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 A267322 Expansion of (1 + x + x^2 + x^4 + 2*x^5)/(1 - x^3)^3. 1
 1, 1, 1, 3, 4, 5, 6, 9, 12, 10, 16, 22, 15, 25, 35, 21, 36, 51, 28, 49, 70, 36, 64, 92, 45, 81, 117, 55, 100, 145, 66, 121, 176, 78, 144, 210, 91, 169, 247, 105, 196, 287, 120, 225, 330, 136, 256, 376, 153, 289, 425, 171, 324, 477, 190, 361, 532, 210, 400, 590, 231, 441, 651 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Triangular numbers alternating with squares and pentagonal numbers. LINKS Ilya Gutkovskiy, Extended illustration of initial terms Eric Weisstein's World of Mathematics, Triangular Number Eric Weisstein's World of Mathematics, Square Number Eric Weisstein's World of Mathematics, Pentagonal Number Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1) FORMULA G.f.: (1 + x + x^2 + x^4 + 2*x^5)/(1 - x^3)^3. a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9). a(3k) = A000217(k+1), a(3k+1) = A000290(k+1), a(3k+2) = A000326(k+1). Sum_{n>=0} 1/a(n) = 2 - Pi/sqrt(3) + Pi^2/6 + 3*log(3) = 5.1269715686... a(n) = (floor(n/3) + 1)*((n+1)*floor(n/3) - 3*floor(n/3)^2 + 2)/2. - Bruno Berselli, Apr 08 2016 EXAMPLE Illustration of initial terms: ========================================================== n:    0   1   2     3     4     5       6       7       8 ----------------------------------------------------------                                                         o                                                       o o                                 o       o   o o o   o o o                     o   o o   o o     o o   o o o   o o o       o   o   o   o o   o o   o o   o o o   o o o   o o o ==========================================================       1   1   1     3     4     5       6       9      12 ---------------------------------------------------------- MATHEMATICA LinearRecurrence[{0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 1, 1, 3, 4, 5, 6, 9, 12}, 70] Table[(Floor[n/3] + 1) ((n + 1) Floor[n/3] - 3 Floor[n/3]^2 + 2)/2, {n, 0, 70}] (* Bruno Berselli, Apr 08 2016 *) PROG (PARI) x='x+O('x^99); Vec((1+x+x^2+x^4+2*x^5)/(1-x^3)^3) \\ Altug Alkan, Apr 07 2016 CROSSREFS Cf. A000217, A000290, A000326, A123596, A124093, A271391. Sequence in context: A047427 A228235 A228895 * A218929 A088875 A022884 Adjacent sequences:  A267319 A267320 A267321 * A267323 A267324 A267325 KEYWORD nonn,easy AUTHOR Ilya Gutkovskiy, Apr 07 2016 STATUS approved

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Last modified December 8 14:38 EST 2019. Contains 329865 sequences. (Running on oeis4.)