A317325 - OEIS

%I #34 Dec 11 2024 06:29:00

%S 0,1,25,3,50,5,75,7,100,9,125,11,150,13,175,15,200,17,225,19,250,21,

%T 275,23,300,25,325,27,350,29,375,31,400,33,425,35,450,37,475,39,500,

%U 41,525,43,550,45,575,47,600,49,625,51,650,53,675,55,700,57,725,59,750,61,775,63,800,65,825,67,850,69

%N Multiples of 25 and odd numbers interleaved.

%C Partial sums give the generalized 29-gonal numbers (A303815).

%C a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 29-gonal numbers.

%H Colin Barker, <a href="/A317325/b317325.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).

%F a(2n) = 25*n, a(2n+1) = 2*n + 1.

%F G.f.: x*(1 + 25*x + x^2)/((1 - x)^2*(1 + x)^2). - _Vincenzo Librandi_, Jul 28 2018

%F a(n) = 2*a(n-2) - a(n-4) for n>3. - _Colin Barker_, Jul 29 2018

%F Multiplicative with a(2^e) = 25*2^(e-1), and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023

%F Dirichlet g.f.: zeta(s-1) * (1 + 23/2^s). - _Amiram Eldar_, Oct 26 2023

%p seq(op([25*n,2*n+1]),n=0..40); # _Muniru A Asiru_, Jul 28 2018

%t With[{nn=30}, Riffle[25 Range[0, nn], 2 Range[0, nn] + 1]] (* _Vincenzo Librandi_, Jul 28 2018 *)

%o (Magma) &cat[[25*n, 2*n + 1]: n in [0..30]]; // _Vincenzo Librandi_, Jul 28 2018

%o (GAP) Flat(List([0..40],n->[25*n,2*n+1])); # _Muniru A Asiru_, Jul 28 2018

%o (PARI) concat(0, Vec(x*(1 + 25*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jul 29 2018

%Y Cf. A008607 and A005408 interleaved.

%Y Column 25 of A195151.

%Y 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).

%Y Cf. A303815.

%K nonn,easy,mult,changed

%O 0,3

%A _Omar E. Pol_, Jul 25 2018