A133629 a(1)=1, a(n) = a(n-1) + (p-1)*p^(n/2-1) if n is even, otherwise a(n) = a(n-1) + p^((n-1)/2), where p=5.

1, 5, 10, 30, 55, 155, 280, 780, 1405, 3905, 7030, 19530, 35155, 97655, 175780, 488280, 878905, 2441405, 4394530, 12207030, 21972655, 61035155, 109863280, 305175780, 549316405, 1525878905, 2746582030, 7629394530, 13732910155, 38146972655, 68664550780

Offset: 1

Author

Hieronymus Fischer, Sep 19 2007

Keywords

• nonn
• easy
• changed

Comments

Partial sums of A133632.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

Crossrefs

Sequences with similar recurrence rules: A027383 (p=2), A087503 (p=3), A133628 (p=4).

Related sequences: A132666, A132667, A132668, A132669.

Other related sequences for different p: A016116 (p=2), A038754 (p=3), A084221 (p=4), A133632 (p=5).

Programs

Maple

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=5*a[n-2]+5 od: seq(a[n], n=1..29); # Zerinvary Lajos, Mar 17 2008

PARI

Vec(x*(1 + 4*x) / ((1 - x) * (1 - 5*x^2)) + O(x^40)) \\ Colin Barker, Nov 25 2016

Python

def A133629(n): return (5+((n&1)<<2))*5**(n>>1)-5>>2 # Chai Wah Wu, Sep 02 2025

Formula

a(n) = Sum_{k=1..n} A133632(k).

The following formulas are given for a general natural parameter p > 1 (p=5 for this sequence).

G.f.: x(1+(p-1)x)/((1-px^2)(1-x)).

a(n) = (p/(p-1))*(p^(n/2)-1) if n is even, otherwise a(n)=(p/(p-1))*((2p-1)*p^((n-3)/2)-1).

a(n) = (p/(p-1))*(p^floor(n/2) + p^floor((n-1)/2) - p^floor((n-2)/2)-1).

a(n) = p^floor(n/2) + (p^floor((n+1)/2)-p)/(p-1).

a(n) = A132669(a(n+1)) - 1.

a(n) = A132669(a(n-1)+1) for n > 0.

A132669(a(n)) = a(n-1)+1 for n > 0.

From Colin Barker, Nov 25 2016: (Start)

a(n) = 5*(5^(n/2) - 1)/4 for n even.

a(n) = (9*5^(n/2-1/2) - 5)/4 for n odd.

a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n > 3.

G.f.: x*(1 + 4*x) / ((1 - x) * (1 - 5*x^2)).

(End)