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%I #21 Jan 26 2024 13:22:12
%S 1,11,51,156,375,771,1421,2416,3861,5875,8591,12156,16731,22491,29625,
%T 38336,48841,61371,76171,93500,113631,136851,163461,193776,228125,
%U 266851,310311,358876,412931,472875,539121,612096,692241,780011,875875,980316,1093831,1216931,1350141,1494000,1649061,1815891,1995071
%N a(n) = (7*n^4 + 5*n^2)/12.
%C a(n) is the sum of terms in the square [1,n]x[1,n] of the natural number array A000027; e.g., the [1,3]x[1,3] square is
%C 1..2..4
%C 3..5..8
%C 6..9..13,
%C so that a(1) = 1, a(2) = 1+2+3+5 = 11, a(3) = 1+2+3+4+5+6+8+9+13 = 51.
%C Partial sums of A063490. - _Omar E. Pol_, Oct 23 2019
%H G. C. Greubel, <a href="/A185505/b185505.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 (5,-10,10,-5,1).
%F a(n) = (7*n^4 + 5*n^2)/12.
%F From _Chai Wah Wu_, Sep 05 2016: (Start)
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
%F G.f.: x*(1 + x)*(1 + 5*x + x^2)/(1 - x)^5. (End)
%F E.g.f.: (1/12)*x*(12 + 54*x + 42*x^2 + 7*x^3)*exp(x). - _G. C. Greubel_, Jul 07 2017
%e a(1)=(7+5)/12, a(2)=(7*16+5*4)/12.
%t Table[(7*n^4+5*n^2)/12, {n,1,60}]
%t LinearRecurrence[{5,-10,10,-5,1},{1,11,51,156,375},50] (* _Harvey P. Dale_, Jan 26 2024 *)
%o (PARI) a(n)=(7*n^4+5*n^2)/12 \\ _Charles R Greathouse IV_, Sep 05 2016
%Y Cf. A000027, A063490.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jan 29 2011
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