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A195605
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a(n) = (4*n*(n+2)+(-1)^n+1)/2 + 1.
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7
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2, 7, 18, 31, 50, 71, 98, 127, 162, 199, 242, 287, 338, 391, 450, 511, 578, 647, 722, 799, 882, 967, 1058, 1151, 1250, 1351, 1458, 1567, 1682, 1799, 1922, 2047, 2178, 2311, 2450, 2591, 2738, 2887, 3042, 3199, 3362, 3527, 3698, 3871, 4050, 4231, 4418, 4607, 4802
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OFFSET
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0,1
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COMMENTS
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Sequence found by reading the numbers in increasing order on the vertical line containing 2 of the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.
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LINKS
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FORMULA
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G.f.: (2+3*x+4*x^2-x^3)/((1+x)*(1-x)^3).
a(n) = a(-n-2) = 2*a(n-1)-2*a(n-3)+a(n-4).
Sum_{n>=0} 1/a(n) = 1/2 + Pi^2/16 - cot(Pi/(2*sqrt(2)))*Pi/(4*sqrt(2)). - Amiram Eldar, Mar 06 2023
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MATHEMATICA
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CoefficientList[Series[(2 + 3 x + 4 x^2 - x^3) / ((1 + x) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{2, 0, -2, 1}, {2, 7, 18, 31}, 50] (* Harvey P. Dale, Jan 21 2017 *)
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PROG
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(Magma) [(4*n*(n+2)+(-1)^n+3)/2: n in [0..48]];
(PARI) for(n=0, 48, print1((4*n*(n+2)+(-1)^n+3)/2", "));
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CROSSREFS
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Cf. A047621 (contains first differences), A016754 (contains the sum of any two consecutive terms).
Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195241 [incomplete list].
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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