A317325 Multiples of 25 and odd numbers interleaved.

0, 1, 25, 3, 50, 5, 75, 7, 100, 9, 125, 11, 150, 13, 175, 15, 200, 17, 225, 19, 250, 21, 275, 23, 300, 25, 325, 27, 350, 29, 375, 31, 400, 33, 425, 35, 450, 37, 475, 39, 500, 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

Offset

0,3

Comments

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

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

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Formula

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

G.f.: x*(1 + 25*x + x^2)/((1 - x)^2*(1 + x)^2). - Vincenzo Librandi, Jul 28 2018

a(n) = 2*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 29 2018

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

Dirichlet g.f.: zeta(s-1) * (1 + 23/2^s). - Amiram Eldar, Oct 26 2023

Maple

seq(op([25*n, 2*n+1]), n=0..40); # Muniru A Asiru, Jul 28 2018

Mathematica

With[{nn=30}, Riffle[25 Range[0, nn], 2 Range[0, nn] + 1]] (* Vincenzo Librandi, Jul 28 2018 *)

Prog

(Magma) &cat[[25*n, 2*n + 1]: n in [0..30]]; // Vincenzo Librandi, Jul 28 2018
(GAP) Flat(List([0..40], n->[25*n, 2*n+1])); # Muniru A Asiru, Jul 28 2018
(PARI) concat(0, Vec(x*(1 + 25*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Crossrefs

Cf. A008607 and A005408 interleaved.

Column 25 of A195151.

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).

Cf. A303815.

Sequence in context: A040612 A040614 A040615 * A040610 A158786 A040611

Adjacent sequences: A317322 A317323 A317324 * A317326 A317327 A317328

Keyword

nonn,easy,mult,changed

Author

Omar E. Pol, Jul 25 2018

Status

approved