0,3

Write 0,1,2,... in clockwise spiral; sequence gives numbers on negative y axis.

Number of divisors of 48^n - J. Lowell (jhbubby(AT)mindspring.com), Aug 30 2008

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer; see p. 23.

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.

T. D. Noe, Table of n, a(n) for n=0..1000

Index entries for sequences related to linear recurrences with constant coefficients

Emilio Apricena, A version of the Ulam spiral

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 344

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

G.f.: A(x) = x(1+7x)/(1-x)^3.

Partial sums of odd numbers 1 mod 8, i.e. 1, 1+9, 1+9+17, ... - Jon Perry (perry(AT)globalnet.co.uk), Dec 18 2004

a(n)=n+8*A000217(n-1) - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 14 2005

1^3 + 3^3*(n-1)/(n+1) + 5^3*[(n-1)(n-2)]/[(n+1)(n+2)] + 7^3*[(n-1)(n-2)(n-3)]/[(n+1)(n+2)(n+3)] + ... = n(4n-3) [Ramanujan]. - Neven Juric, Apr 15 2008

Starting (1, 10, 27, 52,...) = binomial transform of [1, 9, 8, 0, 0, 0,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 30 2008

a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=10 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]

Also, let Dec(n)=decagonal numbers, T(n)=triangular numbers, then Dec(n)=T(n)+7*T(n-1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 28 2009]

Part of the spiral:

16 17 18 19 ...

15 4 5 6 ...

14 3 0 7 ...

13 2 1 8 ...

A001107:=-(1+7*z)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+8 od: seq(a[n], n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 18 2008

lst={}; Do[AppendTo[lst, 4*n^2-3*n], {n, 0, 5!}]; lst...and/or... s=0; lst={s}; Do[s+=n+1; AppendTo[lst, s], {n, 0, 6!, 8}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 25 2008]

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

(Other) sage: [crt(0, n, 4, 3 )*crt(1, n, 4, 3 )/4 for n in xrange(0, 47)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 30 2009]

Cf. A000217, A001106, A051682.

Cf. A007585. a(n)=A033954(-n)=A074377(2n-1).

Cf. A093565 ((8, 1) Pascal, column m=2). Partial sums of A017077.

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

Adjacent sequences: A001104 A001105 A001106 this_sequence A001108 A001109 A001110

Sequence in context: A045177 A043887 A161450 this_sequence A103135 A008468 A119548

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).