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 A330707 a(n) = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4. 4
 0, 1, 3, 7, 13, 20, 28, 38, 50, 63, 77, 93, 111, 130, 150, 172, 196, 221, 247, 275, 305, 336, 368, 402, 438, 475, 513, 553, 595, 638, 682, 728, 776, 825, 875, 927, 981, 1036, 1092, 1150, 1210, 1271, 1333, 1397, 1463 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Essentially four odds followed by four evens. Last digit is neither 4 nor 9. Essentially twice or twin sequences in the hexagonal spiral from A002265.                   21  21  21  22  22  22  22                 21  14  14  14  14  15  15  23               20  13   8   8   8   9   9  15  23             20  13   8   4   4   4   4   9  15  23           20  13   7   3   1   1   1   5   9  16  23         20  13   7   3   1   0   0   2   5  10  16  24           19  12   7   3   0   0   2   5  10  16  24             19  12   7   3   2   2   5  10  16  24               19  12   6   6   6   6  10  17  24                 19  12  11  11  11  11  17  25                   18  18  18  18  17  17  25 . There are 12 twin sequences. 6 of them (A001859, A006578, A077043, A231559, A024219, A281026) are in the OEIS. a(n) is the seventh.   0, 1, 3, 7, 13, 20, 28, 38, 50, ...   1, 2, 4, 6,  7,  8, 10, 12, 13, ...   1, 2, 2, 1,  1,  2,  2,  1,  1, ... period 4. See A014695. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1). FORMULA a(n) = A231559(-n). a(1+2*n) + a(2+2*n) = A033579(n+1). a(40+n) - a(n) = 1210, 1270, 1330, 1390, 1450, ... . See 10*A016921(n). From Colin Barker, Dec 27 2019: (Start) G.f.: x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)). a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4. (End) E.g.f.: (cos(x) + sin(x) + (-1 + 4*x + 3*x^2)*exp(x))/4. - Stefano Spezia, Dec 27 2019 a(n) = ( 3*n^2 + n - 1 + sqrt(2)*sin((2*n+1)*Pi/4) )/4 = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4. - G. C. Greubel, Dec 30 2019 MAPLE seq((3*n^2+n-1+sqrt(2)*sin((2*n+1)*Pi/4))/4, n = 0..60); # G. C. Greubel, Dec 30 2019 MATHEMATICA LinearRecurrence[{3, -4, 4, -3, 1}, {0, 1, 3, 7, 13}, 60] (* Amiram Eldar, Dec 27 2019 *) PROG (PARI) concat(0, Vec(x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)) + O(x^60))) \\ Colin Barker, Dec 27 2019 (MAGMA) [(3*n^2+n-1+ (-1)^Floor(n/2))/4: n in [0..60]]; // G. C. Greubel, Dec 30 2019 (Sage) [(3*n^2+n-1+(-1)^floor(n/2))/4 for n in (0..60)] # G. C. Greubel, Dec 30 2019 CROSSREFS Cf. A001859, A002265, A006578, A014695, A016921, A024219, A033579, A077043, A231559, A281026. Sequence in context: A330407 A014283 A294398 * A033551 A022777 A033154 Adjacent sequences:  A330704 A330705 A330706 * A330708 A330709 A330710 KEYWORD nonn,easy AUTHOR Paul Curtz, Dec 27 2019 STATUS approved

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Last modified August 7 11:08 EDT 2020. Contains 336275 sequences. (Running on oeis4.)