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a(n) = n*(4*n - 3)*(16*n^2 - 12*n - 3).
1

%I #21 Feb 16 2025 08:33:27

%S 0,1,370,2835,10660,28645,63126,121975,214600,351945,546490,812251,

%T 1164780,1621165,2200030,2921535,3807376,4880785,6166530,7690915,

%U 9481780,11568501,13981990,16754695,19920600,23515225,27575626,32140395,37249660,42945085,49269870

%N a(n) = n*(4*n - 3)*(16*n^2 - 12*n - 3).

%C Doubly 10-gonal (or decagonal) numbers.

%H G. C. Greubel, <a href="/A264895/b264895.txt">Table of n, a(n) for n = 0..5000</a>

%H OEIS Wiki, <a href="https://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DecagonalNumber.html">Decagonal Number</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1)

%F G.f.: x*(1 + 365*x + 995*x^2 + 175*x^3)/(1 - x)^5.

%F a(n) = A001107(A001107(n)).

%F Sum_{n>0} 1/a(n) = (sqrt(21)*gamma + sqrt(21)*polygamma(0, 1/4) - 3*polygamma(0, (1/8)*(5 - sqrt(21))) + 3*polygamma(0, (1/8)*(5 + sqrt(21))))/(9*sqrt(21))= 1.00322253307732984...., where gamma is the Euler-Mascheroni constant (A001620), and polygamma is the derivative of the logarithm of the gamma function.

%t Table[n (4 n - 3) (16 n^2 - 12 n - 3), {n, 0, 30}]

%t LinearRecurrence[{5,-10,10,-5,1}, {0, 1, 370, 2835, 10660}, 50] (* _G. C. Greubel_, Sep 07 2018 *)

%o (PARI) vector(100, n, n--; n*(4*n-3)*(16*n^2-12*n-3)) \\ _Altug Alkan_, Nov 27 2015

%o (Magma) [n*(4*n - 3)*(16*n^2 - 12*n - 3): n in [0..30]]; // _Vincenzo Librandi_, Nov 28 2015

%Y Cf. A001107, A002817, A000583, A232713, A063249.

%K nonn,easy,changed

%O 0,3

%A _Ilya Gutkovskiy_, Nov 27 2015