(k+2,n) + P(k+2,n+1) + P(k+2,n+2) where P(k,n) = k*((n-2)*k - (n-4))/2.
%F A130423 a(n) = n*(3*n^2-3*n+8)/2. G.f.: x*(4-2*x+7*x^2)/(1-x)^4. [_Colin Barker_, Apr 30 2012]
%F A130423 a(1)=4, a(2)=14, a(3)=39, a(4)=88, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Harvey P. Dale_, Aug 15 2012
%e A130423 The array begins:
%e A130423 k / A[k,n]
%e A130423 3.|.4.10.19.31..46..64..85.109.136.166....=A005448(n+1).
%e A130423 4.|.5.14.29..50..77.110.149.194.245.302...=A005918(n).
%e A130423 5.|.6.18.39..69.108.156.213.279.354.438...=A129863(n).
%e A130423 6.|.7.22.49..88.139.202.277.364.463.574...
%e A130423 7.|.8.26.59.107.170.248.341.449.572.710...
%e A130423 8.|.9.30.69.126.201.294.405.534.681.846...
%p A130423 P := proc(k,n) n*((k-2)*n-k+4)/2 ; end: A := proc(k,n) add( P(k,i),i=n..n+2) ; end: A130423 := proc(n) A(n+3,n) ; end: seq(A130423(n),n=0..40) ; # _R. J. Mathar_, Jun 14 2007
%t A130423 CoefficientList[Series[(4-2*x+7*x^2)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 28 2012 *)
%t A130423 Table[n (3n^2-3n+8)/2,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{4,14,39,88},40] (* _Harvey P. Dale_, Aug 15 2012 *)
%o A130423 (Magma) I:=[4, 14, 39, 88]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jun 28 2012
%Y A130423 Cf. A000217, A000290, A000326, A000384, A000566, A000567, A005448, A005918, A016825, A017377, A129803, A129863.
%K A130423 easy,nonn
%O A130423 1,1
%A A130423 _Jonathan Vos Post_, May 26 2007
%E A130423 More terms from _R. J. Mathar_, Jun 14 2007