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%I #90 Sep 08 2022 08:44:51
%S 0,7,22,45,76,115,162,217,280,351,430,517,612,715,826,945,1072,1207,
%T 1350,1501,1660,1827,2002,2185,2376,2575,2782,2997,3220,3451,3690,
%U 3937,4192,4455,4726,5005,5292,5587,5890,6201,6520,6847,7182,7525,7876,8235
%N Second 10-gonal (or decagonal) numbers: n*(4*n+3).
%C Same as A033951 except start at 0. See example section.
%C Bisection of A074377. Also sequence found by reading the line from 0, in the direction 0, 22, ... and the line from 7, in the direction 7, 45, ..., in the square spiral whose vertices are the generalized 10-gonal numbers A074377. - _Omar E. Pol_, Jul 24 2012
%D S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
%H Ivan Panchenko, <a href="/A033954/b033954.txt">Table of n, a(n) for n = 0..1000</a>
%H Emilio Apricena, <a href="/A035608/a035608.png">A version of the Ulam spiral</a>.
%H Leo Tavares, <a href="/A033954/a033954_1.jpg">Illustration: V numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = A001107(-n) = A074377(2*n).
%F G.f.: x*(7+x)/(1-x)^3. - _Michael Somos_, Mar 03 2003
%F a(n) = a(n-1) + 8*n - 1 with a(0)=0. - _Vincenzo Librandi_, Jul 20 2010
%F For n>0, Sum_{j=0..n} (a(n) + j)^4 + (4*A000217(n))^3 = Sum_{j=n+1..2n} (a(n) + j)^4; see also A045944. - _Charlie Marion_, Dec 08 2007, edited by _Michel Marcus_, Mar 14 2014
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 7, a(2) = 22. - _Philippe Deléham_, Mar 26 2013
%F a(n) = A118729(8n+6). - _Philippe Deléham_, Mar 26 2013
%F a(n) = A002943(n) + n = A007742(n) + 2n = A016742(n) + 3n = A033991(n) + 4n = A002939(n) + 5n = A001107(n) + 6n = A033996(n) - n. - _Philippe Deléham_, Mar 26 2013
%F Sum_{n>=1} 1/a(n) = 4/9 + Pi/6 - log(2) = 0.2748960394827980081... . - _Vaclav Kotesovec_, Apr 27 2016
%F E.g.f.: exp(x)*x*(7 + 4*x). - _Stefano Spezia_, Jun 08 2021
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(2)) + log(2)/3 - 4/9 - sqrt(2)*arcsinh(1)/3. - _Amiram Eldar_, Nov 28 2021
%F For n>0, (a(n)^2 + n)/(a(n) + n) = (4*n + 1)^2/4, a ratio of two squares. - _Rick L. Shepherd_, Feb 23 2022
%F a(n) = A060544(n+1) - A000217(n+1). - _Leo Tavares_, Mar 31 2022
%e 36--37--38--39--40--41--42
%e | |
%e 35 16--17--18--19--20 43
%e | | | |
%e 34 15 4---5---6 21 44
%e | | | | | |
%e 33 14 3 0===7==22==45==76=>
%e | | | | | |
%e 32 13 2---1 8 23
%e | | | |
%e 31 12--11--10---9 24
%e | |
%e 30--29--28--27--26--25
%t Table[n(4n+3),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,22},50] (* _Harvey P. Dale_, May 06 2018 *)
%o (PARI) a(n)=4*n^2+3*n
%o (Magma) [n*(4*n+3): n in [0..50]]; // _G. C. Greubel_, May 24 2019
%o (Sage) [n*(4*n+3) for n in (0..50)] # _G. C. Greubel_, May 24 2019
%o (GAP) List([0..50], n-> n*(4*n+3)) # _G. C. Greubel_, May 24 2019
%Y Cf. A002620, A033951.
%Y Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
%Y Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
%Y Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, this sequence, A062728, A135705.
%Y Cf. A060544.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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