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A033954 Second 10-gonal (or decagonal) numbers: n*(4*n+3).

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