A257171 Sum of numbers on n-th segment of Ulam's spiral.

1, 5, 9, 13, 27, 36, 62, 78, 120, 145, 207, 243, 329, 378, 492, 556, 702, 783, 965, 1065, 1287, 1408, 1674, 1818, 2132, 2301, 2667, 2863, 3285, 3510, 3992, 4248, 4794, 5083, 5697, 6021, 6707, 7068, 7830, 8230, 9072, 9513, 10439, 10923, 11937, 12466, 13572, 14148, 15350

Offset

0,2

Comments

From Ulam's spiral, consider successive segments s(0) = [1]; s(1) = [2, 3]; s(2) = [4, 5]; s(3) = [6, 7]; s(4) = [8, 9, 10] and so on. a(n) is sum of numbers of the segment s(n). The first differences are A002601 interleaved with A000290. See illustration in the links.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Kival Ngaokrajang, Illustration of initial terms

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

Formula

a(0) = 1; a(1) = 5; a(2) = 9; for n >= 3, a(n) = a(n-1)+((n-1)/2+1)^2, if n = even; otherwise a(n) = a(n-1)+2*((n-1)^2/4+(n-1)+7/4).

From Colin Barker, Apr 17 2015: (Start)

a(n) = (n^3+5*n^2+14*n+16)/8 for n even and n>1.

a(n) = (n^3+4*n^2+11*n+8)/8 for n odd and n>1.

G.f.: (2*x^8-3*x^7-5*x^6+9*x^5+5*x^4-8*x^3+x^2+4*x+1) / ((x-1)^4*(x+1)^3).

(End)

Mathematica

f[n_] := Block[{t = {5, 9}}, For[i = 3, i <= n, i++, If[OddQ@ i, AppendTo[t, t[[i - 1]] + ((i - 1)/2 + 1)^2], AppendTo[t, t[[i - 1]] + 2 ((i - 1)^2/4 + (i - 1) + 7/4)]]]; {1}~Join~t]; f@ 48(* Michael De Vlieger, Apr 17 2015 *)

Prog

(PARI) a(n) = if(n<=0, 1, if(n<=1, 5, if(n<=2, 9, if(Mod(n, 2)==0, a(n-1)+2*((n-1)^2/4+(n-1)+7/4), a(n-1)+((n-1)/2+1)^2))))
for (n=0, 100, print1(a(n), ", "))
(PARI) Vec((2*x^8-3*x^7-5*x^6+9*x^5+5*x^4-8*x^3+x^2+4*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Apr 18 2015

Crossrefs

Cf. A054552, A002061, A000290.

Sequence in context: A267190 A170896 A323106 * A233973 A089977 A024728

Adjacent sequences: A257168 A257169 A257170 * A257172 A257173 A257174

Keyword

nonn,easy

Author

Kival Ngaokrajang, Apr 17 2015

Status

approved