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A317326 Multiples of 26 and odd numbers interleaved.

%I A317326 #61 Dec 11 2024 07:46:03
%S A317326 0,1,26,3,52,5,78,7,104,9,130,11,156,13,182,15,208,17,234,19,260,21,
%T A317326 286,23,312,25,338,27,364,29,390,31,416,33,442,35,468,37,494,39,520,
%U A317326 41,546,43,572,45,598,47,624,49,650,51,676,53,702,55,728,57,754,59,780,61,806,63,832,65,858,67,884,69
%N A317326 Multiples of 26 and odd numbers interleaved.
%C A317326 a(n) is the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 30-gonal numbers (A316729).
%C A317326 Partial sums give the generalized 30-gonal numbers.
%C A317326 More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.
%C A317326 From _Bruno Berselli_, Jul 27 2018: (Start)
%C A317326 Also, this type of sequence is characterized by:
%C A317326 O.g.f.: x*(1 + m*x + x^2)/(1 - x^2)^2;
%C A317326 E.g.f.: x*(2 - m + (2 + m)*exp(2*x))*exp(-x)/4;
%C A317326 a(n) = -a(-n) = (2 + m - (2 - m)*(-1)^n)*n/4;
%C A317326 a(n) = (m/2)^((1 + (-1)^n)/2)*n;
%C A317326 a(n) = 2*a(n-2) - a(n-4), with signature (0,2,0,-1). (End)
%H A317326 Colin Barker, <a href="/A317326/b317326.txt">Table of n, a(n) for n = 0..1000</a>
%H A317326 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A317326 a(2*n) = 26*n, a(2*n+1) = 2*n + 1.
%F A317326 From _Bruno Berselli_, Jul 27 2018: (Start)
%F A317326 O.g.f.: x*(1 + 26*x + x^2)/(1 - x^2)^2.
%F A317326 E.g.f.: x*(-6 + 7*exp(2*x))*exp(-x).
%F A317326 a(n) = -a(-n) = (7 + 6*(-1)^n)*n.
%F A317326 a(n) = 13^((1 + (-1)^n)/2)*n.
%F A317326 a(n) = 2*a(n-2) - a(n-4). (End)
%F A317326 Multiplicative with a(2^e) = 13*2^e, and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F A317326 Dirichlet g.f.: zeta(s-1) * (1 + 3*2^(3-s)). - _Amiram Eldar_, Oct 26 2023
%t A317326 Table[(7 + 6 (-1)^n) n, {n, 0, 70}] (* _Bruno Berselli_, Jul 27 2018 *)
%o A317326 (Julia) [13^div(1+(-1)^n,2)*n for n in 0:70] |> println # _Bruno Berselli_, Jul 28 2018
%Y A317326 Cf. A252994 and A005408 interleaved.
%Y A317326 Column 26 of A195151.
%Y A317326 Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15), A317312 (k=16), A317313 (k=17), A317314 (k=18), A317315 (k=19), A317316 (k=20), A317317 (k=21), A317318 (k=22), A317319 (k=23), A317320 (k=24), A317321 (k=25), A317322 (k=26), A317323 (k=27), A317324 (k=28), A317325 (k=29), this sequence (k=30).
%Y A317326 Cf. A316729.
%K A317326 nonn,mult,easy
%O A317326 0,3
%A A317326 _Omar E. Pol_, Jul 25 2018