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

Author

Kival Ngaokrajang, Apr 17 2015

Keywords

• nonn
• easy

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 A000290 interleaved with 2*A002061. 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).

Crossrefs

Cf. A054552, A002061, A000290.

Programs

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 *)

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

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)