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Generalized 10-gonal numbers: m*(4*m - 3) for m = 0, +- 1, +- 2, +- 3, ...
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%I #85 Feb 28 2022 04:13:11

%S 0,1,7,10,22,27,45,52,76,85,115,126,162,175,217,232,280,297,351,370,

%T 430,451,517,540,612,637,715,742,826,855,945,976,1072,1105,1207,1242,

%U 1350,1387,1501,1540,1660,1701,1827,1870,2002,2047,2185,2232,2376,2425

%N Generalized 10-gonal numbers: m*(4*m - 3) for m = 0, +- 1, +- 2, +- 3, ...

%C Also called generalized decagonal numbers.

%C Odd triangular numbers decremented and halved.

%C It appears that this is zero together with the partial sums of A165998. - _Omar E. Pol_, Sep 10 2011 [this is correct, see the g.f., _Joerg Arndt_, Sep 29 2013]

%C Also, A033954 and positive members of A001107 interleaved. - _Omar E. Pol_, Aug 04 2012

%C Also, numbers m such that 16*m+9 is a square. After 1, therefore, there are no squares in this sequence. - _Bruno Berselli_, Jan 07 2016

%C Convolution of the sequences A047522 and A059841. - _Ilya Gutkovskiy_, Mar 16 2017

%C Numbers k such that the concatenation k5625 is a square. - _Bruno Berselli_, Nov 07 2018

%C Exponents in expansion of Product_{n >= 1} (1 + x^(8*n-7))*(1 + x^(8*n-1))*(1 - x^(8*n)) = 1 + x + x^7 + x^10 + x^22 + .... - _Peter Bala_, Dec 10 2020

%H Vincenzo Librandi, <a href="/A074377/b074377.txt">Table of n, a(n) for n = 0..1000</a>

%H Neville Holmes, <a href="https://web.archive.org/web/20120511031019/http://www.comp.utas.edu.au/users/nholmes/sqncs/gmtrc.htm#A074377">More Gemometric Integer Sequences</a>. [Wayback Machine copy]

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

%F (n(n+1)-2)/4 where n(n+1)/2 is odd.

%F G.f.: x*(1+6*x+x^2)/((1-x)*(1-x^2)^2). - _Michael Somos_, Mar 04 2003

%F a(2*k) = k*(4*k+3); a(2*k+1) = (2*k+1)^2+k. - _Benoit Jubin_, Feb 05 2009

%F a(n) = n^2+n-1/4+(-1)^n/4+n*(-1)^n/2. - _R. J. Mathar_, Oct 08 2011

%F Sum_{n>=1} 1/a(n) = (4 + 3*Pi)/9. - _Vaclav Kotesovec_, Oct 05 2016

%F E.g.f.: exp(x)*x^2 + (2*exp(x) - exp(-x)/2)*x - sinh(x)/2. - _Ilya Gutkovskiy_, Mar 16 2017

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) - 4/9. - _Amiram Eldar_, Feb 28 2022

%t CoefficientList[Series[x(1 +6x +x^2)/((1-x)(1-x^2)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Sep 29 2013 *)

%t LinearRecurrence[{1,2,-2,-1,1}, {0,1,7,10,22}, 50] (* _G. C. Greubel_, Nov 07 2018 *)

%o (PARI) a(n)=(2*n+3-4*(n%2))*(n-n\2)

%o (PARI) concat([0],Vec(x*(1 + 6*x + x^2)/((1 - x)*(1 - x^2)^2) +O(x^50))) \\ _Indranil Ghosh_, Mar 16 2017

%o (Magma) [n^2+n-1/4+(-1)^n/4+n*(-1)^n/2: n in [0..50]]; // _Vincenzo Librandi_, Sep 29 2013

%Y Cf. A011848, A014493, A074378, A118277, A165998.

%Y Cf. A001107 (10-gonal numbers).

%Y Column 6 of A195152.

%Y Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), this sequence (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

%Y Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [_Bruno Berselli_, Jul 25 2016]

%K nonn,easy

%O 0,3

%A _W. Neville Holmes_, Sep 04 2002

%E New name from _T. D. Noe_, Apr 21 2006

%E Formula in sequence name from _Omar E. Pol_, May 28 2012