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 A257171 Sum of numbers on n-th segment of Ulam's spiral. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified July 29 08:46 EDT 2021. Contains 346340 sequences. (Running on oeis4.)