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 A033954 Second 10-gonal (or decagonal) numbers: n*(4*n+3). 54
 0, 7, 22, 45, 76, 115, 162, 217, 280, 351, 430, 517, 612, 715, 826, 945, 1072, 1207, 1350, 1501, 1660, 1827, 2002, 2185, 2376, 2575, 2782, 2997, 3220, 3451, 3690, 3937, 4192, 4455, 4726, 5005, 5292, 5587, 5890, 6201, 6520, 6847, 7182, 7525, 7876, 8235 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Same as A033951 except start at 0. See example section. 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 REFERENCES S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250. R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99. LINKS Ivan Panchenko, Table of n, a(n) for n = 0..1000 Emilio Apricena, A version of the Ulam spiral. Leo Tavares, Illustration: V numbers Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = A001107(-n) = A074377(2*n). G.f.: x*(7+x)/(1-x)^3. - Michael Somos, Mar 03 2003 a(n) = a(n-1) + 8*n - 1 with a(0)=0. - Vincenzo Librandi, Jul 20 2010 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 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 a(n) = A118729(8n+6). - Philippe Deléham, Mar 26 2013 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 Sum_{n>=1} 1/a(n) = 4/9 + Pi/6 - log(2) = 0.2748960394827980081... . - Vaclav Kotesovec, Apr 27 2016 E.g.f.: exp(x)*x*(7 + 4*x). - Stefano Spezia, Jun 08 2021 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 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 a(n) = A060544(n+1) - A000217(n+1). - Leo Tavares, Mar 31 2022 EXAMPLE 36--37--38--39--40--41--42 | | 35 16--17--18--19--20 43 | | | | 34 15 4---5---6 21 44 | | | | | | 33 14 3 0===7==22==45==76=> | | | | | | 32 13 2---1 8 23 | | | | 31 12--11--10---9 24 | | 30--29--28--27--26--25 MATHEMATICA Table[n(4n+3), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 7, 22}, 50] (* Harvey P. Dale, May 06 2018 *) PROG (PARI) a(n)=4*n^2+3*n (Magma) [n*(4*n+3): n in [0..50]]; // G. C. Greubel, May 24 2019 (Sage) [n*(4*n+3) for n in (0..50)] # G. C. Greubel, May 24 2019 (GAP) List([0..50], n-> n*(4*n+3)) # G. C. Greubel, May 24 2019 CROSSREFS Cf. A002620, A033951. Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988. Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951. 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. 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. Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, this sequence, A062728, A135705. Cf. A060544. Sequence in context: A341401 A320694 A261465 * A159227 A081274 A038764 Adjacent sequences: A033951 A033952 A033953 * A033955 A033956 A033957 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified December 2 07:04 EST 2023. Contains 367510 sequences. (Running on oeis4.)