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 A114254 Sum of all terms on the two principal diagonals of a 2n+1 X 2n+1 square spiral. 8
 1, 25, 101, 261, 537, 961, 1565, 2381, 3441, 4777, 6421, 8405, 10761, 13521, 16717, 20381, 24545, 29241, 34501, 40357, 46841, 53985, 61821, 70381, 79697, 89801, 100725, 112501, 125161, 138737, 153261, 168765, 185281, 202841, 221477, 241221 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..10000 Project Euler, Problem 28. Number Spiral Diagonals Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA O.g.f.: 3/(-1+x) + 16/(-1+x)^2 + 44/(-1+x)^3 + 32/(-1+x)^4 = (1 + 21*x + 7*x^2 + 3*x^3)/(-1+x)^4. - R. J. Mathar, Feb 10 2008 a(n) = 1 + 10*n^2 + (16*n^3 + 26*n)/3. [Corrected by Arie Groeneveld, Aug 17 2008] EXAMPLE For n = 1, the 3 X 3 spiral is . 7---8---9 | 6 1---2 | | 5---4---3 . so a(1) = 7 + 9 + 1 + 5 + 3 = 25. . For n = 2, the 5 X 5 spiral is . 21--22--23--24--25 | 20 7---8---9--10 | | | 19 6 1---2 11 | | | | 18 5---4---3 12 | | 17--16--15--14--13 . so a(2) = 21 + 25 + 7 + 9 + 1 + 5 + 3 + 17 + 13 = 101. MATHEMATICA Array[1 + 10 #^2 + (16 #^3 + 26 #)/3 &, 36, 0] (* Michael De Vlieger, Mar 01 2018 *) PROG (PARI) a(n) = 1 + 10*n^2 + (16*n^3 + 26*n)/3; \\ Joerg Arndt, Mar 01 2018 CROSSREFS Cf. A016754, A054569, A053755, A054554 for diagonals from origin. Cf. A325958 (first differences). Sequence in context: A356533 A221274 A042220 * A042222 A158551 A044276 Adjacent sequences: A114251 A114252 A114253 * A114255 A114256 A114257 KEYWORD easy,nonn AUTHOR William A. Tedeschi, Feb 06 2008, Mar 01 2008 STATUS approved

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Last modified April 24 18:17 EDT 2024. Contains 371962 sequences. (Running on oeis4.)