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

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

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