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

Author

Omar E. Pol, Jul 25 2018

Keywords

• nonn
• easy
• mult

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).

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.

Programs

GAP

Flat(List([0..40],n->[25*n,2*n+1])); # Muniru A Asiru, Jul 28 2018

Magma

&cat[[25*n, 2*n + 1]: n in [0..30]]; // Vincenzo Librandi, Jul 28 2018

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 *)

PARI

concat(0, Vec(x*(1 + 25*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

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