A054487 - OEIS

%I #31 Jan 05 2025 19:51:36

%S 1,14,90,390,1320,3762,9438,21450,45045,88660,165308,294372,503880,

%T 833340,1337220,2089164,3187041,4758930,6970150,10031450,14208480,

%U 19832670,27313650,37153350,49961925,66475656,87576984,114316840

%N a(n) = (3*n+4)*binomial(n+7, 7)/4.

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

%H Vincenzo Librandi, <a href="/A054487/b054487.txt">Table of n, a(n) for n = 0..1000</a>

%H I. Adler, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.

%H E. I. Emerson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F G.f.: (1+5*x)/(1-x)^9.

%F From _G. C. Greubel_, Jan 19 2020: (Start)

%F a(n) = 6*binomial(n+8, 8) - 5*binomial(n+7, 7).

%F E.g.f.: (20160 +262080*x +635040*x^2 +540960*x^3 +205800*x^4 +38808*x^5 +3724*x^6 +172*x^7 +3*x^8)*exp(x)/20160. (End)

%F a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - _Wesley Ivan Hurt_, Jun 07 2021

%p seq( (3*n+4)*binomial(n+7,7)/4, n=0..40); # _G. C. Greubel_, Jan 19 2020

%t CoefficientList[Series[(1+5x)/(1-x)^9, {x,0,40}], x] (* _Vincenzo Librandi_, Jul 30 2014 *)

%t Table[6*Binomial[n+8,8] -5*Binomial[n+7,7], {n,0,40}] (* _G. C. Greubel_, Jan 19 2020 *)

%t LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,14,90,390,1320,3762,9438,21450,45045},30] (* _Harvey P. Dale_, Jul 19 2022 *)

%o (Magma) [((3*n+4)*Binomial(n+7,7))/4: n in [0..40]]; // _Vincenzo Librandi_, Jul 30 2014

%o (PARI) a(n) = (3*n+4)*binomial(n+7, 7)/4; \\ _Michel Marcus_, Sep 07 2017

%o (Sage) [(3*n+4)*binomial(n+7, 7)/4 for n in (0..40)] # _G. C. Greubel_, Jan 19 2020

%o (GAP) List([0..40], n-> (3*n+4)*Binomial(n+7, 7)/4 ); # _G. C. Greubel_, Jan 19 2020

%Y Cf. A034265.

%Y Cf. A093563 ((6, 1) Pascal, column m=8).

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, May 06 2000

%E Corrected and extended by _James A. Sellers_, May 10 2000