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 A259181 a(n) = n*(n+1)*(n+2)*(n+3)*(2*n^2+6*n+7)/360. 4
 0, 1, 9, 43, 147, 406, 966, 2058, 4026, 7359, 12727, 21021, 33397, 51324, 76636, 111588, 158916, 221901, 304437, 411103, 547239, 719026, 933570, 1198990, 1524510, 1920555, 2398851, 2972529, 3656233, 4466232, 5420536, 6539016, 7843528, 9358041, 11108769 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS After 0, second bisection of A129548. This sequence is also the total number of squares of all sizes in i X i subsquares in an n X n grid, whereas A000330 simply gives the number of all sizes of squares in an n X n grid. See illustrations. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Luce ETIENNE, illustration of initial terms Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1). FORMULA a(n) = (1/6)*Sum_{i=0..n} (i+1)*(i+2)*(2*i+3)*(n-i)^2. a(n) = Sum_{i=0..n} A000290(n-i)*A000330(i+1). G.f.: x*(1 + x)^2 / (1 - x)^7. - Colin Barker, Nov 08 2015 a(n) = (A000539(n+1) - A000217(n+1))/30. - Yasser Arath Chavez Reyes, Feb 24 2024 EXAMPLE a(0) = 0; a(1) = 1*1; a(2) = 4*1+1*5 = 9; a(3) = 9*1+4*5+1*14 = 43. PROG (PARI) vector(100, n, n--; n*(n+1)*(n+2)*(n+3)*(2*n^2+6*n+7)/360) \\ Altug Alkan, Nov 08 2015 (PARI) concat(0, Vec(-x*(x+1)^2 / (x-1)^7 + O(x^100))) \\ Colin Barker, Nov 08 2015 CROSSREFS Cf. A000217, A000290, A000330, A000539, A129548. Cf. A060060: (1/6)*Sum_{i=0..n} (i+1)*(i+2)*(2*i+3)*i^2. Sequence in context: A332373 A181945 A244869 * A330088 A281773 A220676 Adjacent sequences: A259178 A259179 A259180 * A259182 A259183 A259184 KEYWORD nonn,easy AUTHOR Luce ETIENNE, Nov 08 2015 STATUS approved

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Last modified July 24 02:45 EDT 2024. Contains 374575 sequences. (Running on oeis4.)