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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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]

a(n)=8*n+a(n-1)-15 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]

Part of the spiral:

16 17 18 19 ...

15 4 5 6 ...

14 3 0 7 ...

13 2 1 8 ...

For n=2, a(2)=8*2+0-15=1; n=3, a(3)=8*3+1-15=10; n=4, a(4)=8*4+10-15=27 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]

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.

Sequence in context: A045177 A043887 A161450 this_sequence A103135 A008468 A119548

Adjacent sequences: A001104 A001105 A001106 this_sequence A001108 A001109 A001110

nonn,easy,nice,new

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