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A033954 Second 10-gonal (or decagonal) numbers: n*(4*n+3). 33
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

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

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: x(7+x)/(1-x)^3. - Michael Somos, Mar 03 2003

a(n) = 8*n+a(n-1)-1 with a(0)=0. - Vincenzo Librandi, Jul 20 2010

For n>0, Sum{i=0..n} (a(n)+i)^4 +  (4*A000217(n))^3 = Sum{i=n+1..2n} (a(n)+i)^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

EXAMPLE

16 17 18 19 ...

15 4 5 6 ...

14 3 0 7 ...

13 2 1 8 ...

For n=1, a(1)=8*1+0-1=7; n=2, a(2)=8*2+7-1=22; n=3, a(3)=8*3+22-1=45. - Vincenzo Librandi, Jul 20 2010

MATHEMATICA

s=0; lst={s}; Do[s+=n++ +7; AppendTo[lst, s], {n, 0, 7!, 8}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)

PROG

(PARI) a(n)=4*n^2+3*n

CROSSREFS

Same as A033951 except start at 0.

a(n)=A001107(-n)=A074377(2n).

Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.

Cf. A002620.

Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, this sequence, A062728, A135705.

Sequence in context: A275642 A171441 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 February 19 08:54 EST 2018. Contains 299330 sequences. (Running on oeis4.)